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UNIVERSITY MATHEMATICS t(1994)
S. Zoch
c. t( x )2012
UNIVERSITY MATHEMATICS
3rd edition
S. Zoch c. 2011
c. 8/15/2007 10 A.M.
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UNIVERSITY MATHEMATICS t(1994)
S. Zoch
c. t( x )2012
Introduction
Topics cover most of basic arithmetic, introductory and intermediate algebras, university and college algebra with a little
geometry, trigonometry, calculus, and probability including statistics.
Thanks to all the great students over all the decades for your good study habits.
These notes are for you and all others interested in the right understanding and application of physical sciences,
mathematics, education, business, philosophy, technology and arts.
Under Revision t(8/17/12) edit
Index
Logic p.65-66
Set Theory p.45-60
Probability and Statistics p.122-143
1a. Integers and whole numbers
Let
R
be the set of real numbers and hold that
numbers so that
N  1 , 2 , 3 ,... .
N
is the set of natural
The set of integers is the collection denoted and equal to the following
Z   x : x  N    0    x : x  N  with
scalar multiplication and
1 , 1  R
1  x  x
where  is real
.
It is written as the set
Z  ..., 3 , 2 , 1 , 0 ,1 , 2 , 3 ,... .
An Integer is a discrete unit, the deficit of a unit, or its absence represented by
expression as a real number (written as an Indian numeral with an implied base
of ten) that is not a fraction in its reduced form with any denominator other than
the real number one.
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The Set of Integers is the collection of all positive and negative natural numbers
including the number zero.
7
  7   21  Z .
3
1
so that the number minus seven is an integer.
0 0
 0  0.010  0.010  00.000   0   0  1  0 
1
10
1
0 
Z
5
.
so that the number zero is an integer.
7 7
 7  7.010  7.0  07.000   7   7  1  7  14  Z
1
10
1
2
so that the number seven is an integer.
2.7 is not an Integer. -3.17 is not an integer.
The number ½ is not an integer. The number 7/3 is not an integer. The number
-11/4 is not an integer.
No irrational number is an integer whence pi is not an integer.
Note that
2
1
0
1
2
7  710  7.010  7.0  0 10  0 10  7 10  0 10  0 10 is the
base ten decimal expansion of the integer seven.
No proper fraction is an integer. Any improper fraction is non integer when it is
reduced and the denominator is not the number one.
7 7
 0  7  0  7  0  7 for all
x
0  0  0
x R
with
x  0 because
and the fundamental theorem of Arithmetic that is a case of the
fundamental Theorem of Algebra.
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Integers are usually not written as fractions and represent the absence, deficit or
presence of whole amounts.
Any Integer can be written as a fraction by expressing it with a one in the
denominator.
Laws of signs
Please know your laws of sign very well as they are relatively easy to learn and
always applicable even required with the use of a calculator.
There are two laws of signs each with two components so that the total number
of laws is four.
I. MULTIPLICATION AND DIVISION-Optics Laws of Signs
Given an expression to multiply or divide two real numbers :
A. If they have the same signs the result is positive
B. If they have unlike signs the result is negative.
In these cases one can think of images and their negatives in film production.
The cases of multiplication:
3 x 4 = 12
3 x - 4 = -12
-3 x 4 = -12
-3 x - 4 = 12
And for division:
12/3=4
-12/3=-4
12/-3=-4
-12/-3=4
2. ADDITION AND SUBTRACTION-Accounting Laws of Signs
Given an expression where no multiplication or division is indicated only to add or
subtract two real numbers:
A. If they have the same signs combine them and give the result this similar sign.
B. If they have unlike signs then take their difference and give this result the sign
of the number which is farther from zero on the real line.
In this case one can think of owing as negative and having as positive.
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2+3=5
2 - 3 = -1
-2 + 3 = 1
-2 - 3 = -5
2a. Fractions and proportions
Given a fraction of the form
them into
b
a
b
it means take a unit or units and divide it or
number of equal parts and select
Given a fraction of the form
numerator and
b
a
b
where
b
a
number of them.
is not zero
a
is called the
is called the denominator.
We never divide by zero and if it is the case we say the expression is undefined.
A fraction is called proper if the numerator is less than the denominator.
A fraction is called improper if the numerator is greater than the denominator.
A fraction is reduced to lowest terms or reduced if the numerator and
denominator have no common factors other than the number one.
A fraction is simplified if it is reduced. Any answer that is a fraction must always
be reduced.
A fraction can be reduced and proper or improper at the same time.
Proper and improper have nothing to do with a fraction being reduced.
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Theory
Any improper fraction can be written as a mixed number.
Theorem of Arithmetic
Given two real numbers
and
s
d
numbers
q
and
s
d
q 
r
d
.
q
is real called the quotient and
r
with
d
is not zero then there exists real
so that
r
is real called the remainder.
If a fraction is improper it may also be reduced at the same time.
7
5
is reduced and may be written as the mixed number 1
2
5
or from the
fundamental theorem 1  2 .
5
If you are working with fractions make everything look like a fraction.
In equations we can clear out the fractions and with expressions we may not be
able to clear them out.
Cancellation Property
Let a, x, and b be free and b is not zero.
ax
bx

xa
bx

ax  a
xb b
Given a fraction of the form
a b
and
b a
a
b
where b is not zero it also means
.
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Given a fraction of the form 

a
b

a
b

a
b
a
b
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then
for calculations.
Distribute negative numbers across terms where they are indicated by their
position in front of parenthesis.
Given a fraction multiplied by a variable it may also be written or calculated as
numerator of fraction times variable divided by the denominator of the fraction.
a  ax
x
b
b
PROPERTIES OF FRACTIONS
Let a, x, c. d, and b be free and b and d are not zero.
a 
1
a
a 0
where
a a
1
0 0
a
where
a 0
0 0
1
a
0

undefined
0
0

undefined
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UNIVERSITY MATHEMATICS t(1994)
a c
b d

a c
b d

a d
b c
a c
b d

ad bc
bd
a c
b d

ad bc
bd
S. Zoch
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ac
bd

ad
bc
The formulas above do not require the least common denominator or LCD to use
them.
In all cases a common denominator can be used which may be the product of
denominators given as easily as the least common denominator to accomplish all
tasks such as clearing fractions from forms of equations and adding or
subtracting fractions in expressions.
Least Common Denominator or LCD
Given some fractions their least common denominator is the amount that is the
smallest or has the least number of prime factors so that each denominator of
every fraction will divide it evenly.
3a. Decimals and basis
Any real number can be converted into a decimal number with any natural base
(usually base of ten) and every decimal number can be converted into a real
number.
The place values of decimal numbers is extremely important knowledge as it
helps us understand their real values and such speech is required to write or
translate decimal numbers into whole numbers, proper fractions, or improper
fractions that can be stated as mixed numbers.
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The significance of place values is relevant for their understanding and skill.
When we add or subtract real numbers we must line up place values and move
down the decimal in our work directly for the result.
When we multiply two decimal numbers we multiply as normal with the
distribution times table method yet to produce the correct result or answer take
the decimal from the right end of the response and move the decimal to the left
the total number of places that is the sum of all significant place values to the
right of the decimals in each given factor (that was multiplied) and the correct
answer can be hand written.
When we divide two real numbers we must divide by a whole number.
When trying to create a decimal number from a proper or improper fraction make
the numerator in the long division box look like a decimal by writing it like a
decimal with a point and as many zeros as you may need for rounding or
termination.
When working with decimals answers are probably going to look like decimals
and when working with fractions answers may probably going to look like
fractions.
As the form of truth is not always unique in its expression and truth can be the
satisfaction of directions some problems have only one right answer yet many
have a multitude of equally valid answers that can take many equivalent forms.
It is important for students to be able to recognize valid forms of truth in many
instances and also to recognize its equivalent formats or statements in all
academic areas.
DECIMALS OR BASE TEN REPRESENTATIONS
7 7
1
 7.010  7.0  0 10  0 10  7 10  0 10  0 10
2
10
1
0
2
is the
base ten decimal expansion of the integer seven.
Let n and m be whole numbers.
Select a base ten decimal number according to the following expansion:
a 10n  a 10n1  ...  a 102  a 101  a 100  b 101  b 102  ...  b m 10m
n
n 1
2
1
0
1
2
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Any decimal number can be converted into a real number and any real number
can be converted into a decimal number.
You should know the names and meanings of the base ten decimal positions
especially the ones to the right of the decimal.
Always question what is the given statement or given data of a mathematics
problem.
Remember there is a big difference between an expression and an equation.
A solution is always an equation.
A simplification of an expression can be another expression or written as an
identity equation.
ROUNDING AND PLACE VALUES
BINARY NUMBERS OR NON BASE TEN REPRESENTATIONS
Let n and m be whole numbers.
Select a binary number or base two number according to the following
expansion:
a 2n  a 2n1  ...  a 22  a 21  a 20  b 21  b 22  ...  bm 2m
n
n 1
2
1
0
1
2
Computers and machines use base two, eight, or sixteen expansions as these
are powers of two so that the dichotomous nature of electronic components can
be represented as on or off or one or zero.
4a. Exponents, square roots and
scientific notations and rational
exponents
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Let x be free and n is a fixed whole number then
x x
So that
x

n
2  2  2  2  8 , 5  5  5  25
2
3
n
n
x  x x x
for all i=1,2,3,…,n or
i
x
1

1
x
n
so that
2
3
3

2
1
x  x x x
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n
1
2
3
 x n
with
 x with n factors of x.
and
3  3  3  3  27
3
.
 13  1 .
2 8
A number is in scientific notation if it is written as a product of a number between
one and ten or one itself and an integer power of ten.
y
If

n
x
then
n
y x
.
Theory
Exponents
There are many ways to work with exponents.
The hardest part of mathematics is making all the choices yet your answer in the
end should be equivalent to any one else’s no matter what methods you use or
choices you make it your work does not violate axioms, definitions, properties or
theorems yet this is the beauty and teaches us to make viable decisions on our
own.
An answer is simplified if it has no negative exponents, numeric bases have been
evaluated and if it is a fraction then it is reduced.
Let x and y be free.
x x
1
0
0
undefined
x 1
0
when
x 0
.
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x x  x
m
n
S. Zoch
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m n
xm n  xmn
x
y
x
n
m
n
y
x

m
n
1
x

n
n
 x
 
 y
 y
 
 x
n
RADICALS OR RATIONAL EXPONENTS and Square Roots
Mathematics, sciences and music have many standards and agreement that are
held by everybody no matter who they are, where they are from and what they
may otherwise believe so that this demonstrates that worldwide agreements exist
as these held valid in many cases for thousands of years and will be upheld in
the future such as the order of operations agreement which makes expressions
well defined and computable and Greek letter pi represents the value of any
circles circumference divided by its diameter.
If
y

 x
n
m

If
y
y
If
5
If
n
x
then

m
n
x
n
y x
we have

x
x

25
then
then
y
and if n is even then
2
x 0
and then
2
y x
.
.
.
5  25
2
1
x
x 0
.
Scientific Notation
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2.718 10
3
27.18 10
4
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is in scientific notation and represents the number 0.002178 .
is not in scientific notation because 27.18 is not between one
and ten and not the number one and represents number 0.002178 .
2
0.2718 10 is not in scientific notation because 0.2718 is not between one
and ten and not the number one and represents number 0.002178 .
Given a number in scientific notation move the decimal to the left the number of
places indicated by the integer power of ten if this power is negative or to the
right the number of places indicated by the integer power of ten if this power is
positive to find its real value.
5a. Ratio and Proportion
RATIOS AND PROPORTIONS
A ratio is an expression of a fraction.
The ratio of a to b is denoted a/b or a : b .
A proportion is the statement or equation that two fractions or ratios are
equivalent.
If
a c
b d
then
a d b c
.
Example
a/b =c/d
½ = 3/6 so (1)(6) = (3)(2)
If a/b =c/d then ad = bc .
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THEOREM OF ALGEBRA
Given any two non equal rational numbers a and b then
a/b= q+r/b
where q is real called the quotient and r is real called the remainder.
Example
5/3= 1 + 2/3
If the remainder is zero or r = 0 then we say the division is even and b and q are
factors of a
as a / b = q .
We have 12/4=3 because 12/4=3+0/4
as q=0 .
If a/b=c then a=bc and we say b and c are factors of a.
Given any real number x and any natural number n we say that nx is a multiple of
x.
Given x = 3 then its multiples are:
3
6
= (1)(3)
= (2)(3)
9
12
…
= (3)(3)
= (4)(3)
PRIME NUMBERS
A natural number greater than one is prime if and only if it can be written as a
multiplication with the natural numbers one and itself and this is the only way to
do it.
Not every prime number is odd because two is prime.
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Not every odd number is prime because nine is odd and not prime.
It is exceedingly hard to tell if a random large number is prime.
This is an open question of number theory.
UNITS, METRICS, AND CONVERSIONS
Units are arbitrary conventions created by people to quantify properties from
observations using scientific studies and measurements such as physics,
chemistry, and biology.
Use of the british system only promotes the imperialist machinations of European
and english conquerors in the Americas specifically in the United States of
America.
The metric system is a modern and almost worldwide standard used in many
countries such as Mexico and most countries overseas. Every student should
learn the metric system and know how to convert between many different units
and systems.
UNIVERSAL CONSTANTS
The amounts of pi and e are universal constants.
6a. Percentages and rounding
Percents and numbers are not the same things. Percents have the percent
symbol % or unit and numbers do not.
Any real number can be converted into a percent and every percent may be
converted into a real number.
25%=0.25
25% is a percent and 0.25 or ¼ are the real numbers it expresses.
To convert a percent into a real number either move the decimal twice to the left
or multiply it by 1/100 and remove the percent symbol.
To convert a real number into a percent move the decimal twice to the right or
multiply by 100 and remove the percent symbol.
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Percent formula
A P
B 100
A is the amount (it is not a percent).
B is the base (it is not a percent and always comes after the word of in problems)
P is the percent (the formula converts the percentage into a real number)
7a. Averages (means, medians, and
mode) basic statistics
The outcomes from an experiment of statistics is called the data set S.
The mean of a data set is the average of its elements or that is add all elements
and divide by the number of elements.
The mode of a data set S is the most frequently occurring element(s) or does not
exist if there is no most frequently occurring element of the given data set S.
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A data set may not have a mode or may have one or a finite number of modes.
The median of a data set S with an odd number of elements is the middle
element when the elements of S are listed from left to right in an ordering.
If the number of elements of a data set S is even then its median is the average
of the middle two elements where all elements of S are listed in order from left to
right.
It is interesting to note that when the number of elements of a data set S is even
that its median may not be included in the data set S itself.
If the number of elements of a data set is odd its median is always included in the
data set.
Probability is the use of combinatorics, set theory, and lattice theory and can also
be considered with subsets of the Euclidean N Spaces.
Probability is a young area of mathematics and has existed only for the last 200
years or so at this point and therefore many text books vary greatly with
definitions.
Probability has its most accurate applications in quantum theoretical physics and
is used in actuarial sciences, economics, game theory, numerical analysis, and
chemistry.
Probability produces a source of understanding random events which may not be
comprehended in their totalities.
Experiments and Sample Spaces
The sample space of any experiment for probability is a set of real numbers or
non real elements or a set of n tuples from a product space of sets which may or
may not contain real numbers and any have other unreal elements called a data
set or sample space .
Events are cases of outcomes for given experiments.
Probability uses set theory, ratios, and the fundamental principles of counting
(combinatorics) to describe chances or likelihoods for given events or
experiments in theory.
Probability uses thought experimentation or creative imagination.
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Any probability of an event for a given experiment is the ratio of number of ways
the event may occur as an outcome from the experiment to the total number of
ways the experiment itself may conclude.
FUNDAMENTAL PRINCIPLE OF COUNTING
Permutations and Combinations
A permutation is a list of elements where the order from left to right makes a
difference and the number and type of elements listed does matter.
A combination is a list of elements where the order from left to right makes no
difference and the number and type of elements listed does matter.
The number of permutations is always greater than or equal to the number of
combinations for the same given elements.
When creating the sample space of a given experiment it is best to consider does
the order of choices when considering outcomes make a difference or no
difference and is there replacement or no replacement when making choices for
steps or stages to derive outcomes (that is can the same choice be made more
than once at any step (replacement) or only once ( no replacement) at any step).
Empirical vs. Theoretical
Theoretical Probability
We assume for any given set A that all equivalent forms of any element x of A
are only counted once or considered as equivalent when we list or write the
elements of A.
Empirical Probability or Statistics
We assume for any given set A that all equivalent forms and quantities of any
element x of A are all listed and counted each time they occur in A when we
consider and write the elements of A. The set A is called a data set and is a
subset of the real numbers or some product space of real numbers.
E represents the collection of all outcomes of experiment e given some specific
condition(s) on outcomes of e that may or may not be equal to all possible
outcomes of e with its generalized conditions represented by S.
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INCLUSIVE AND EXCLUSIVE EVENTS
Two events E and F are mutually inclusive if their set theoretic intersection is not
empty.
Two events E and F are mutually exclusive if their set theoretic intersection is
empty.
CONDITIONAL PROBABILITIES
Statistics
Statistics is the empirical application of theories from probability.
Statistics is one of the youngest areas of mathematics and has existed only for
the last 70 years or so at this point and therefore many text books vary greatly
with definitions and examples are usually numeric and complicated in nature.
It is useful to know how statistics are used, manipulated, and interpreted .
It is also useful to consider who is creating and using statistics as they can create
bias in the experiments, results, interpretations, and applications.
The sample space of any experiment for statistics is always a set of real numbers
or a set of n tuples from a product space of real numbers called a data set.
Given any event or experiment which can occur as a series of steps or stages
where at every step there is a finite number of choices to proceed to the next
step then the total number of outcomes or conclusions to the event or experiment
is the product of the number of choices which can be taken at each step or stage
to conclude the event or experiment.
Let N be the set of natural numbers.
Given a set
Let
A
let
U,U ,U
1
2
3
A
be its cardinality [s].
,...,U n
be sets and n is a natural number then we can define
a new set called their product as
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n
U  U U U
i 1
i
1
2
3
 ... U n 
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  x , x ,..., x  : x U
1
2
n
i
i
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for all i  1, 2,3,..., n

.
Given an event or experiment e that can be considered as a finite number of
steps where at each step there are any number of choices or options define its
sample space S as the collection of all outcomes of event e.
Given a set A we define its power set W(A) as the set of all subsets from A.
Given an event or experiment e where it can be considered as a finite number of
steps n with n is a natural number and where at each step there is a set of
choices or options.
Let Ai be the collection of all options for the ith step of experiment e and define
the sample space S of e as
n
S
  Ai 
i 1
  x , x ,..., x  : x  A
1
2
n
i
i
for all i  1, 2,3,..., n

.
Note that in the definition above the number of elements of any Ai may not be
finite.
Fundamental Principle of Counting
Given an event or experiment e where it can be considered as a finite number of
steps where at each step there are a finite number of choices or options then the
total number of outcomes of experiment e is equal to the product of the number
of choices for each step in e to complete the occurrence of event e.
[e]
Given an event or experiment e that can be considered as a finite number of
steps n and assume at each step there are a finite number of choices or options
define its sample space S as the collection of all outcomes of event e.
8a. Substituting values and
equations
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It may be true that the Universe does not always follow pemdas when
calculating and/or evaluating expressions in nature. Pemdas is an
agreement of man and woman so that the value of algebraic
expressions is well defined and therefore also viable in computer
calculations.
It is true too that a computer may be programmed that would violate
pemdas where computations of expressions are concerned and still
be functional.
Pemdas does demonstrate that we all can get along and agree no
matter where we are from, what we believe otherwise, or what we
look like all over the world for thousands of years where mathematics
and sciences are concerned.
Given an expression, equation, or inequality for any free variable, term, or any
parenthesis imagine scraping it away or removing it and where it had been
replace its absence by parenthesis.
Imagine moving into this parenthesis or substituting a real number (given or
evaluated), another expression or a variable and follow the order of operations
and laws of signs.
Parenthesis are required by an axiom when we use substitution and replace
because one times anything is itself and anything raised to the 1st power is itself
so that
x   x  1  x

1
and
x
   x   1
x 
1
.
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Theory
One of the most important skills in algebra is algebraic substitution and
replacement.
Using it allows us to correctly substitute values into formulas and accurately
produce viable results as in economics, finance, and sciences with calculators
and computers.
I. MULTIPLICATION AND DIVISION-Optics Laws of Signs
Given an expression to multiply or divide two real numbers :
C. If they have the same signs the result is positive
D. If they have unlike signs the result is negative.
In these cases one can think of images and their negatives in film production.
2. ADDITION AND SUBTRACTION-Accounting Laws of Signs
Given an expression where no multiplication or division is indicated only to add or
subtract two real numbers:
A. If they have the same signs combine them and give the result this similar sign.
B. If they have unlike signs then take their difference and give this result the sign
of the number which is farther from zero on the real line.
In this case one can think of owing as negative and having as positive.
PEMDAS is a worldwide agreement held so that arithmetic is well defined.
1. View only the amount contained by the inner most parenthesis, brackets,
braces, under radicals, or inside absolute values. (Those which are inner most
are contained by the most other sets of parenthesis, brackets, braces, radicals,
or absolute values.)
2. Evaluate exponents only in these inner most.
3. Multiply or divide as it goes from left to right only in these inner most.
4. Add or subtract as it goes from left to right only in these inner most.
5. Repeat for what parenthesis contains the inner most amount from above until
the expression is a single number expressed as a single term and unfactored.
If there is nothing which contains the parenthesis or the steps yield a redundancy
in the inner most then view the entire statement of the expression.
Parenthesis
Exponents
Multiply or Divide ( as you view from left to right )
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Add or Subtract ( as you view from left to right )
Parenthesis, brackets, and braces all mean the same thing which is whatever
amount is represented directly outside of and next to them (if no indication it is
the number one or if a minus sign it is the number minus one) will be multiplied
by what they contain and what they contain is raised to the 1st power unless
otherwise indicated.
A sign is only affected by an exponent if and only if it is contained inside
parenthesis.
9a. Setting up equations and word
problems
Equality is called a relation.
An equation is the statement that two expressions are equal.
Equations always begin with the equal symbol showing and expressions never
should begin with the equal symbol showing.
Expressions turn into equations as we evaluate them or simplify.
It is very important to recognize the difference between equations and expression
as their direction statements usually are different.
1. Read the problem all the way through without stopping for totality before
you begin to try to solve it.
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2. Make a basic comprehension of the problem in real world terms so that
you can reason a solution.
3. Draw any related pictures, diagrams, or figures if possible. Start to sketch
out the problem.
4. List the given relevant or essential data on your paper with units and
determine what information may not be relevant.
5. List any implied or given formulas and or equations on your paper.
6. Make sure all units are uniform.
7. Make a let statement to define the unknown or requested amount(s) as a
variable and derive other unknown amount as expressions using this
variable where required.
8. Create an equation which is equivalent to the word problem in
Mathematics from our case in English.
9. Make sure the equation is of only one variable type so it can be solved.
This may require substitutions.
10. Solve it.
11. Consider that all requested amounts are exhibited by re-reading the
problem and that you have shown all required answers. Otherwise you
may have to calculate more results. This may require you to re read the
problem statement again.
12. Include units for answers where they are required.
13. Check your answer(s) and make sure it (they) appear to make sense to
you.
14. Organize or arrange your results in a presentable manner clearly
indicating your answer.
Word problems can be stated in an unclear or ambiguous way so if this is the
case try one interpretation and if it does not produce good results try another
interpretation to produce logical and complete results. Never give up.
Theory
Given an equation with at most one free variable called x having nonzero
coefficients then its solution set is the collection of all values or real numbers that
can be substituted for x so that evaluation done by the order of operations and
the laws of sign yields or produces a true statement.
An equation is conditional if it has a finite number of solutions.
An equation is a contradiction if it has no solutions.
It will produce a false statement like 0=2 when you try to solve it and the answer
is no solution.
An equation is an identity if it has an infinite number of solutions.
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It will produce a true statement like 0=0 when you try to solve it and the answer is
all real numbers.
Each equation has a left hand side and a right hand side.
Let a and b be free algebraic expressions then we have any equation can be
represented as the symbolic statement
a=b .
A math problem should have the two components of the direction and the given
statement.
The direction of an equation is usually to solve and the direction of an expression
is usually to simplify.
The number of equations is uncountably infinite and some equations cannot be
expressed in finite space so we use variables to represent these equations.
Given an equation it is called radical if there exists a free variable with an
exponent that is non integer.
Given an equation it is called rational if there exists a free variable with an
exponent that is a negative integer.
PROPERTIES OF EQUALITY
If a = b then a + c = b + c .
If a = b then a - c = b - c .
If a = b then
ac= bc.
If a = b and c is not zero
then
a/c = b/ c
.
If a = b then b = a .
NON LINEAR EQUATIONS
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Quadratic Equations
An equation of the form
ax  bx  c  0
2
where a is not zero is called
non linear or quadratic.
Every quadratic equation is non linear. Every non linear equation is not
necessarily quadratic.
Quadratic equation implies it is non linear.
Non linear equation does not imply it is quadratic.
QUADRATIC FORMULA
Given an equation of the form
b  b  4ac

x
2a
ax  bx  c  0
2
the solutions are given by
2
.
PRINCIPLE OF ZERO PRODUCTS
If the product of a finite number of factors is zero then any one of the factors
could be zero.
If ab=0 then either a=0 or b=0 .
STEPS FOR SOLVING NON LINEAR EQUATIONS
1. Clear fractions in an equation by multiplying both sides by the least common
denominator. or
2. Clear parenthesis by distributing.
3. Combine like terms on each side separately .
4. Decide if the equation is linear or non linear.
5. If it is non linear make a choice to collect all terms on one side and zero on
the other side and accomplish this.
6. Factor the non zero side.
7. Set each factor to zero and solve these as linear equations.
8. If the non zero side will not factor and if the equation is quadratic then use the
quadratic formula.
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10a. Basic operations and
polynomials and linearity
An equation, expression, or inequality is polynomial with respect to a free
variable if and only if any exponent of the free variable is a whole number, no
base contains a free variable in its expression and it is not transcendental (or it is
the real number zero or the zero equation identity) otherwise it is called non
polynomial.
W  0 ,1 , 2 , 3 ,... is the set of whole numbers.
An expression which is a polynomial and a single term is called a monomial.
Given a polynomial its degree is the greatest exponent or power of any free
variable or the greatest sum of the exponents for all free variables if there are
more than one.
2
Given the polynomial expression 3x it is a monomial because it is a single
term and 3 is called the coefficient, x is called the base and 2 is called its
exponent or power.
Given an expression to add or subtract polynomials we follow the order of
operations by distributions of one or minus one to their terms then collect like
terms.
Given an expression where it is indicated to multiply two other expressions of
polynomials we distribute every term in the first factor to every term in the second
factor then collect like terms if possible.
Dividing polynomials requires understanding of the Fundamental Theorem of
Algebra and the long division algorithm.
Theory
THEOREM OF ALGEBRA
Given two polynomials
polynomials
q
s
d
.
q 
r
d
and
r
s
and
d
with
d
is not zero then there exists
so that
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TERMS AND FACTORS FOR EXPRESSIONS
Given an expression its terms are separated by plus or minus symbols and
the factors of its terms are separated by multiplications and if there is no
separation by plus or minus symbols the expression is a single term.
Every factor of any term for an expression may have one or a finite number of
terms itself.
Given an expression like terms have exactly the same variables as factors raised
to exactly the same exponents.
We combine like terms by adding/subtracting their coefficients.
Simplified for an expression can mean many things.
A fraction is simplified if its numerator and denominator contain no common
factors.
An expression is simplified if it contains no negative exponents and like terms are
combined.
NOTATIONS AND WRITING MATHEMATICS
A single variable expression is assumed to be positive if it has no positive sign, a
positive sign, or no negative sign written in front of its first factor as we view its
expression or representation.
x = +x = (1)(x)
A single variable expression is assumed to be negative if it has a negative sign
written in front of its first variable as we view its expression or representation.
-x
No sign of addition, subtraction, multiplication, and/or division can be directly
written next to each other unless they are either separated by a free or fixed
variable or parenthesis.
If there is a negative sign written outside of parenthesis it means multiplication of
the number minus one with the rest of the written expression.
-x = (-1)(x)
Remember that
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x x x
1
1

S. Zoch
 1  x   a  x  a  x 1  nx
n

x 0
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.
VARIABLES
A variable is a letter , symbol , object, event, place, shape, color, pattern ,
person, name or any combination or grouping of these that represents one
(fixed) or many (free) other
letter(s) , symbol (s) , object (s), event (s),
pattern (s), person (s), number (s) , set (s), category (ies) , group (s), path (s),
time (s), taste (s) , odor (s), feeling (s) , idea (s), statement (s), sound (s) ,
shape (s), color (s), state (s) , unit (s) , movement (s) , observation (s) , place (s),
touch (es), sight (s), quantity (ies), TO WHICH IT OR THEY MAY OR MAY NOT
BE EQUIVALENT.
A variable is a symbolic name for an object or event so that it can be called upon
even if it has infinite properties or is difficult to describe.
A variable is a symbolic name that represents one object or event or possibly
many other objects or events from a set to which the variable may or may not be
equal or look the same.
As there exists numbers with a large amount of entries or infinite number of non
zero decimal positions or entries which therefore can not be drawn, listed, or
written in finite time and/or finite area we must use variables, names or symbols
to express them.
A variable is either free or fixed (constant) .
A variable is chosen to be free or fixed (constant) with a let statement from a
certain standard observer.
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A free variable is a symbol or group of symbols which can represent many items
of the list written above at all times.
As an example we say let x be a free real number so that in any place of the
symbol x where it might be located we could substitute a number like 43.
A fixed (constant) variable is a symbol or group of symbols which only represents
one item of the list written above at every time.
Any letter from an alphabet or real number is a fixed variable or constant.
The symbol 8 is a fixed variable because it only represents the amount of 8
items.
Many of the skills of Algebra are accomplished using the properties of the
numbers zero and one called algebraic manipulations.
An expression is the finite sum, difference, product and/or quotient of a finite
number of free and/or fixed variables or any single variable.
Expressions represent a real number and always begin without an equal symbol
when given.
Each real number can express itself in an infinite number of ways with the binary
operations of adding, subtracting, multiplying, and/or dividing.
TERMS AND FACTORS FOR EXPRESSIONS
Given an expression its terms are separated by plus or minus symbols and
the factors of its terms are separated by multiplications and if there is no
separation by plus or minus symbols the expression is a single term.
Every factor of any term for an expression may have one or a finite number of
terms itself.
Given an expression like terms have exactly the same free variables as factors
raised to exactly the same exponents.
We combine like terms by adding/subtracting their coefficients.
Simplified for an expression can mean many things.
A fraction is simplified if its numerator and denominator contain no common
factors.
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An expression is simplified if it contains no negative exponents and like terms are
combined.
Coefficients, bases, and powers
Polynomials
An expression, equation or inequality is called polynomial if and only if the
exponent of any free variable is a whole number, no base contains an exponent
with a free variable, and it is not transcendental otherwise it is non polynomial.
An expression which is a polynomial and a single term is called a monomial.
Given a monomial its degree is the exponent or power of any free variable or the
sum of the exponents for all free variables if there are more than one.
Given a polynomial with more than one term its degree is the largest degree of
any of its terms.
11a. Factoring polynomials and
solving non linear equations
Given an expression it is factored if and only if it is written as a product, with at
least two factors, and as a single term otherwise it is unfactored.
If an expression is not factored we call it unfactored.
Any expression is either factored or unfactored when it is given and never exists
in a mixed state.
Any amount may be factored out of an expression with more than one term but
usually we factor out the greatest common factor.
GCF
Given an expression with more than one term its greatest common factor is the
amount which is the largest or has the most prime factors so that it can divide
each term of the given expression evenly.
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Theory
To factor an expression we use parenthesis and write the amount we want to
multiply or factor (take) out next to the expression that is a multiple of the original
given expression using parenthesis.
ALGEBRAIC FORMS
OR FORMATS OF EXPRESSIONS
Much useful information can be derived from the algebraic forms of equations
and expressions.
Understanding and skill with application over the forms or formats of equations
and expressions allows us to manipulate abstractions in our modern lives and
business to our benefit such as finance, legality, and productivity with technology
and resources.
Given any expression we can ask about its given or written form is it factored or
unfactored at some time.
Given any expression we can ask how many terms does it have the way it is
being written or expressed.
Given an expression with a single term or any term of all other expressions we
can ask how many factors are written at a time and what are the factors of
terms?
Steps to Factor
1. Factor out a GCF if other than the number one.
2. Trinomial use ac method with grouping if necessary.
3. Difference of perfect squares use formula.
4. Sum or difference of perfect cubes us formula.
5. Four or more terms always factor by grouping.
DIFFERENCE OF TWO PERFECT SQUARES
x y
2
2
  x 

y  x  y 
The left hand side of the equation above is unfactored and the right side is the
factored format.
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DIFFERENCE OR SUM OF TWO PERFECT CUBES
x y
3
  x 

y  x  xy  y 

3
  x 

y  x  xy  y 
3
3
x y
2
2
2
2
The left hand side of the equations above are unfactored and the right side is the
factored format.
Simplify means:
1. Perform the indicated operations.
2. Evaluate.
3. Calculate.
4. Add, subtract, multiply or divide.
5. Combine like terms.
6. Distribute across parenthesis.
7. No answer should have negative exponents.
8. Any numeric base should be evaluated.
9. No radical of even index may have a negative radicand.
10. No fraction can have a radical or i in its denominator.
11. Any answer that is a fraction must always be reduced or cancelled.
12.
12a. Linear equations a
LINEAR EQUATIONS
An equation of the form
ax  b  0
where a is not zero, a and b are fixed
and x is free is called linear in one free variable .
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Any algebraic equation, expression, or
inequality is linear (with respect to a free
variable) if and only if (it is the real number
zero or the zero equation) any exponent a
free variable is zero or one, no base with
nonzero coefficient contains a free variable
with a non zero coefficient in its exponent
expression and it is not transcendental
otherwise it is called non linear.
Theory
STEPS FOR SOLVING LINEAR EQUATIONS
1. Clear fractions in an equation by multiplying both sides by the least common
denominator. or
2. Clear parenthesis by distributing.
3. Combine like terms on each side separately .
4. Decide if the equation is linear or non linear.
5. If it is linear make a choice to collect all terms with the factor of x on one side
and all other terms without the factor of x on the other side and accomplish this
using the properties of equality.
6. Make sure the coefficient of the variable is the number one.
Make sure the only factor and exponent of x is one so that it says x = ‘a number’.
13a. Linear equations and graphing
DOMAINS OF EQUATIONS AND EXPRESSIONS
Given an equation or expression with at most two free variables x and y having
nonzero coefficients define its domain as the collection of all values which may
be substituted for x so that evaluation makes a statement which is defined.
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Defined means no division by zero and no negatives under even indexed
radicals.
The domain is usually all real numbers unless there is a variable in the
denominator of a fraction and/or a variable under a radical.
If there is a variable in the denominator of a fraction set this expression of the
denominator equal to zero and solve it.
These solutions are not in the domain so that the domain in this case is all reals
except these solutions.
If there is a variable under a radical set the expression under the radical greater
than or equal to zero and solve such inequality. These solutions are in the
domain so that the domain is the set of only these solutions.
RANGE
Given an equation with at most two free variables x and y having non zero
coefficients its range is defined as the collection or set of all numbers y which
are calculated through evaluation with the order of operations agreement as
values of x are substituted from the domain.
Given any collection of points in a plane its domain is the collection of all x values
of their coordinates and the range is the collection of all y values of their
coordinates.
Two distinct points are all that is required to create a line.
An equation of the form Ax + By = C where A, B, and C are fixed and A and B
are not both zero with x and y are free is called linear with respect to x and y.
Given an equation of the form Ax + By = C we define its solution set as the
collection of all (s,t) or points is a plane so that A(s) +B(t) =C is a true statement.
SLOPE VALUES OF NON VERTICAL LINES
Every line has a slope value associated with it which is a number that is either
positive, negative, zero, or undefined.
The slope value of a line tells us about the geometry of its graph like how much
the line is slanted and in what direction it is slanted, no slant (flat), or a vertical
line.
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Given two distinct points in a plane
S. Zoch
x , y 
1
1
and
x , y 
2
2
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we define the slope
value of the line that passes through them denoted and equal to the following :
m

y y
x x
2
1
2
1
.
Theory
There is a huge relationship through the definition of solution sets for equations
between algebra and geometry.
DOMAINS OF EQUATIONS AND EXPRESSIONS
Given an equation or expression with only the free variable x we define its
domain as the collection of all values which may be substituted for x so that
evaluation makes a statement which is defined.
Defined means no division by zero and no negatives under even indexed
radicals.
The domain is usually all real numbers unless there is a variable in the
denominator of a fraction and/or a variable under a radical.
If there is a variable in the denominator of a fraction set this expression of the
denominator equal to zero and solve it.
These solutions are not in the domain so that the domain in this case is all reals
except these solutions.
If there is a variable under a radical set the expression under the radical greater
than or equal to zero and solve such inequality. These solutions are in the
domain so that the domain is the set of only these solutions.
Given any collection of points in a plane its domain is the collection of all x values
of their coordinates and the range is the collection of all y values of their
coordinates.
THE REAL PLANE
The real plane is defined as a set of ordered pairs representing its points as
coordinates denoted and equal to the following
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R R  R
R
R
2

   x , y  :

 
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
x, y  R  .

2
is called the Euclidean two space or the real plane.
2
is the Cartesian product of two sets of real numbers.
The scheme to construct the ordering of a plane was first conceived by Rene
DesCartes.
A plane when it is considered is a purely mathematical object so that it requires
our imagination and symbols to conceive the totality of the concept.
Given an ordered pair ( x , y ) we may plot it or graph it by starting at the origin of
a plane and going x units right if x is positive or x units left if x is negative then
from that place y units up if y is positive or y units down if y is negative and then
making a mark or dot.
PLANAR RELATIONS
OR GRAPHS OF EQUATIONS
Horizontal lines are flat straight across and vertical lines go straight up and down.
FINDING EQUATIONS OF LINES
Coordinates of a point in a plane and a slope value are required to produce an
equation of a line.
y = mx+b
STEPS FOR GRAPHING LINES IN A PLANE
1. Plot points using a chart with x then y or usually an alphabetical order.
2. Find the x intercept by substituting y = 0 into the given equation if any
exist.
3. Find the y intercept by substituting x = 0 into the given equation if any
exist.
4. Choose a value for x or for y (not both at once) and substitute it into the
given equation and solve for the other value and list these as a point x
value then y value.
5. Create and label x and y axis and scale units appropriately .
6. Plot these points and draw a line between them using arrows at the ends.
FUNCTIONS
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An equation which uses at most two free variables called x and y is a function if
and only if for any substitution of a value of x into the equation evaluation will
yield only one value for y.
Given any two dimensional graph or collection of points in a plane it is a function
if there does not exist a vertical line which intersects it or them in more than one
place.
Any singleton point in a plane is a function.
Any non vertical line is a function.
Any circle of positive radius in not a function.
Given an equation with at most x and y are free then if y is a function of x and
we say f (x) = y where f is identified as the name of the equation .
f (x) is called function notation and produces names and ordered pairs for
complicated equations.
GRAPHING EQUATIONS IN THE PLANE
1. Plot points using a chart with x then y or usually an alphabetical order.
2. Find the x intercepts by substituting y = 0 into the given equation if any
exist.
3. Find the y intercepts by substituting x = 0 into the given equation if any
exist.
4. Find the vertical asymptotes from the domain of the given equation.
5. Find the horizontal asymptotes .
6. Create and label x and y axis and scale units appropriately .
7. Sketch the graph using intercepts and asymptotes.
14a. Exponents and radicals
EXPONENTS
There are many ways to work with exponents.
The hardest part of mathematics is making all the choices yet your answer in the
end should be equivalent to any one else’s.
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Let x and y be free.
x x
1
0
undefined
0
x 1
when
x x  x
m n
0
m
n
x 0
.
xm n  xmn
x
y
x
n
m
n
y
x


m
n
1
x
n
n
 x
 
 y
 y
 
 x
n
RADICALS OR RATIONAL EXPONENTS
 x
n
If
y
m

n

x
x
m
n
then
n
y x
.
EXPONENTIAL FUNCTIONS
An equation with at most two free variables say x and y having nonzero
coefficients is called exponential if and only if x is an exponent and y is not.
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y b
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x
15a. Rational expressions and
equations
A rational expression is a technical name for a fraction so that all fraction
properties apply to rational expressions.
Any real number can be written as a rational expression as it is a polynomial and
because it is equal to itself divided by one and one is a polynomial.
THEOREM OF ALGEBRA
Given any two non equal rational numbers a and b then
a/b= q+r/b
where q is real called the quotient and r is real called the remainder.
Example
5/3= 1 + 2/3
If the remainder is zero or r = 0 then we say the division is even and b and q are
factors of a
as a / b = q .
We have 12/4=3 because 12/4=3+0/4
as q=0 .
If a/b=c then a=bc and we say b and c are factors of a.
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RATIONAL FUNCTIONS
A function y = f(x) is called rational if f(x) is or can be written as one polynomial
divided by another non zero polynomial.
PARABOLAS
LOGARITHMIC FUNCTIONS
Let x and y be free and b is a non zero real number.
An equation with at most only the two free variables say x and y is called
logarithmic if and only if y is an exponent and x is not.
x b
y
Logarithmic notation is a way to express such equations.
If
log x  y
b
then
b
y

x
.
EXPONENTIAL FUNCTIONS
An equation with at most only the two free variables say x and y is called
exponential if and only if x is an exponent and y is not.
y b
x
Long division of polynomials
16a. Set Theory
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REAL NUMBERS
Real numbers are those amounts which we deal with on a daily basis either in
accounting, travel, medicine, engineering, and/or architecture .
SUBSETS OF REAL NUMBERS
The set of natural numbers is the collection of elements
1,2,3,4,5,…
The set of whole numbers is the collection of elements
0,1,2,3,4,…
The set of integers is the collection of elements
…,-4,-3,-2,-1,0,1,2,3,4,…
The set of rational numbers is the collection of all elements of the form
a/b so that a and b are integers and b is not zero.
The set of irrational numbers is the collection of all elements of the form whose
decimal expansions do not terminate and do not repeat.
SET OF REAL NUMBERS
The set of real numbers is the union of the set of rational numbers with the set of
irrational numbers.
Let R represent the set of real numbers.
There are an uncountably infinite number of real numbers.
The natural numbers are countably infinite because it would take a person or
machine an infinite amount of time to list them.
The real numbers are uncountably infinite because it would not be possible for a
person(s) or machine(s) to list them in any time and it would not be possible for
an infinite number of persons and/or machines to list them in any time.
Given any real number it is either rational or irrational.
Every natural number is also a whole number.
N W
Every whole number is also an Integer.
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Z
Every integer number is also a rational number.
Z Q
Every whole number is not also a natural number. Every integer number is not
also a whole number. Every rational number is not also an integer number.
No real number is both rational and irrational.
Let the finite cardinality of a set be denoted by n.
Let the countable infinite cardinality of a set be denoted by

.
Let the uncountable infinite cardinality of a set be denoted by c.
The cardinality of the set of natural numbers, whole numbers, and integers is
.
The cardinality of the set of irrational numbers and real numbers is c.
We hold that
 c

and
c 2
c
.
Given any real number we find its opposite by changing its sign.
Every real number has an opposite except zero which can be either plus or
minus.
The real numbers have an ordering on the real line from smallest to greatest from
left to right with the inequality relation.
Given any real number x we define its absolute value as the distance from itself
to zero on the real line denoted
x
.
PROPERTIES OF REAL NUMBERS AND OPERATIONS
The real binary operations are the calculations we normally apply to any pair of
real numbers such as adding, subtracting, multiplying, and dividing.
Commutative
Addition Commutes
A+B = B+A so that
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A+B+C = C+B+A = A+C+B = C+A+B = B+C+A = B+A+C
…
Multiplication Commutes
AB = BA so that
ABC = CBA = ACB = CAB = BCA = BAC
…
Distributive
Distribution over additions
A(B+C)= AB+AC
2(3+4)= 2·3+2·4
Distribution over subtractions
A(B-C)= AB-AC
2(3-4)= 2·3-2·4
Associative
Associativity of Addition
(A+B)+C = A+(B+C)
(2+3)+4 = 2+(3+4)
Associativity of Multiplication
(AB)C = A(BC)
(2·3)4 = 2(3·4)
Multiplicative Identity
One is the multiplicative identity because anything multiplied by one is itself.
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(1)(x) = (x)(1) = x
(1)(2) = (2)(1) = 2
Additive Identity
Zero is the additive identity because anything added to zero is itself.
0+x= x+0=x
0+2= 2+0=2
Multiplicative Inverse
Given any real number not equal to zero called x we define its inverse or
reciprocal as 1/x.
Any nonzero real number multiplied by its reciprocal is one.
Additive Inverse or Opposites
NOTATIONS AND WRITING MATHEMATICS
A single variable expression is assumed to be positive if it has no positive sign, a
positive sign, or no negative sign written in front of its first factor as we view its
expression or representation.
x = +x = (1)(x)
A single variable expression is assumed to be negative if it has a negative sign
written in front of its first variable as we view its expression or representation.
-x
No sign of addition, subtraction, multiplication, and/or division can be directly
written next to each other unless they are either separated by a free or fixed
variable or parenthesis.
If there is a negative sign written outside of parenthesis it means multiplication of
the number minus one with the rest of the written expression.
-x = (-1)(x)
Remember that
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x x x
1
1

S. Zoch
 1  x   a  x  a  x 1  nx
n

x 0
c. t( x )2012
.
SET THEORETIC OPERATIONS AND
RELATIONS
Given two sets A and B we say they are equal and write A = B if and only if they
are both subsets of each other or A  B
and
B A .
Given a set A and another set B we can create a new set called their union which
includes all elements of A or B written A  B .
We write or denote the following:
A B

 x : x  A or x  B 
.
Given a set A and another set B we can create a new set called their intersection
which includes only elements in common to A and B written A  B .
We write or denote the following:
A B

 x : x  A and x  B 
.
examples
Given a set A and a set B where A is a subset of B then we define the
complement of A with respect to B as the set of all elements which are elements
of B and not A denoted B/A,
A
, or
A
/
.
DeMORGAN’S LAWS
The complement of the union of sets is the intersection of their complements.
The complement of the intersection of sets is the union of their complements.
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PRODUCT SPACES OF SETS
Let A and B be sets then we can define a new set called their product denoted
and equal as

   x , y  :

 
A B

x  A and y  B 
.
Let A, B and C be sets then we can define a new set called their product as

   x , y , z  :

 
A  B C
Let
U,U ,U
1
2
3
,...,U n

x  A , y  B , z  C  .
be sets then we can define a new set called their
product as
U U U
1
2
3
 ... U n 
 x , x ,..., x  : x U
1
2
n
i
i
for all i  1,2,3,...,n
.
Given a set A we define its power set W(A) as the set of all subsets from A.
We assume for any given set A that all equivalent forms of any element x of A
are only counted once when we consider, list, or write the elements of A.


Example
Let
3 , 2 , w ,8 , 
A then
 2 , 3 , w ,8 = 3 , 2 , w ,8 , 2  A =  2 , w 3 , 3 w ,8 , w =


 8 , 2 , 3 , w , 8  =


COMPLEX NUMBERS
 2 , 3 , 2 , w , 3 ,8  = …….
Complex numbers are those amounts which we deal with on a basis either in
electronics, computers, graphics, engineering, and physics .
Let x be a free complex variable.
The set of real number is a subset of the set of complex numbers.
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Every real number is by definition also complex.
The set of complex numbers is not a subset of the set of real numbers.
Every complex number is not necessarily also real.
Let
i  1
2
be an imaginary variable so that
i
.
We define the set of complex numbers denoted and equal to the following:
C




a bi : a,b  R and i  1 
2
.
Categories of equations based on their solution sets
An equation is conditional if it has a finite number of solutions.
An equation is a contradiction if it has no solutions.
It will produce a false statement like 0=2 when you try to solve it and the answer
is no solution.
An equation is an identity if it has an infinite number of solutions.
It will produce a true statement like 0=0 when you try to solve it and the answer is
all real numbers.
The number of equations is uncountably infinite and some equations cannot be
expressed in finite space so we use variables to represent these equations.
Given an equation it is called radical if there exists a free variable with an
exponent that is non integer.
Given an equation it is called rational if there exists a free variable with an
exponent that is a negative integer.
18a. Format of Problem Statements
or Queries
A math problem should have two components consisting of a direction and a
given statement. The directions should be satisfied to achieve an answer.
The directions are applied to the given statement with analysis.
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ooooooooooooooooooooooooooooooooooooooooooooo
MOST BASIC IDEAS, DEFINITIONS, AND
THEOREMS OF MATHEMATICS, SET
THEORY, AND LOGIC
Set theory is one of the most comprehensive, basic, and most
powerful areas of mathematics.
Areas of mathematics include uses of Set Theory’s definitions and /or
theorems such as Probability, Real Analysis, Complex Analysis,
Category Theory, Boolean Algebras, Graph Theory, Abstract Algebra,
Non Standard Analysis, Lattice Theory, Ramsey Theory, Dimension
Theory, Number Theory, and Topology.
Any equations, laws or formulae which are fundamental to
physics have many mathematical implications beyond their
known or accepted interpretative meanings and these
cannot be discarded in reality as their conditional truth is
verified by experimental evidence and the laws and axioms
of mathematics exist beyond the realms of physical sciences
and computers.
Nearly all the areas of mathematics listed above also use Algebra
and many have their own unique algebraic definitions called relations
yet most algebraic principles are common to all.
Every area of Mathematics includes many uses of Logic and Sets.
All of the areas of Philosophy and Mathematics depend on given
premises, relations, and/or axioms.
Only the areas of Philosophy known as First Order Predicate
Calculus, Universal Algebras, and Logic may not require the use of
Set Theory.
Philosophy is an ancestor of Mathematics and Logic.
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AXIOMS
The axioms of Mathematics are similar to those of Logic and required
to create basis.
AXIOM OF OPPOSITES
Every real number has an opposite except zero and any real number
added to its opposite must equal to zero. Zero added or subtracted
from any real number a must equal to a.
AXIOM OF PARALLEL LINES
We assume all distinct lines which are parallel never cross or
intersect in any two dimensional Euclidean plane where they exist.
AXIOM OF ARCHIMEDES
Given any two non equal real numbers there is always another
distinct real number between them in value one of which is their
average.
AXIOM OF CHOICE
Given any infinite collection or set we may select an infinite proper
subset of singletons.
Given any two infinite sets A and B we may select an infinite proper
subset of pairs or any singleton element from their product
A B

   x , y  :

 

x  A and y  B 
.
Given any three infinite sets A, B, and C we may select an infinite
proper subset of triples or 3 tuples or any singleton element from their
product
A  B C

   x , y , z  :

 

x  A , y  B , z  C 
.
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Given any finite number of infinite sets U 1, U 2, U 3 ,...,U n
we may
select an infinite proper subset of n tuples or any singleton element
from their product
U U U
1
2
3
 ... U n 
 x , x ,..., x  : x U
1
2
n
i
i
for all i  1,2,3,...,n

.
Given any countable number of infinite sets we may select an
infinite proper subset of countable tuples or any singleton element
from their product.
Given any uncountable number of infinite sets we may select an
infinite proper subset of uncountable tuples or any singleton element
from their product.
…
.
.
AXIOM OF VARIABLE
We may use symbolic names called variables to represent objects or
events to which the variable may or may not be equal or the same
object or event to any standard observer.
AXIOM OF VARIABLE OCCUPATION
We assume that if two variables which may or may not occupy the
same space and appear to many standard observers to be equivalent
can be used in an equivalent way for every expression, task and/or
calculation preformed by any standard observers.
AXIOM OF VARIABLE TRANSCENDENCE
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The use of one variable to represent an object(s) or event(s) may be
replaced with any other variable at any time to produce the same
representation.
Usually this is done with a let statement.
AXIOM OF VARIABLE SUBSTITUTION
Given an expression with free variables we may substitute amounts
equal to these so that the new expression is always equivalent to the
original expression under any evaluation.
AXIOM OF LINEAR ORIENTATION
AXIOM OF ORIENTATION FOR PLANES
Right hand rule.
SET THEORY
Given a finite number of sets U 1, U 2, U 3 ,...,U n with n is a
natural number then define their Cartan Join as the
collection
C  U VU V U V ...V U V U   a , a , a ,..., a  : a U for all i  1, 2,3,..., n 
1
2
n 1
3
n
1
2
3
n
i
i
or
n
C
 VU i .
i 1
The Cartan Join of sets commutes.
Given a set A

and if U i  A for all i=1,2.3,…,n and n is a
natural number so that
n

A
then the Cartan Join
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C  U VU V U V ...V U V U
1
2
3
n 1
is equal to the power set of
S. Zoch
n

A
 a , a , a ,..., a  : a  A
1
2
3
n
i
c. t( x )2012
for all i  1, 2,3,..., n

.
A set is a collection of objects called elements.
As there exists sets with a number of elements which can not be
drawn, listed, or written in finite time and/or finite area because they
contain some element which is too large to draw or print or they
contain a countably infinite or uncountably infinite number of
elements we must use variables, names, or symbols to express them.
Let A and x be free set theoretic and elemental variables.
A represents a set and x represents an element of a set.
Given a set A if x is an element of A we write
an element of A we write
Example
Let A 
x A
and if x is not
x A .
 2 , 3 , w ,8 , 
so
8 A
or
3  A
and
3A
.
Given a set A and a set B where A=B then B=A.
A set is either empty, finite, or infinite.
  
is an example of an empty set.
 2 , 3 , w ,8 ,  is an example of a finite set.
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The set of natural numbers
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 1 , 2 , 3 ,... 
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is an example of an
infinite set.
The elements of sets can be listed in any order to have the same set
such as
2 , 3 , w ,8 ,= 3 , 2 , w ,8 ,= 2 , 3 ,8 , w ,…
The elements of sets can be any type of unusual objects which may
or may not be real however usually for us they will be real numbers.
Given a set A and another set B if every element of A
is also an element of B we say A is a subset of B and
write A  B .
Example
Let C  e , 3 , w , 6 , 5  and
C

D
D

e , h , 3 , 4 w ,11 , 6 , 5 
so
.
Given a set we define its cardinality, number, or order as
the number of elements it contains.
Given any set A we define its cardinality as the number of elements it
contains denoted A .
SET THEORETIC OPERATIONS AND
RELATIONS
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Given two sets A and B we say they are equal and write A = B if and
only if they are both subsets of each other or A  B
and
B

A
.
Given a set A and another set B we can create a new set called their
union which includes all elements of A or B written A  B .
We write or denote the following:
A B

 x : x  A or x  B 
.
Given a set A and another set B we can create a new set called their
intersection which includes only elements in common to A and B
written A  B .
We write or denote the following:
A B

 x : x  A and x  B 
.
examples
Given a set A and a set B where A is a subset of B then we define the
complement of A with respect to B as the set of all elements which
are elements of B and not A denoted B/A, A , or A / .
DeMORGAN’S LAWS
The complement of the union of sets is the intersection of their
complements.
The complement of the intersection of sets is the union of their
complements.
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Given a finite number of sets U 1, U 2, U 3 ,...,U n with n is a
natural number then define their Cartan Join as the
collection
C  U VU V U V ...V U V U   a , a , a ,..., a  : a U for all i  1, 2,3,..., n 
1
2
n 1
3
n
1
2
3
n
i
i
or
n
 VU i .
C
i 1
PRODUCT SPACES OF SETS
Let A and B be sets then we can define a new set called their
product denoted and equal as

   x , y  :

 
A B

x  A and y  B 
.
Let A, B and C be sets then we can define a new set called their
product as
A  B C
Let

   x , y , z  :

 
U,U ,U
1
2
3
,...,U n

x  A , y  B , z  C  .
be sets then we can define a new set called
their product as
U U U
1
2
3
 ... U n 
 x , x ,..., x  : x U
1
2
n
i
i
for all i  1,2,3,...,n
.
Given a set A we define its power set W(A) as the set of all subsets
from A.
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We assume for any given set A that all
equivalent forms of any element x of A are only
counted once when we consider, list, or write the
elements of A.


Example
Let
3 , 2 , w ,8 , 
A then
 2 , 3 , w ,8 = 3 , 2 , w ,8 , 2  A =  2 , w 3 , 3 w ,8 , w =




8 , 2 , 3 , w , 8 
=
 2 , 3 , 2 , w , 3 ,8  = …….
VENN DIAGRAMS
The collection of all sets U is not a set and is called a class.
The universe of all sets is denoted U.
A Venn Diagram is a two dimensional graph, picture, or
subset of a (Euclidean) plane representing some given sets
and their conjunction and disjunction properties.
Let A be a set then
we draw, take, or identify a circle of positive radius in the plane called
C
A
to represent A with its circumference and all elements of A are
written, listed or graphed as symbols in the interior of the circle
C
A
.
Given distinct elements of the universal set U , call them sets A and B
,
we draw or take distinct circles in the plane called C A and C B
where the boundaries and interiors of the these two circles do not
intersect if A intersected with B is empty or A and B are mutually
exclusive.
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If A intersected with B is non empty and A is not B then we let the
interiors of C A and C B intersect in as a non empty region and
write all elements in common to A and B in this region because A and
B are mutually inclusive.
If A is contained in B then A is a subset of B and the interior and
boundary of circle C A is contained in the interior of circle C B .
If B is contained in A then the interior and boundary of circle
contained in the interior of circle C A .
C
B
is
If A = B then C A = C B and all element of A and B are only listed
once each in the interior of the circle .
The universe of all sets U is a class and contains all categories of
VENN DIAGRAMS. Any finite collection of finite sets may be
represented by a visible Venn diagram.
example
REAL NUMBERS
Real numbers are those amounts which we deal with on a daily basis
either in accounting, travel, medicine, engineering, and/or
architecture .
SUBSETS OF REAL NUMBERS
The set of natural numbers is the collection of elements
1,2,3,4,5,…
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The set of whole numbers is the collection of elements
0,1,2,3,4,…
The set of integers is the collection of elements
…,-4,-3,-2,-1,0,1,2,3,4,…
The set of rational numbers is the collection of all elements of the
form
a/b so that a and b are integers and b is not zero.
The set of irrational numbers is the collection of all elements of the
form whose decimal expansions do not terminate and do not repeat.
SET OF REAL NUMBERS
The set of real numbers is the union of the set of
rational numbers with the set of irrational numbers.
Let R represent the set of real numbers.
There are an uncountably infinite number of real numbers.
The natural numbers are countably infinite because it would take a
person or machine an infinite amount of time to list them.
The real numbers are uncountably infinite because it would not be
possible for a person(s) or machine(s) to list them in any time and it
would not be possible for an infinite number of persons and/or
machines to list them in any time.
Given any real number it is either rational or irrational.
Every natural number is also a whole number.
N
W
Every whole number is also an Integer.
W
Z
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Every integer number is also a rational number.
Z
Q
Every whole number is not also a natural number. Every integer
number is not also a whole number. Every rational number is not also
an integer number.
No real number is both rational and irrational.
Let the finite cardinality of a set be denoted by n.
Let the countable infinite cardinality of a set be denoted by

.
Let the uncountable infinite cardinality of a set be denoted by c.
The cardinality of the set of natural numbers, whole numbers, and
integers is  .
The cardinality of the set of irrational numbers and real numbers is c.
We hold that
 c

and
c 2
c
.
Given any real number we find its opposite by changing its sign.
Every real number has an opposite except zero which can be either
plus or minus.
The real numbers have an ordering on the real line from smallest to
greatest from left to right with the inequality relation.
Given any real number x we define its absolute value as the distance
from itself to zero on the real line denoted x .
PROPERTIES OF REAL NUMBERS
AND OPERATIONS
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The real binary operations are the calculations we normally apply to
any pair of real numbers such as adding, subtracting, multiplying, and
dividing.
Commutative
Addition Commutes
A+B = B+A so that
A+B+C = C+B+A = A+C+B = C+A+B = B+C+A = B+A+C
…
Multiplication Commutes
AB = BA so that
ABC = CBA = ACB = CAB = BCA = BAC
…
Distributive
Distribution over additions
A(B+C)= AB+AC
2(3+4)= 2·3+2·4
Distribution over subtractions
A(B-C)= AB-AC
2(3-4)= 2·3-2·4
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Associative
Associativity of Addition
(A+B)+C = A+(B+C)
(2+3)+4 = 2+(3+4)
Associativity of Multiplication
(AB)C = A(BC)
(2·3)4 = 2(3·4)
Multiplicative Identity
One is the multiplicative identity because anything multiplied by one
is itself.
(1)(x) = (x)(1) = x
(1)(2) = (2)(1) = 2
Additive Identity
Zero is the additive identity because anything added to zero is itself.
0+x= x+0=x
0+2= 2+0=2
Multiplicative Inverse
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Given any real number not equal to zero called x we define its inverse
or reciprocal as 1/x.
Any nonzero real number multiplied by its reciprocal is one.
Additive Inverse or Opposites
NOTATIONS AND WRITING
MATHEMATICS
A single variable expression is assumed to be positive if it has no
positive sign, a positive sign, or no negative sign written in front of its
first factor as we view its expression or representation.
x = +x = (1)(x)
A single variable expression is assumed to be negative if it has a
negative sign written in front of its first variable as we view its
expression or representation.
-x
No sign of addition, subtraction, multiplication, and/or division can be
directly written next to each other unless they are either separated by
a free or fixed variable or parenthesis.
If there is a negative sign written outside of parenthesis it means
multiplication of the number minus one with the rest of the written
expression.
-x = (-1)(x)
Remember that
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x x x
1
1

S. Zoch
 1  x   a  x  a  x 1  nx

n
x 0
c. t( x )2012
.
DECIMALS OR BASE TEN
REPRESENTATIONS
Let n and m be whole numbers.
Select a base ten decimal number according to the following
expansion:
a 10n  a 10n1  ...  a 102  a 101  a 100 . a 101  b 102
n
n 1
2
1
0
1
2
 ...  b
m
m 10
Any decimal number can be converted into a real number and any
real number can be converted into a decimal number.
You should know the names and meanings of the base ten decimal
positions especially the ones to the right of the decimal.
Always question what is the given statement or given data of a
mathematics problem.
Remember there is a big difference between an expression and an
equation.
A solution is always an equation.
A simplification of an expression can be another expression or written
as an identity equation.
ROUNDING AND PLACE VALUES
BINARY NUMBERS OR NON BASE TEN
REPRESENTATIONS
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Let n and m be whole numbers.
Select a binary number or base two number according to
the following expansion:
an 2n  an 12n1  ...  a2 22  a121  a0 20 . b121  b2 22  ...  bm 2m
Computers and machines use base two, eight, or sixteen
expansions as these are powers of two so that the
dichotomous nature of electronic components can be
represented as on or off or one or zero.
LOGIC AND BOOLEAN ALGEBRA
Logic is a pure science and tries to replicate the unavoidable
directions of existence and events using categories.
Let p and q be free logical statements or variables.
Any logical statement p is called definite if p may be only true or
false and is a assigned a value of zero if it is false and a value of one
if it is true called its truth value.
A false statement is called a contradiction.
A true statement is called a tautology.
Given a definite logical statement we define its negation as the
statement with the opposite of the given statements truth value.
LOGICAL QUANTIFIERS
Conjunction or Union or Join
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p or q
written
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pV q
Disjunction or Intersection or Meet
p and q
written
pq
Values of logical statements involving quantifiers can be
listed in diagrams called truth tables.
If the statement q is implied by the existence of the statement p
we can say p implies q
or
if p then q
written
p
q .
If the statement p exists as result of the existence of the statement q
we can say p is implied by q
or
p only if q
written
p
q .
LOGICAL EQUIVALENCE
If p implies q and q implies p then we say
p if and only if q and write p  q and consider that p is the same
as q or that they are logically equivalent .
Logic and mathematics are inherently connected with philosophy.
Most statements of mathematics include a given statement, directions
to apply to the given statement and the result(s).
It is important to recognize the context and format of a given
statement so that directions can be applied to produce accurate
results.
De Morgan’s Laws for Logical Statements
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FRACTIONS
Any answer which if a fraction must always be reduced.
A fraction is reduced to lowest terms or reduced if the numerator and
denominator have no common factors.
If you are working with fractions make everything look like a fraction.
In equations we can clear out the fractions and with expressions we
may not be able to clear them out.
Given a fraction of the form
a
b
where b is not zero
a is called the
numerator and b is called the denominator.
We never divide by zero and if it is the case we say the expression is
undefined.
Given a fraction of the form
a
b
it means take a unit and divide it
into b number of equal parts and select from these a number of them.
A fraction is called proper if the numerator is less than the
denominator.
A fraction is called improper if the numerator is greater than
the denominator.
A fraction can be reduced and proper or improper at the same time.
If a fraction is improper it may also be reduced.
Reduced and proper and improper are not the same definitions.
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Any improper fraction can be written as a mixed number.
Cancellation Property
Let a, x, and b be free and b is not zero.
ax
bx

xa
bx
ax  a
xb b

Given a fraction of the form
a b
and
a
b
.
b a
Given a fraction of the form 

a
b

a
b

a
b
where b is not zero it also means
a
b
then
.
Distribute negative numbers across terms where they are indicated
by their position in front of parenthesis .
Given a fraction multiplied by a variable it may also be written or
calculated as numerator of fraction times variable divided by the
denominator of the fraction.
PROPERTIES OF FRACTIONS
Let a, x, c. d, and b be free and b and d are not zero.
a 
1
a
where
a 0
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a a
1
0 0
a
where
a 0
0 0
1
a
0

undefined
0
0

undefined
a c
b d

a c
b d

a d
b c
a c
b d

ad bc
bd
a c
b d

ad bc
bd
ac
bd

ad
bc
The formulas above do not require the least common denominator or
LCD to use them.
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In all cases a common denominator can be used which may be the
product of denominators given as easily as the least common
denominator to accomplish all tasks such as clearing fractions from
forms of equations and adding or subtracting fractions in expressions.
Least Common Denominator or LCD
DIVISION
THE REAL LINE
Every set has a well order.
As the collection of real numbers is an uncountably infinite set it has a
well order.
The ordering of the real numbers is shown on a real line when we fix
an origin as a point on the line identified with the real number zero
and all positive numbers on one side of this point developed from a
given unit and all negative numbers on the other side of this point
denoted by negative multiples of the given unit.
INFINITIES
LAWS OF SIGNS
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Please know your laws of sign very well as they are relatively easy to
learn and always used.
MULTIPLICATION AND DIVISION
Given an expression to multiply or divide two
real numbers if they have the same signs the
result is positive and if they have unlike signs
the result is negative.
ADDITION AND SUBTRACTION
Given an expression where no multiplication or
division is indicated only to add/subtract two real
numbers:
If they have the same signs combine them and
give the result this similar sign.
If they have unlike signs then take their
difference and give this result the sign of the
given number which is farther from zero on the
real line.
In this case one can think of owing as negative and having as
positive.
2+3=5
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2 - 3 = -1
-2 + 3 = 1
-2 - 3 = -5
VARIABLES
A variable is a letter , symbol , object, event, place, shape, color,
pattern , person, name or any combination or grouping of these that
represents one (fixed) or many (free) other
letter(s) , symbol (s) ,
object (s), event (s), pattern (s), person (s), number (s) , set (s),
category (ies) , group (s), path (s), time (s), taste (s) , odor (s), feeling
(s) , idea (s), statement (s), sound (s) , shape (s), color (s), state (s)
, unit (s) , movement (s) , observation (s) , place (s), touch (es), sight
(s), quantity (ies), TO WHICH IT OR THEY MAY OR MAY NOT BE
EQUIVALENT.
A variable is a symbolic name for an object or event so that it can be
called upon even if it has infinite properties or is difficult to describe.
A variable is a symbolic name that represents one object or event or
possibly many other objects or events from a set to which the variable
may or may not be equal or look the same.
As there exists numbers with a large amount of entries or infinite
number of non zero decimal positions or entries which therefore can
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not be drawn, listed, or written in finite time and/or finite area we must
use variables, names or symbols to express them.
A variable is either free or fixed (constant) .
A variable is chosen to be free or fixed (constant) with a let statement
from a certain standard observer.
A free variable is a symbol or group of symbols which can represent
many items of the list written above at all times.
As an example we say let x be a free real number so that in any place
of the symbol x where it might be located we could substitute a
number like 43.
A fixed (constant) variable is a symbol or group of symbols which only
represents one item of the list written above at every time.
Any letter from an alphabet or real number is a fixed variable or
constant.
The symbol 8 is a fixed variable because it only represents the
amount of 8 items.
ALGEBRA
Algebra is the predecessor of Abstract Algebra.
Abstract Algebra and Combinatorics are the predecessors of Number
Theory.
Many of the skills of Algebra are accomplished using the
properties of the numbers zero and one called algebraic
manipulations.
An expression is the finite sum, difference, product and/or
quotient of a finite number of free and/or fixed variables or
any single variable.
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Expressions represent a real number and always begin without an
equal symbol when given.
Each real number can express itself in an infinite number of ways
with the binary operations of adding, subtracting, multiplying, and/or
dividing.
All the following are expressions of the symbol to represent the
amount of 8 real objects:
2+2+2+2
(2)(2)(2)
(2)(4)
8-0
9-1
10-2
11-3
12-4
13-5…
8/1
16/2
32/4
64/8 …
TERMS AND FACTORS FOR
EXPRESSIONS
Given an expression its terms are separated by plus or minus
symbols and
the factors of its terms are separated by multiplications and if there is
no separation by plus or minus symbols the expression is a single
term.
Every factor of any term for an expression may have one or a finite
number of terms itself.
Given an expression like terms have exactly the same free variables
as factors raised to exactly the same exponents.
We combine like terms by adding/subtracting their coefficients.
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Simplified for an expression can mean many things.
A fraction is simplified if its numerator and denominator contain no
common factors.
An expression is simplified if it contains no negative exponents and
like terms are combined.
Coefficients, bases, and powers
FACTORED EXPRESSIONS
Given an expression it is factored if and only if it is written as a
product, with at least two factors, and as a single term.
If an expression is not factored we call it unfactored.
Any expression is either factored or unfactored when it is given and
never exists in a mixed state .
Any amount may be factored out of an expression but usually we
factor out the greatest common factor.
GCF
Given an expression with more than one term its greatest common
factor is the amount which is the largest or has the most prime factors
so that it can divide each term of the given expression evenly.
To factor an expression we use parenthesis and write the amount we
want to multiply or factor (take) out next to the expression that is a
multiple of the original given expression using parenthesis.
ORDER OF OPERATIONS
Every one uses the order of operations agreement
worldwide.
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Given an expression with no free variables evaluate it as follows:
Numerators and Denominators of fractions are evaluated separately .
1. View only the amount contained by the inner most
parenthesis, brackets, braces, under radicals, or inside
absolute values. (Those which are inner most are contained
by the most other sets of parenthesis, brackets, braces,
radicals, or absolute values.)
2. Evaluate exponents only in these inner most.
3. Multiply or divide as it goes from left to right only in these
inner most.
4. Add or subtract as it goes from left to right only in these
inner most.
5. Repeat for what parenthesis contains the inner most
amount from above until the expression is a single number
expressed as a single term and unfactored. If there is
nothing which contains the parenthesis or the steps yield a
redundancy in the inner most then view the entire statement
of the expression.
Parenthesis
Exponents
Multiply
or Divide ( as you view from left to right )
Add
or Subtract ( as you view from left to right )
P.E.M.D.A.S.
Parenthesis, brackets, and braces all mean the same thing which is
whatever amount is represented directly outside of and next to them
will be eventually be multiplied by what they contain.
If there is nothing represented directly outside of and next to them
then it is assumed to be the number one.
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PERCENTS
Percents and numbers are not the same things. Percents have the
percent symbol % and numbers do not.
Any real number can be converted to a percent and every percent
may be converted to a real number.
ALGEBRAIC SUBSTITTION
One of the most important skills in algebra is algebraic substitution.
Using it allows us to correctly substitute values into formulas and
accurately produce results as in economics, finance, and sciences.
FORMULA
INEQUALITY
An inequality is the statement that two expressions may not be equal.
Let a, b, and c be fixed real numbers.
Let x be a free real variable.
Given a real number a and another real number called b where a is
not equal to b we say that a is less than b if and only if a is to the left
of b on the real line and write a < b .
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2< 3
Given a real number a and another real number called b where a is
not equal to b we say that a is greater than b if and only if a is to the
right of b on the real line and write a > b .
-2 > -3
If the arrow points to the left ( < ) it says less than if it points to the
right
( > ) it says greater than. Given two unequal real numbers the arrow
points to the one which is leftmost on the real line.
If we want to say a real number c is positive we may state c > 0 .
If we want to say a real number c is negative we may state c < 0 .
If a < b then b > a .
Examples
2 < 3 so 3 > 2 .
-3 < -2 so -2 > -3 .
a < b
says a is less than b
.
a  b
says a is less than or equal to b
a > b
says a is greater than b
a  b
says a is greater than or equal to b
.
.
.
PROPERTIES OF INEQUALITY
If a < b then a + c < b + c .
If a < b then a - c < b - c .
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If a < b then a c < b c
and a c > b c
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if c > 0
if c < 0 .
If a < b and c is not zero
then
a/c < b/ c
if c > 0
and
a/c > b/ c
if c < 0
.
INEQUALITIES WITH SETS TO MAKE
INTERVALS ON THE REAL LINE
Intervals are infinite subsets of the real line.
Let a, b, and c be fixed real numbers.
Let x be a free real variable free.
x < b says x is less than b and we use parenthesis to represent
this inequality when graphing or writing intervals.
x  b says x is less than or equal to b and we use brackets to
represent this inequality when graphing or writing intervals.
x > b says x is greater than b and we use parenthesis to
represent this inequality when graphing or writing intervals.
x  b says x is greater than or equal to b and we use brackets
to represent this inequality when graphing or writing intervals.
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x < b and x > b
x  b and x  b
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use parenthesis ( ) for intervals.
use brackets [ ] for intervals.
We always use parenthesis next to the positive or negative infinity
symbols when we state intervals which involve them.
Using inequalities we may create sets called intervals as subsets of
the real line.
Solution
Interval
Set
Graph
ooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooo
x < a

 , a




x
:
x a




 , 3




x
:
x 3



Example
x < 3
ooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooo
x  a



(  , a ]
x
:



x



x a
Example
(  , 5 ]
x  -5
:
x  5



ooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooo
x > a

a , 




x
:
x a




2 , 




x
:
x 2



Example
x > 2
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ooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooo



[ a ,  )
x  a
x
:



x



x a
Example
[ 3 ,  )
x  -3
x  3
:



ooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooo
a< x < b

a , b




x
a x b
:



Example
-4 < x < 3

4 , 3




x
4  x  3
:



ooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooo
a< x  b
(a,b]



x
:
a x b



Example
-4 < x  3



( -4 , 3 ]
x
:
4  x  3



ooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooo
x < a or x > b



x
:
x  a or x  b

 , a
 b
, 




Example
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
x < 2 or x > 3



x
:
x  2 or x  3
 , 2
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, 
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



Examples
[-1, 5]  [0, 2] = [0, 2]
[-1, 5]  (0, 2) = (0, 2)
[-1, 5]  (0, 2] = (0, 2]
[-1, 5]  [0, 2) = [0, 2)
[-1, 5]  [0, 2] = [-1, 5]
[-1, 5]  (0, 2) = (0, 2)
[-1, 1]  [1, 3] =
1
[-1, 1)  (1, 3] = 
[-1, 4]  [2, 6] = [2, 4]
(-  , 5]  [-3,  ) = [-3, 5]
(-  , 4)  (4,  ) = 
(-  , 4]  [4,  ) =
4
(-  , 4]  [4,  ) = (-  ,  )
(-  , 8)  (2,  ) = (2, 8)
(-  , 8)  [2,  ) = [2, 8)
(-  , 8]  (2,  ) = (2, 8]
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ABSOLUTE VALUES AND
INEQUALITY OR EQUALITY
If
x
< a
then
-a < x < a .
If
x
> a
then
x > a or
If
x a
then
either
x a
x <-a
or
.
x  a
.
NON LINEAR INEQUALITIES
1.
2.
3.
4.
5.
6.
7.
8.
9.
Clear parenthesis by distributions.
Combine like terms on each side separately .
Do not clear fractions.
Move all terms to one side and zero on the other side.
Factor the non zero side and if it is a fraction factor the
numerator and the denominator.
Set each factor equal to zero and solve for roots.
make a chart using the roots of step 6 to create regions on the
real line.
choose a test value from one of the regions to substitute into
the original statement.
if it produces a true statement the region where it was selected
from is part of the solutions otherwise adjacent region(s) are
solutions.
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EQUALITY
Equality is called a relation.
An equation is the statement that two expressions are equal.
Equations always begin with the equal symbol showing and
expressions never begin with the equal symbol showing.
It is very important to recognize the difference between equations and
expression as their direction statements usually are different.
PROPERTIES OF EQUALITY
If a = b then a + c = b + c .
If a = b then a - c = b - c .
If a = b then
ac= bc.
If a = b and c is not zero
then
a/c = b/ c
.
If a = b then b = a .
THEOREM OF ALGEBRA
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Given any two non equal rational numbers a and b
then
a/b= q+r/b
where q is real called the quotient and r is real called
the remainder.
Example
5/3= 1 + 2/3
If the remainder is zero or r = 0 then we say the division is even and
b and q are factors of a
as a / b = q .
We have 12/4=3 because 12/4=3+0/4
as q=0 .
If a/b=c then a=bc and we say b and c are factors of a.
Given any real number x and any natural number n we say that nx is
a multiple of x.
Given x = 3 then its multiples are:
3
6
= (1)(3)
= (2)(3)
9
12
…
= (3)(3)
= (4)(3)
RATIOS AND PROPORTIONS
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A ratio is an expression of a fraction .
The ratio of a to b is denoted a/b or a : b .
A proportion is the statement or equation that two fractions or ratios
are equivalent.
Example
a/b =c/d
½ = 3/6 so (1)(6) = (3)(2)
If a/b =c/d then ad = bc .
PRIME NUMBERS
A natural number greater than one is prime if and only if it can be
written as a multiplication with the natural numbers one and itself and
this is the only way to do it.
Not every prime number is odd because two is prime.
Not every odd number is prime because nine is odd and not prime.
It is exceedingly hard to tell if a random large number is prime.
This is an open question of number theory.
19. UNITS, METRICS, AND
CONVERSIONS also standards of
Mathematics
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Units are arbitrary conventions created by people to quantify
properties from observations using scientific studies and
measurements such as physics, chemistry, and biology.
Use of the british system only promotes the imperialist machinations
of European and english conquerors in the Americas specifically in
the United States of America.
The metric system is a modern and almost worldwide standard used
in many countries such as Mexico and most countries overseas.
Every student should learn the metric system and know how to
convert between many different units and systems.
UNIVERSAL CONSTANTS
The amounts of pi and e are universal constants.
DEGREES AND RADIANS
DOMAINS OF EQUATIONS AND
EXPRESSIONS
Given an equation or expression with only the free variable x we
define its domain as the collection of all values which may be
substituted for x so that evaluation makes a statement which is
defined.
Defined means no division by zero and no negatives under even
indexed radicals.
The domain is usually all real numbers unless there is a variable in
the denominator of a fraction and/or a variable under a radical.
If there is a variable in the denominator of a fraction set this
expression of the denominator equal to zero and solve it.
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These solutions are not in the domain so that the domain in this case
is all reals except these solutions.
If there is a variable under a radical set the expression under the
radical greater than or equal to zero and solve such inequality.
These solutions are in the domain so that the domain is the set of
only these solutions.
RANGE
Given an equation with at most two free variables call them x and y
its range is defined as the collection or set of all numbers y which are
calculated through evaluation with the order of operations agreement
as values of x are substituted from the domain.
Given any collection of points in a plane its domain is the collection of
all x values of their coordinates and the range is the collection of all y
values of their coordinates.
ALGEBRAIC FORMS
OR FORMATS OF EXPRESSIONS
Much useful information can be derived from the algebraic forms of
equations and expressions.
Understanding and skill with application over the forms or formats of
equations and expressions allows us to manipulate abstractions in
our modern lives and business to our benefit such as finance, legality,
and productivity with technology and resources.
Given any expression we can ask about its given or written form is it
factored or unfactored at some time.
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Given any expression we can ask is it written as a proper fraction or
is it written as a whole number.
Given any expression we can ask how many terms does it have the
way it is being written or expressed.
Given an expression with a single term or any term of all other
expressions we can ask how many factors are written at a time and
what are the factors of terms?
EXPONENTS
There are many ways to work with exponents.
The hardest part of mathematics is making all the choices yet your
answer in the end should be equivalent to any one else’s.
Let x and y be free.
x x
1
0
undefined
0
x 1
0
when
x x  x
m
n
x 0
.
m n
xm n  xmn
x
y
x
n
m
n


y
x
m
n
1
x
n
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n
 x
 
 y
 y
 
 x
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n
RADICALS OR RATIONAL EXPONENTS
 x
n
m

x
m
n
SCIENTIFIC AND OTHER NOTATIONS
A number is written in scientific notation if it is written as a product of
a number between one and ten and an integer power of ten.
COMPLEX NUMBERS
Complex numbers are those amounts which we deal with on a basis
either in electronics, computers, graphics, engineering, and physics .
Let x be a free complex variable.
The set of real number is a subset of the set of complex
numbers.
Every real number is by definition also complex.
The set of complex numbers is not a subset of the set of real
numbers.
Every complex number is not necessarily also real.
Let
i
be an imaginary variable so that
i  1
2
.
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We define the set of complex numbers denoted and equal to the
following:
C




a bi : a,b  R and i  1 
2
.
PHYSICS
Physics is the most direct application of mathematics to our physical
observable universe.
Physics uses mathematics to understand relations between
observable and measurable phenomena.
VECTORS
A vector has direction and magnitude (force).
Vectors are represented by rays of finite length.
The direction of the vector represents the origination and position of
an applied force and its magnitude or length represents the strength
of the force in a given unit (usually Newton’s).
ai+bj
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THOUGHT EXPERIMENTS
A thought experiment is an event conceived in one’s mind to derive a
possible outcome which may not necessarily be done in practice and
which may be impossible in practice.
GEOMETRY
Geometry is one of the oldest areas of mathematics also Astronomy
is ancient and vastly created by the Egyptians, Aztecs, Native
Americans, Greeks, and Chinese. Modern day geometry is called
Topology.
Geometry is a very pure and elegant study.
Units of measurement for one dimensional objects are called lengths.
Length is measured using a given linear unit such as meters.
Units of measurement for two dimensional objects are called areas.
Area is measured using a given two dimensional square unit such as
square meters.
Units of measurement for three dimensional objects are called
volumes.
Volume is measured using a given three dimensional cubic unit such
as meters cubed .
Rays, arcs, points, lines, and segments
A point is a zero dimensional object.
Two rays intersecting only at their endpoints with an angle of 180
degrees between them is equivalent to a line.
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Rays mimic light expanding from a source.
Because of Einstein’s theory of General Relativity and the fact that
black holes exist we know that lines and planes are purely
mathematical objects as gravity can bend the rays of light and warp
the fabric of spaces.
TRIANGLES
The sum of interior angles of any triangle must equal 180 degrees.
CIRCLES
RECTANGLES
Every square is a rectangle yet not all rectangles are squares.
TRIGONOMETRY
Trigonometry is very useful for navigations, satellite orbits, and space
travel.
The properties of triangles are so numerous and useful that we call
their study and application trigonometry.
TRIGONOMETRIC FUNCTIONS
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Given the diagram above we define the trigonometric ratios as
written:
We have Y is the hypotenuse.
sin  
 B

opposite
hypotenuse
cos  
 G

adjacent
hypotenuse
tan  
  sin   
sec  

Y
Y
cos 
1
cot 
1
Y
cos   G



B
G

opposite
adjacent
hypotenuse
adjacent
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csc  

cot  
  cos   
1
Y
sin   B
sin 

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hypotenuse
opposite
1
tan 

G 
B
adjacent
opposite
Let x represent degrees and y is real then :
y = sin(x)
y = cos(x)
y = tan(x)
y = sec(x)
y = csc(x)
y = cot(x)
Polynomials
An expression is called polynomial if and only if the exponents of any
free variables are whole numbers or the exponents are not fractions
and/or negative numbers.
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An expression which is a polynomial and a single term is called a
monomial.
Given a monomial its degree is the exponent or power of any free
variable or the sum of the exponents for all free variables if there are
more than one.
Given a polynomial with more than one term its degree is the largest
degree of any of its terms.
FACTORING Polynomials
DIFFERENCE OF TWO PERFECT
SQUARES
2

x y
2
  x 

y  x  y 
The left hand side of the equation above is unfactored and the right
side is the factored format.
DIFFERENCE OR SUM OF TWO
PERFECT CUBES
x y
3
3
  x 

y  x  xy  y 
2
2
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x y
3
3
  x 

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y  x  xy  y 
2
2
The left hand side of the equations above are unfactored and the
right side is the factored format.
TRINOMIALS
FACTORING BY GROUPING
EQUATIONS AND EQUALITY
An equation is the statement that two expressions are equal.
Each equation has a left hand side and a right hand side.
All equations may be expressed using variables.
The number of equations is uncountably infinite.
Let a and b be free algebraic expressions then we have any equation
can be represented as the symbolic statement
a=b .
The solution set for an equation is a real number or a
collection of real numbers which can be substituted for the
free variable(s) so that evaluation of each side of the
equation through the order of operations yields a true
statement or tautology.
A math problem should have the two components of the
direction and the given statement.
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The direction of an equation is usually to solve and the
direction of an expression is usually to simplify.
Simplify has many meanings.
Simplify means:
13.Perform the indicated operations.
14.Evaluate.
15.Calculate.
16.Add, subtract, multiply or divide.
17.Combine like terms.
18.Distribute across parenthesis.
19.No answer should have negative exponents.
20.Any numeric base should be evaluated.
21.No radical of even index may have a negative
radicand.
22. No fraction can have a radical or i in its denominator.
23. Any answer that is a fraction must always be reduced
or cancelled.
24.
Categories of equations based on
their solution sets
An equation is conditional if it has a finite number of solutions.
An equation is a contradiction if it has no solutions.
It will produce a false statement like 0=2 when you try to solve it and
the answer is no solution.
An equation is an identity if it has an infinite number of solutions.
It will produce a true statement like 0=0 when you try to solve it and
the answer is all real numbers.
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The number of equations is uncountably infinite and some equations
cannot be expressed in finite space so we use variables to represent
these equations.
Given an equation it is called radical if there exists a free variable with an
exponent that is non integer.
Given an equation it is called rational if there exists a free variable with an
exponent that is a negative integer.
LINEAR EQUATIONS
An equation of the form
ax  b  0
where a is not zero, a and b
are fixed and x is free is called linear in one free variable .
An equation, expression, or inequality is linear with respect to a free
variable if and only if any exponent of a free variable is zero or one
and no exponent contains a free variable in its expression and the
equation is not transcendental otherwise it is called non linear.
STEPS FOR SOLVING LINEAR EQUATIONS
1. Clear fractions in an equation by multiplying both sides by
the least common denominator. or
2. Clear parenthesis by distributing.
3. Combine like terms on each side separately .
4. Decide if the equation is linear or non linear.
5. If it is linear make a choice to collect all terms with the
factor of x on one side and all other terms without the factor
of x on the other side and accomplish this using the
properties of equality.
6. Make sure the only factor and exponent of x is one so
that it says x = ‘a number’.
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NON LINEAR EQUATIONS
Quadratic Equations
ax  bx  c  0
2
An equation of the form
where a is not zero is
called non linear or quadratic.
Every quadratic equation is non linear. Every non linear equation is
not necessarily quadratic.
Quadratic equation implies it is non linear.
Non linear equation does not imply it is quadratic.
QUADRATIC FORMULA
Given an equation of the form
ax  bx  c  0
2
the solutions are
given by
b  b  4ac

x
2a
2
.
PRINCIPLE OF ZERO PRODUCTS
If the product of a finite number of factors is zero then any one of the
factors could be zero.
If ab=0 then either a=0 or b=0 .
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STEPS FOR SOLVING NON LINEAR
EQUATIONS
1. Clear fractions in an equation by multiplying both sides by
the least common denominator. or
2. Clear parenthesis by distributing.
3. Combine like terms on each side separately .
4. Decide if the equation is linear or non linear.
5. If it is non linear make a choice to collect all terms on one
side and zero on the other side and accomplish this.
6. Factor the non zero side.
7. Set each factor to zero and solve these as linear
equations.
8. If the non zero side will not factor and if the equation is
quadratic then use the quadratic formula.
STEPS TO SOLVE WORD PROBLEMS
14.
Read the problem all the way through without
stopping for totality before you begin to try to solve it.
15.
Make a basic comprehension of the problem in
real world terms so that you can reason a solution.
16.
Draw any related pictures, diagrams, or figures if
possible. Start to sketch out the problem.
17.
List the given relevant or essential data on your
paper with units and determine what information may
not be relevant.
18.
List any implied or given formulas and or
equations on your paper.
19.
Make sure all units are uniform.
20.
Make a let statement to define the unknown or
requested amount(s) as a variable and derive other
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unknown amount as expressions using this variable
where required.
21.
Create an equation which is equivalent to the word
problem in Mathematics from our case in English.
22.
Make sure the equation is of only one variable
type so it can be solved. This may require substitutions.
23.
Solve it.
24.
Consider that all requested amounts are exhibited
by re- reading the problem and that you have shown all
required answers. Otherwise you may have to
calculate more results. This may require you to re read
the problem statement again.
25.
Include units for answers where they are required.
26.
Check your answer(s) and make sure it (they)
appear to make sense to you.
14. Organize or arrange your results in a presentable
manner.
Word problems can be stated in an unclear or ambiguous way so if
this is the case try one interpretation and if it does not produce good
results try another interpretation to produce logical and complete
results. Never give up.
THE REAL PLANE
The real plane is defined as a set of ordered pairs representing its
points as coordinates denoted and equal to the following
R R  R
R
R
2
2
2

   x , y  :

 

x, y  R 

.
is called the Euclidean two space or the real plane.
is the Cartesian product of two sets of real numbers.
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The scheme to construct the ordering of a plane was first conceived
by Rene DesCartes.
A plane when it is considered is a purely mathematical object so that
it requires our imagination and symbols to conceive the totality of the
concept.
Given an ordered pair ( x , y ) we may plot it or graph it by starting at
the origin of a plane and going x units right if x is positive or x units
left if x is negative then from that place y units up if y is positive or y
units down if y is negative and then making a mark or dot.
PLANAR RELATIONS
OR GRAPHS OF EQUATIONS
There is a huge relationship through the definition of solution sets for
equations between algebra and geometry.
Given an equation which uses at most two free variables called x and
y we define its graph or solution set as the collection of all ordered
pairs or points in the plane ( s , t ) so that substitution of s for x and t
for y is evaluated to produce a true statement.
The solution set of an equation is also its graph.
CIRCLES
LINES
Two distinct points are all that is required to create a line.
An equation of the form Ax + By = C where A, B, and C are fixed and
A and B are not both zero with x and y are free is called linear with
respect to x and y.
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Given an equation of the form Ax + By = C we define its solution set
as the collection of all (s,t) or points is a plane so that A(s) +B(t) =C
is a true statement.
Horizontal lines are flat straight across and vertical lines go straight
up and down.
SLOPE VALUES OF NON VERTICAL LINES
Every line has a slope value associated with it which is a number that
is either positive, negative, zero, or undefined.
The slope value of a line tells us about the geometry of its graph like
how much the line is slanted and in what direction it is slanted, no
slant (flat), or a vertical line.
Given two distinct points in a plane
x , y 
1
1
and
x , y 
2
2
we define
the slope value of the line that passes through them denoted and
equal to the following :
m

y y
x x
2
1
2
1
.
FINDING EQUATIONS OF LINES
Coordinates of a point in a plane and a slope value are required to
produce an equation of a line.
y = mx+b
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LINEAR APPLICATIONS
Linear regression and Least Squares Fit of data
Given at least two distinct points or a finite collection of distinct points
in a plane we define their Least Squares Fit to be the equation of the
line which is a minimum distance from each point.
To find the Least Squares fit of a finite collection of points we usually
input their coordinates into a calculator or computer program.
Given two distinct points in a plane their least squares fit is the line
that passes through them.
STEPS FOR GRAPHING LINES IN A
PLANE
7. Plot points using a chart with x then y or usually an alphabetical
order.
8. Find the x intercept by substituting y = 0 into the given equation
if any exist.
9. Find the y intercept by substituting x = 0 into the given equation
if any exist.
10.
Choose a value for x or for y (not both at once) and
substitute it into the given equation and solve for the other
value and list these as a point x value then y value.
11.
Create and label x and y axis and scale units
appropriately .
12.
Plot these points and draw a line between them using
arrows at the ends.
FUNCTIONS
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An equation which uses at most two free variables called x and y is a
function if and only if for any substitution of a value of x into the
equation evaluation will yield only one value for y.
Given any two dimensional graph or collection of points in a plane it is
a function if there does not exist a vertical line which intersects it or
them in more than one place.
Any singleton point in a plane is a function.
Any non vertical line is a function.
Any circle of positive radius in not a function.
Given an equation with at most x and y are free then if y is a function
of x and we say f (x) = y where f is identified as the name of the
equation .
f (x) is called function notation and produces names and ordered
pairs for complicated equations.
OPERATIONS ON FUNCTIONS AND
THEIR COMPOSITIONS
Given functions f(x) and g(x) we may define new functions called their
sum, difference, product, and quotient.
(f+g)(x)= f(x) + g(x)
(f-g)(x)= f(x) - g(x)
(fg)(x)= f(x)g(x)
(f/g)(x)= f(x)/g(x)
where g(x) is not zero.
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Given functions f(x) and g(x) we may define new function called their
composition written and equal to :
 f  g  x  f g x
If
 f  g  x   x
then we say
f
and
.
and
g
g  f  x   x
are inverses of each other and write
g

f
1
.
VERTICAL AND HORIZONTAL SHIFTS
AND TRANSLATIONS OF FUNCTIONS
Given a function y = f(x) and a real number c we have the following:
1.
y = -f(x)
is a reflection of f(x) about the x axis.
2.
y = f(-x)
is a reflection of f(x) about the y axis.
3.
y = f(x) + c is a shift of f(x) c units up if c is positive and down
if c is negative.
4. y = f(x+c)
c is negative.
is a shift of f(x) c units left if c is positive and right if
RATIONAL FUNCTIONS
A function y = f(x) is called rational if f(x) is or can be written as one
polynomial divided by another non zero polynomial.
PARABOLAS
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LOGARITHMIC FUNCTIONS
Let x and y be free and b is a non zero real number.
An equation with at most only the two free variables say x and y is
called logarithmic if and only if y is an exponent and x is not.
x b
y
Logarithmic notation is a way to express such equations.
If
log x  y
b
then
b
y

x
.
EXPONENTIAL FUNCTIONS
An equation with at most only the two free variables say x and y is
called exponential if and only if x is an exponent and y is not.
y b
x
LOGARITHMIC AND EXPONENTIAL
APPLICATIONS
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ECONOMICS
Economics is the scientific analysis of business, goods, services,
capital, and markets using Real Analysis, Game Theory, and Graph
Theory.
All decimal numbers which represent dollar amounts should always
be rounded to the nearest penny or .01 (hundredths) .
Be able to use currency exchange rates.
The units in economics are usually real and numeric.
One economic unit which uses imaginary quantities is electricity.
An economic product or output is fractionalizable if it may be divided
into parts and will not loose its character (identity), structure, and
usefulness.
An example of a product which is fractionalizable (or not discrete) is a
liter of ice cream because there does exist a product or out put unit
marketable as ½ of the liter.
An economic product or output is unfractionalizable (or discrete) if it
may not be divided into any number of parts because it will loose its
character (identity), structure, or usefulness.
An example of a product which is unfractionalizable or discrete is a
television because there does not exist a product or output unit viable
as ½ a television.
Almost all economic products are either fractionalizable or
unfractionalizable.
Economic products can mostly be considered as elements of finite
product spaces of raw goods and services.
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Any given unit of every quantity, variable, economic service,
or product (output) called a quantitative unit is pure if and
only if it is never written or equal to a sum and/or mixture
resulting from any procedure of combining proportional
amounts of other distinct quantitative units or free variables
where each proportion of these quantities is distinguishable
from all others for almost every standard observer at all
times before the quantitative unit exists.
A given unit of every quantity, variable, economic service, or
product (output) is not pure called partitive or stochastic if it
is always written or equal to a sum and/or mixture resulting
from a procedure of combining proportional amounts of at
least two quantitative units or free variables where each
proportion of these quantities is distinguishable to almost
every standard observer at all times.
A caterer has the following price structure for banquets. The first 23
meals are charged the basic price per meal. The next 22 meals are
discounted by $3 each and all additional meals are each reduced by
$4. If the total cost for 80 meals comes to $800, what is the basic
price per meal?
Suppose the price of Crest toothpaste dropped from $1.50 to $1.00
when the tube size was reduced from 100ml to 70ml . Calculate the
percent change in tube volume, tube price, and unit price.
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Given an initial value of $4200 and a final value of $3200 with an
income yield of 5% find the income, Capital gain yield, and rate of
total return for the investment. Calculate yields and rates of return
to the nearest .01%.
Given an income of $11000 and an income yield of 8% with a Capital
gain yield of 20% find the final and initial values, and the rate of total
return for the investment. Calculate yields and rates of return to the
nearest .01%.
The Oxen Farm has 450 acres of land allotted for raising corn and
wheat. The cost to cultivate corn is $44 per acre. The cost to
cultivate wheat is $33 per acre. The owners have $16,700 available
to cultivate these crops. How many acres of each crop should the
owners plant?
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SUPPLY AND DEMAND
At a price per unit of $ 55 for a VCR consumers demand 100 VCRs
and at a price per unit of $95 consumers demand 50 VCRs in a
given market. Find the equation of the line that represents the least
squares fit where x is the price per unit and y is the quantity of units
demanded by the market and graph it.
At a price per unit of $ 45 for a DVD a company wants to sell 30
DVDs and at a price per unit of $100 they want to sell 150 DVDs in a
given market. Find the equation of the line that represents the least
squares fit where x is the price per unit and y is the quantity of units
supplied by the market and graph it.
PROFIT
Revenue less costs equals profit.
P=R–C
INTEREST AND COMPOUND INTEREST
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The simple interest I produced by an investment called a principle P
into an account with a simple rate or interest called r for an amount of
time in years t is given by
I
 Pr t
.
The monetary amount in account A with simple interest I produced by
an initial investment principle quantity P into the account with interest
rate r for an amount of time in years t is given by
A  P  Pr t .
Find the amount of an investment for a principle of $8800
compounded quarterly after 11.5 years with a rate of interest 5%.
Find the amount of an investment for a principle of $8000
compounded monthly after 10.5 years with a rate of interest 5%.
Find the amount of an investment for a principle of $3000
compounded continuously at the rate of 3% after four years.
Find the amount of an investment for a principle of $4000 with simple
interest rate of 5.2% after three months.
ANNUITIES AND AMORTIZATIONS
An employee savings plan allows any employee to deposit $25 at
the end of each month into a savings account earning 6% annual
interest compounded monthly. Find the future value of this
savings plan if an employee makes the deposits for ten years.
ACCOUNTING
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FINANCE
STOCKMARKETS AND BANKING
GAME THEORY
CHEMISTRY
BIOLOGY
Calculus is directly applied in rates of change for Biology.
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MAXIMA AND MINIMA OF GRAPHS
INCREASING AND DECREASING
GRAPHS
GRAPHING EQUATIONS IN THE PLANE
7. Plot points using a chart with x then y or usually an
alphabetical order.
8. Find the x intercepts by substituting y = 0 into the given
equation if any exist.
9. Find the y intercepts by substituting x = 0 into the given
equation if any exist.
10.
Find the vertical asymptotes from the domain of the given
equation.
11.
Find the horizontal asymptotes .
12.
Create and label x and y axis and scale units
appropriately .
6. Sketch the graph using intercepts and asymptotes.
CALCULUS
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Calculus is the study of real limits, areas, equations, infinity, and
relationships between rates of change in events and real numbers.
The predecessor of Calculus is called Analysis which can be real,
complex, or numeric.
LIMITS
INTEGRALS
CONTINUITY
A function
y  f (x)
is continuous at
a
in the domain of
f
if and only if :
1.
2.
f (a) exists
lim f ( x) exists
x a
3.
lim f ( x)  f (a)
x a
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REAL THREE SPACE
The real three space is defined as a set of all ordered triples
representing its points as coordinates denoted and equal to the
following
R R R  R
3

   x , y , z  :

 

x, y, z  R 

.
3
is called the Euclidean three space or the real three space.
3
is the Cartesian product of three sets of real numbers.
R
R
Three distinct points are all which is required to create a plane.
EQUATIONS OF PLANES
An equation of the form
Ax  By  Cz  D
is called linear in three free variables and represents a unique plane
in R3 .
SYSTEMS OF LINEAR EQUATIONS
ax  by  c
dx  ey  f
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SYSTEMS OF LINEAR INEQUALITIES
ax  by  c
dx  ey  f
SYSTEMS OF EQUATIONS
SYSTEMS OF NON LINEAR EQUATIONS
REAL (EUCLIDEAN) N SPACE
The real n space where n is a natural number is defined as a set of all
ordered n tuples representing its points as coordinates denoted and
equal to the following
R  R  R  R  ...  R  R
n

 x , x ,..., x  : x  R for all i  1,2,3,...,n  .
1
2
n
i
R
n
is called the Euclidean n space or the real n space.
R
n
is the Cartesian product of n sets of real numbers.
SEQUENCES AND SERIES
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A sequence is a list of numbers or elements in order from left to right.
a , a , a ,...
1
2
3
Any element of a sequence is a real number represented by ai
called the ith term or element of the sequence where i=1,2,3,…
A sequence is constant if all the elements are equal.
A sequence is finite if the list is finite
a1, a2, a3 ,..., an with n is natural.
A sequence is called infinite if the list or number of its elements is
infinite
a , a , a ,...
1
2
3
We can think of a sequence as a function from the natural numbers to
the real numbers.
Sequence Notations
ARITHMETIC SEQUENCES
GEOMETRIC SEQUENCES
SERIES
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MATRICES
Matrices may be applied to any case of linear expressions or
equations with a finite number of free variables. Matrices are used in
logistical situations, economics, probabilities, and aviation.
A matrix is a rectangular array of entries.
Let A and M be a free matrix theoretic variable.
Let M mxn be the collection of all matrices with m number of columns
and n number of rows where m and n are natural numbers.
We say A is an m by n matrix written m x n with
m is the number of rows and n is the number of columns for the
matrix A.
A matrix is square if the number of columns equals the number of
rows or m=n.
1  0.5
2
1


A 0 7

9  4
1
0

I 4 0

0
3 1
6 8
1 0

0 1
0 0 0
1 0 0
0 1 0

0 0 1
is an example of a 4 x 4 matrix which is square.
is called the 4 x 4 identity matrix.
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Matrix Notations and Operations
DISCRETE MATHEMATICS
NUMBER THEORY
Number theory employs prime numbers to create ciphers for
encoding and encryption.
GRAPH THEORY
Economics
t(7/10/08)
Introduction
Financial institutions such as banks should be careful when rounding
[f1] monetary units such as the dollar. Rounding can be used either
to the benefit or detriment of a banks and/or clients. Rounding is
necessary for monetary units and most units that are finite. Rounding
of currency units where it is required in formulae applications ( such
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as computation of interest of a given principle) can produce imaginary
[i] or fictitious fraction unit parts or increments or deficits for any given
individual currency unit such as the dollar unit.
Manipulation of interest rates and compounding [f] should be strictly
controlled by the federal government to decrease the possibilities of
abuse by financial institutions and corporations against the average
person and their families.
Formulae of finance, economics, or accounting where lending is
concerned and in every other instance of their applications to the sets
of monetary units are discontinuous [top] over the set of real numbers
as no unit of money or currency can have every real increment of all
its fraction parts in any mint’s [m] production for all countries or
organizations over any time except in imagination or theory [t].
Events in U.S. and world economies is the demonstration of the
difference in theoretical accounting and empirical accounting
reflected in the differences between theoretical probability and
empirical statistics.
Probability
Introduction
Let N be the set of natural numbers.
Given a set A let A be its cardinality [s].
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Let
U,U ,U
1
2
3
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be sets and n is a natural number then we
,...,U n
can define a new set called their product as
n
U  U U U
i 1
i
1
2
3
 ... U n 
  x , x ,..., x  : x U
1
2
n
i
i
for all i  1, 2,3,..., n

.
Given an event or experiment e that can be considered as a
finite number of steps where at each step there are any
number of choices or options define its sample space S as
the collection of all outcomes of event e.
Given a set A we define its power set W(A) as the set of all subsets
from A.
Given an event or experiment e where it can be considered
as a finite number of steps n with n is a natural number and
where at each step there is a set of choices or options.
Let Ai be the collection of all options for the ith step of
experiment e and define the sample space S of e as
n
S
  Ai 
i 1
  x , x ,..., x  : x  A
1
2
n
i
i
for all i  1, 2,3,..., n

.
Note that in the definition above the number of elements of
any Ai may not be finite.
Fundamental Principle of Counting
Given an event or experiment e where it can be considered
as a finite number of steps where at each step there are a
finite number of choices or options then the total number of
outcomes of experiment e is equal to the product of the
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number of choices for each step in e to complete the
occurrence of event e.
[e]
Given an event or experiment e that can be considered as a
finite number of steps n and assume at each step there are a
finite number of choices or options define its sample space S
as the collection of all outcomes of event e.
Let Ai be the collection of all options for the ith step of
experiment e and assume it is a finite set for any i.
Define the sample space S of e as
n
S
  Ai 
i 1
  x , x ,..., x  : x  A
1
2
n
i
i
for all i  1, 2,3,..., n
 and note the
cardinality of S follows by the Fundamental Principle of Counting if
the order of the steps in experiment e matter.
Theorem of determined cardinalities for
finite experimentation’s sample space under
permutation
Given
n
S  A
i 1
i
and if the order of the steps in experiment e matters
n
in the production of its outcomes then
S

i 1
A
i
.
Theorem of determined cardinalities for
finite experimentation’s sample space under
combinations
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n
Given
  Ai and if the order of the steps in experiment e does not
S
i 1
matter in some cases for the production of e’s outcomes then
n
S

A
i 1
i
.
Given a sample space S
n
  Ai let
i 1
T
S
be its tree diagram or
lattice [L] given by the ordinate relationship by subscription with the
relation of inequality applied to consecutive tuple elements for all
n
x  A
i 1
i
.
Let x  S then x is an outcome of experiment e with the
Cartesian product set S as its basis [b].
x
represents a unique path or branch in the tree diagram
T
S
if and
only if x is considered as a permutation of tuple entries on the product
n
A
i 1
i
and the order of steps in e has meaning in the outcome that x
represents in experiment e.
x
represents the number of combinations on its ordinate tuple
values on the product
n
A
i 1
i
in the tree diagram
occurrence for the tuple entries in
for experiment e.
x
T
S
if the order of
has no meaning in its outcome
Or
x
represents a unique path or branch in the tree diagram
T
S
if and
only if x as an outcome of experiment e is a unique.
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T
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does not represent a unique path or branch in the tree diagram
if x as an outcome of experiment e is not a unique.
S
n
  Ai let
Given a sample space S
E S
i 1
E
be an event so that
.
represents the collection of all outcomes of experiment e given
some specific condition(s) on outcomes of e that may or may not be
equal to all possible outcomes of experiment e with its given
generalized initial conditions represented by S .
E
n
  Ai let
Given a sample space S
E S
i 1
E
then we define the probability [p] of
E
P E  
S
be an event so that
denoted and equal to
E
.
n
  Ai let
Given a sample space S
E be an event so that
E  S then there exist sets E , E , E ,..., E so that E  A
all i=1,2,3,…,n so that E  E and E   E .
i 1
1
2
3
i
for
n
n
i 1
Given a sample space S
i
n
i
i 1
n
  Ai let
i 1
E
and
i
F
be events so
E , F  S then define the probability of occurrence of event E
F denoted and equal to
P  E  F   P  E   P  F / E   P  F   P  E / F  where
that
and
P  F / E  is the conditional probability of the occurrence for event
F given that event E has occurred and P  E / F  is the
conditional probability of the occurrence for event E given that
event F has occurred [h].
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Probability is the use of combinatorics, set theory, and lattice theory
and can also be considered with subsets of the Euclidean N Spaces.
Probability is a young area of mathematics and has existed only for
the last 200 years or so at this point and therefore many text books
vary greatly with definitions.
Probability has its most accurate applications in quantum theoretical
physics and is used in actuarial sciences, economics, game theory,
numerical analysis, and chemistry.
Probability produces a source of understanding random events which
may not be comprehended in their totalities.
Experiments and Sample Spaces
The sample space of any experiment for probability is a set of real
numbers or non real elements or a set of n tuples from a product
space of sets which may or may not contain real numbers and any
have other unreal elements called a data set or sample space .
Events are cases of outcomes for given experiments.
Probability uses set theory, ratios, and the fundamental principles of
counting (combinatorics) to describe chances or likelihoods for given
events or experiments in theory.
Probability uses thought experimentation or creative imagination.
Any probability of an event for a given experiment is the ratio of
number of ways the event may occur as an outcome from the
experiment to the total number of ways the experiment itself may
conclude.
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FUNDAMENTAL PRINCIPLE OF
COUNTING
Given any event or experiment which can occur as a series
of steps or stages where at every step there is a finite
number of choices to proceed to the next step then the total
number of outcomes or conclusions to the event or
experiment is the product of the number of choices which
can be taken at each step or stage to conclude the event or
experiment.
Permutations and Combinations
A permutation is a list of elements where the order from left to right
makes a difference and the number and type of elements listed does
matter.
A combination is a list of elements where the order from left to right
makes no difference and the number and type of elements listed
does matter.
The number of permutations is always greater than or equal to the
number of combinations for the same given elements.
When creating the sample space of a given experiment it is best to
consider does the order of choices when considering outcomes make
a difference or no difference and is there replacement or no
replacement when making choices for steps or stages to derive
outcomes (that is can the same choice be made more than once at
any step (replacement) or only once ( no replacement) at any step).
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Empirical vs. Theoretical
Theoretical Probability
We assume for any given set A that all equivalent forms of any
element x of A are only counted once or considered as equivalent
when we list or write the elements of A.
Empirical Probability or Statistics
We assume for any given set A that all equivalent forms and
quantities of any element x of A are all listed and counted each time
they occur in A when we consider and write the elements of A. The
set A is called a data set and is a subset of the real numbers or some
product space of real numbers.
INCLUSIVE AND EXCLUSIVE EVENTS
Two events E and F are mutually inclusive if their set theoretic
intersection is not empty.
Two events E and F are mutually exclusive if their set theoretic
intersection is empty.
CONDITIONAL PROBABILITIES
Statistics
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Statistics is the empirical application of theories from probability.
Statistics is one of the youngest areas of mathematics and has
existed only for the last 70 years or so at this point and therefore
many text books vary greatly with definitions and examples are
usually numeric and complicated in nature.
It is useful to know how statistics are used, manipulated, and
interpreted .
It is also useful to consider who is creating and using
statistics as they can create bias in the experiments, results,
interpretations, and applications.
The sample space of any experiment for statistics is always a set of
real numbers or a set of n tuples from a product space of real
numbers called a data set.
The outcomes from an experiment of statistics is called the data set
S.
The mean of a data set is the average of its elements or that is add
all elements and divide by the number of elements.
The mode of a data set S is the most frequently occurring element(s)
or does not exist if there is no most frequently occurring element of
the given data set S.
A data set may not have a mode or may have one or a finite number
of modes.
The median of a data set S with an odd number of elements is the
middle element when the elements of S are listed from left to right in
an ordering.
If the number of elements of a data set S is even then its median is
the average of the middle two elements where all elements of S are
listed in order from left to right.
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It is interesting to note that when the number of elements of a data
set S is even that its median may not be included in the data set S
itself.
If the number of elements of a data set is odd its median is always
included in the data set.
Nomality
Given a sample space S
  let
E be an event so that
E  S then we define the Nomality of E denoted and equal to
and if E  S define that N  E   1
N  E   1  1
S E S/E
.
Implying that the empty set is with non zero nomaility at every event
all the time and that any event of one less cardinality than the sample
space is certain in propensity at some times.
Empty set is a subset of every set and the possibility that an event
may not occur or become completed is possible almost always.
Nomality of an event is the reciprocal of the cardinality for the event’s
complement in a finite sample space.
Theorem
Given a sample space
then 0  N
 E   12
S
or
  let
E be an event so that E
N  E  1 .
S
Theorem
Given a sample space S   let E be an event so that E  S
E  1 , E  S 1 or E   then N  E   P  E  .
if
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Theorem
Given a sample space S   let E be an event so that
then 0  N  E   N  S / E   2 .
E S
Theorem
Given a sample space S   let E be an event so that E  S
E   or E  S 1 then 0  N  E   N  S / E   1 .
if
Theorem
Given a sample space S   let E be an event so that
then N  E   N  S / E   1 if and only if
E
 2 and
S
E S
4 .
Theorem
Given a sample space
then
N E   1
S
S
  let
E
be an event so that   E  S
0 .
Given a non empty sample space S   in existence implies
an event or experiment that happens as a finite number of
steps where at every step there is a finite number of choices
or the case of an outcome or no outcome observed over
finite time by given standard observer(s) and conditions.
Nomality suggests the null event (that no step or choice is
made at some time or no observation is confirmed nor
denied) is not unlikely always at any event’s conception,
construction or consideration over future time or in the past
for all conditions.
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Given a sample space
S
  let
S. Zoch
E
be an event so that   E  S
then nomality and probability are undefined or
N E   1
0
c. t( x )2012
P E  
0
0
and
.
Probability suggests that no outcome is produced at some
time or the null event is never likely at any event’s
consideration where outcomes are generated or observed at
any times by every observer and conditions.
Given a sample space
then
0
P E  
S
S
  let
E
be an event so that   E  S
0 .
Theorem
Given a sample space S   let E be an event so that
then N  E   N  S / E   2 if and only if
E  1 and S
E S
2 .
Nomality is a much better measure of propensity for events
where specified from the generalization of the event’s
conditions.
Nomality does not disagree with probability except in exact
real number values and specifically only in the case of an
event with one less cardinality than the sample space or the
null event.
Nomality still resides in a theory of fairness and equal
likelihood that is not always evident in the real world or the
games and experiments of men.
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Theory of probability must account for ratios that are a priori,
manipulated, incalculable or immeasurable.
Much information is gained by comparing the probability and
nomality of events.
Theorem
Given a sample space S   let
then lim N  E   0 .
E
be an event so that
E S
E
be an event so that
E S
S / E 
Theorem
Given a sample space
then
lim N  E   
S
  let
.
S / E 0
Formality
Given a sample space S let E1, E 2, E 3,..., E n be events so
that
E i  S for all i=1,2,3,…,n and
S
n
E  E   for
that F    0 and
let F  E  be a real number

i 1
any i  j then define for all i=1,2,3,…,n
F  E i   0 if E i   or if E i  
E
i
with
i
j
i
0  F  E   1 . Define the formality of S as the positive
real number S   F  E  or   0 .
so that
i
n
F
i
i 1
F
Theorem
Given a sample space
S
then
0 S
F

S
.
Formality of Probability theorem
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Given a finite sample space S
events so that

let
E i  S for all i=1,2,3,…,n and
c. t( x )2012
E , E , E ,..., E
1
S
2

3
n
be
n
i 1
E
i
with
E  E   for any i  j then define for all i=1,2,3,…,n that
F   0 and
F  E   0 if E   or if E   let F  E   P  E 
S   F  E   1 or   0 .
i
j
i
i
i
i
i
n
F
i 1
i
F
Note that the theorem of formal probability stated above implies that
S represents the equally likely outcomes of an experiment or event.
n
In reality every one knows that
S
F
 F
i 1
 E  1
i
when actual
observations of outcomes for most experiments are produced and
observed outside of theory and in most laboratories or conditions.
Formality suggests the null event (that no step or choice is
made at some time or no observation is confirmed nor
denied) is unlikely always at any event’s conception,
construction or consideration over future time or in the past
for all conditions.
If S   then S F  0 implies there is a non event that will not occur
for any propensity to observe outcomes.
Theorem of Real Probability
Formality of any finite or empty sample space
assignments
subsets.
E
of
S
S
is free from all
into mutually exclusive events or disjoint
Axiom of Fairness
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Where any result is equally likely to be observed as only one
outcome of an experiment by any observer every time at all
conditions the event or experiment producing this result is called fair
and the outcome or a trial is called singular.
It is interesting to consider the limits of nomality and probability as the
number of trials of an experiment goes to infinity or S   for a
singular outcome or otherwise.
example
Consider the case of rolling a fair six sided die. We assume it is fair
so that there is only one chance that any side may end up over
another.
It may also be assumed that a die can be constructed with six sides
so that only one side is always likely at any roll for anybody
anywhere.
example
Lottery can be fixed so that the outcome is known at every draw by
cheating.
example
S  1 , 2 , 3 , 4  with E  2 , 4  then N  E   P  E  and
N  E   N  S / E  1 .
Let
S  1 , 2 , 3 , 4  implies an experiment such as:
Roll a fair four sided die with 1,2,3,4 on the sides respectively where
any side is equally likely to be observed as only one outcome of any
roll of the die by any observer every time. The nomality and
probability the roll is an even number are equivalent in this case.
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Primal Statute
Given a real number greater than five called x then the
probability it is a prime number is less than or equal to 2/5
and the probability that it is not a prime number is greater
than or equal to 3/5 with respect to base ten expression of x
in finite time and arbitrarily large values of x.
Proof
Let N= 0,1,2,…,9 and B=1,3, 7, 9 and B’=0,2,4,5,6,8
Then P(B)=4/10=2/5<3/5=6/10=P(B’).
Prime Propensity
Given a natural number x written as base n expansion with n
is a natural number greater than two then as n increases the
probability of determining if x is a prime number in finite time
with respect to the given base of n decreases for arbitrarily
large values of x.
Prime Base Ten Limit
Given a natural number x greater than nine expressed as
base ten then if its ones position is not one, three , seven, or
nine then it is not a prime number.
Proof
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Rules of divisibility by counter example.
Theorem three end prime
Given a natural number x then if its ones digit base ten is a
three then x is a prime number or a product of three and a
prime number with its ones digit is a one.
Theorem nine end prime
Given a natural number x then if its ones digit base ten is a
nine then x is a prime number or a product of three and a
prime number with its ones digit is a three.
Theorem seven end prime
Given a natural number x then if its ones digit base ten is a
seven then x is a prime number or a product of three and a
prime number with its ones digit is nine.
Theorem one end prime ( other)
Given a natural number x then if its ones digit base ten is a
one then x is a prime number or a product of three and a
prime number with its ones digit is seven.
Three prime divisibility t1
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Given a natural number x written base ten then if it has its
ones digit is one, three, seven, or nine then x is divisible by
three or x is a prime number.
Theorem of prime governance t2
Given a natural number x if its ones digit base ten is one,
three, seven, or nine then x is a prime number or the product
of three and a prime number with its ones digit base ten is
either one, three, seven, or nine.
Theorem ones end
Given a natural number x not equal to one if its ones position
is a one and the sum of its digits is not divisible by three then
it is a prime number or it is the nth power of a prime number
with n is a natural number.
Definition epsilon
Given a natural number n then define the nth prime ordant
as the collection denoted and equal to
w
n

 p : p is a
prime number with p  x y and x, yn  .
Definition row
Given a natural number n then w is the number of prime
numbers so that x+y=p and p is a prime number and x and y
are less than or equal to n.
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The number w is the cardinality of wn .
Definition phi
Given a natural number n then s is the number of ways a
prime number p can be expressed so that x+y=p and x and
y are less than or equal to n.
Example
If n=6 then wn  2,3,5,7,11 , s=8 and
w=5 .
Example
If n=5 then wn  2,3,5,7 , s=6 and
w=4 .
Question 110
Given a natural number n then what is the number of ways s
a prime number p can be expressed so that x+y=p and x
and y are less than or equal to n?
Question 111
Given a natural number n then what is the number of prime
numbers w so that x+y=p and p is a prime number and x and
y are less than or equal to n?
Question 112
Given a natural number n what is the probability of selecting
two natural numbers less than or equal to n so that their sum
is a prime number?
Example
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Select two natural numbers less than or equal to 6. What is the
probability that the sum of the numbers is a prime number?
If n=6 than the probability of selecting two natural numbers
less than or equal to six that will add to a prime number is
8/21 .
Example
Select two natural numbers less than or equal to 5. What is the
probability that the sum of the numbers is a prime number?
If n=5 than the probability of selecting two natural numbers
less than or equal to six that will add to a prime number is
6/15 or 3/5 .
Imaginary Graphs and relations to
Probability
Given a graph it is non polar, polar or multipolar.
A graph is non polar if it has only one vertex so that every
edge is a loop or multiloop.
A graph is polar if it is well defined on a finite number of
vertices with edges that are only two element subsets or
tuples that represent it edges.
A graph is multipolar if it is well defined on a finite number of
vertices with edges that are only r element subsets or r
tuples that represent its given number of edges and r may be
greater than two.
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Given a graph it is either empty finite or infinite.
If a graph is finite and polar then it is in the category that is
accepted in current well definition of mathematics and its
communities.
Otherwise the extension of graphs’ categorical definition may
be well defined as implied by the existing category of
probability of events that exist in a finite number of steps
where order does not matter with replacement that is not a
permutation, combination and outside of fpc where order
matters.
Cn2=(nxn-n)/2
binary watchit Theorem
or number of polar finite graphs
Given a natural number n greater than one then the number
of ways to select 2 objects from n objects with replacement
so that the order of selection does not matter is

w  n n n .
2
2
n
2
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Proof
The number of ways to select two objects from n objects
where order matters with replacement is nxn by fpc.
The number nxn includes counting of n pairs of two objects
selected from the given n with replacement as n=1xn by fpc .
nxn-n is the number of pairs of two objects selected from n
objects without replacement where order matters.
nxn-n is divisible by two as only two objects are selected as
pairs from the given n objects.
(nxn-n)/2 is the number of ways to select two objects from n
object where order does not matter without replacement ( or
the number of combinations cn2).
Whence wn2=(nxn-n)/2+n is the number of ways to select
two objects from n objects where order does not matter with
replacement .
wn2 is called the binary watchit .
Wn2 is the number of graphs that are not mutipraphs
including loops and cn2 is the number of graphs that are not
multigraphs and are loop free.
The number of non polar graphs is unitary or one and may
be homeomorphic to the infolate.
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Conjecture
Watchit number Conjecture
or number of multi polar finite graphs
imaginary graphs or multipolar
graphs
Given a natural number n greater than one then the number
of ways to select r objects from n objects where r is less than
or equal to n with replacement so that the order of selection
does not matter is

w  n n n .
r!
r
n
r
Wnr implies the existence for definition to extend the
categories of graph theory into graphs that have edges that
may not act or relate through only two vertices’ connections.
Time Free Event Theorem
If a non empty graph has an edge that is a loop then it is
time free.
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Event Particle Theorem
If the tree graph Gs of a sample space has a path that is a
loop then its outcomes are time free and the graph is a non
trivial Euler circuit that is nonplanar and homeomorphic to a
singleton in R3 with natural topology also possessing non
trivial fundamental group and having fractal dimension if the
event requires an infinite number of steps to every outcomes
completion called an infolate.
Consecutive Sums
The set of all numbers that are equal to the sum of two
consecutive natural numbers called the set of consecutive
sums C does not contain the natural numbers one and two
so that it is not equal to the set of natural numbers N. It is a
fact that not every natural number is equal to the sum of two
consecutive natural numbers like 6 for example.
Prime consecutive sum exclusion theorem
P is not equal to C where P is the set of prime numbers and
C is the set of consecutive natural number sums.
Proof
Not every prime number is the sum of two consecutive
natural numbers as two is an example because it is prime
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and 2=1+1 so that it is not in C. Not every element of C is
prime as 15 is and example as 15=7+8 and 15 is not prime.
Consecutive Sums Conjecture
The sum of two consecutive natural numbers is prime if and
only if it is not divisible by a prime number less than the sum
itself.
One implication of the consecutive sums theorem/conjecture
may be equivalent to the prime factorization theorem for
natural numbers. The other implication suggests the
probability is great that the sum of two consecutive natural
numbers is prime where the summands are each selected
randomly.
Summand Prime Probability corollary
When the value of the sum of two natural numbers increases
the probability that it is a prime number decreases.
If the Riemann hypothesis is true then the Summand Prime
probability corollary will be a consequence of the Riemann
Theorem or a slight variation of its given statement of the
hypothesis probably including the number four.
T(2/10/2012)
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Calculators, Computer
Software, and Technology
Each student should be proficient with at least one scientific
calculator.
A calculator should not be required for the student to comprehend
algebraic expressions and how their symbolic forms are changed
through representations with the given binary operations.
A calculator should not be required for the student to comprehend
how variables are used in abstraction to signify other objects.
A calculator should not be required to multiply any two integers
between zero and twelve or find their squares through 12 and cubes
through 5.
All students should be proficient with more than one brand and type
of calculator and computer program.
It is desirable that students could accurately use any or no calculator
to produce results at all times.
Every person should be able to use any calculator by trail and error.
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Use of only one type or brand of software or calculator only
limits one’s user capabilities and promotes that brand or type
exclusively.
Learn all technologies, their ranges of applications, and of their
interconnections.
Be able to use any computer, print, scan, fax, and archive your
electronic data.
Use and know a large part of the vast depth of knowledge available
through the world wide web.
STUDY SKILLS, LEARNING METHODS,
AND STRATEGIES FOR STUDENTS
Please make sure you understand your best learning style(s) and are
able to adapt to any teacher’s methods with ease.
Don’t make excuses to get out of your responsibilities.
Do what is expected of you from all your teachers without question.
Make sure you are prepared for all your classes at all times.
Explore all learning styles new to you and ideas such as
public speaking, writing/drawing/art, theatre, music, sports,
and dance.
You should be able to read any language.
You should be able to read at least one written and at least
one symbolic measurement (metric) language such as
numbers (basic arithmetic) .
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Ability to read, speak, and write in more than one language
is a great resource to levy for a job that you want.
Know your skills and be able to demonstrate this in a professional
manner.
Ability to speak in more than one language should be required by all
schools.
Ability to physically write or express writing
in hand written format without technology in
at least one language is essential.
Ability to physically write in any form, language, and system is optimal
and never to be forgotten or dismissed by any person or people.
Ability to physically write with all programs and machines in at least
one language is usually ok.
Type writer skills are most useful in our modern day technology
driven environment and a personal favorite course of the author in
High School.
Proficiency with at least one word processing program is desirable
and usually expected.
Produce projects using all sorts of electronic methods and media to
reach any audience.
Practice presenting many presentations orally and graphically before
an audience with different methods.
Proficiency with any technology and computer program is the best
way to manage any career.
Be able to write descriptively, creatively, confidently, and correctly
with proficient ease and abundance.
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Try to dress well and be fresh and clean whenever possible.
Always have an answer even if it is ‘I’ll get back to you on that’, and
do it if you have to say it.
Be yourself and develop many styles, hobbies, and skills as you are
able and make ways to do so more often.
Always be respectful and kind to your teachers and friends and instill
these values in yourself and all your parents, children, cousins, and
siblings whenever possible.
Never hate or be jealous as these are a waste of time and unhealthy.
Never hate mathematics and/ or science or say this especially in the
presence of sons, daughters, brothers, sisters, nephews, or nieces.
Respect all copyright and legal agreements.
Have a good dictionary close at hand as the internet is not reliable for
definitions and most software programs also lack word definitions and
only offer spelling options.
Have respect towards all your fellow classmates and
instructors at all times and treat each other kindly.
Your parents, instructors, co workers, elders, friends,
teachers, and mentors are always trying to help you and do
not ever deserve rudeness, hatred, cruelty, or disrespect of
any kind as these actions can only put barriers before your
learning and growing showing you in a most unfavorable
light and not impressing your peers, family, or potential
employers in a productive manner.
Always remember and apply all the good things you ever learned and
keep an open mind for other knew great ideas and listen to those who
know more.
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Don’t ever take mean angry words, or gestures to heart or worry so
that you have no burden(s) and give no power to their speakers
and/or gesturers.
Stand up for yourself, your family, and those you love all the time
without failure so that people know you are upstanding and having
great principles, strengths, and good ethics.
Never get bored as this means you are boring.
Use your time efficiently and productively always.
Develop superb ethics, morals, and principles and strive to learn
more always from many people and the past and never judge.
Read all kinds of literature and know the news or world events.
Do not believe everything you see and read all the time and question
sources, motives, and politics of journalist, corporations, and news
programs sometimes.
Learn sales and marketing strategies and schemes and do not be
manipulated by them to your disadvantage by commercial entities,
businesses, and corporations.
Stand up and protect all your rights and know them.
Considering that people should have more rights as technology
develops where it may threaten our well beings, freedoms, and
livelihoods help develop and speak your mind so that human rights
are truly inclusive and just every day.
Always vote and participate and understand politics to take care of
our Nation and societies for our futures in world peace.
Have more than one role model or hero so that you are never let
down in your aspirations.
An example of a great hero is ...
Remember that perfect attendance, excellent attendance, and
superbly studious behavior may be considered by most teachers.
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Try and take good notes to practice the most useful skill of
handwriting.
Please try to attend all your lectures in courses.
Have no tardiness and do not make interruptions in classes as this is
distracting and inconsiderate to other students.
Please raise your hand if you have questions or be polite when
speaking.
Do not use obscenities in the classroom as this is unprofessional and
not good speech offending many.
Don’t be distracting, inconsiderate, or the center of attention in class
time as there exist other students beside you and such behavior is
selfish.
If you need attention find productive ways to get it in your own time
and take a drama or speech class and express yourself in a healthy
way.
Know your assignments, deadlines, and exam dates as a responsible
student.
Do not cheat or plagiarize. Learn to be more creative and
resourceful.
A second opinion or a proofread of assignments can be most
valuable as it is often extremely hard to find or note our own
inadequacies and mistakes.
Make up any work missed due to every tardiness and absence and
be responsible for all information such as notes.
Skim or scan through all assignments at least if you do not have time
for careful study. Try all the time to get the big idea(s) or picture(s).
Remember school is your responsibility and commitment and in the
end you can only rely on yourself especially in graduate school.
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Remember that school does not last forever and if you make the
required commitment and do your work it will end in short time with
your graduation, laud, and great profit.
Never give up on any dreams or goals you have determined for
yourself as persistence is a quality you want.
Please try to respect and not judge any one else’s religion, morals, or
ethics
as it is not good business. You will never know or understand
completely any person’s full past, culture, and/or circumstances so
that disrespect and judgment are unwarranted or unfounded.
Always talk to people.
Know how to write letters and send post through the mail as it is more
efficacious than any electronic format or procedure to make contact.
Know the limits of video and audio recordings and their applications.
Have the best respectable behavior in social situations with regards
to technologies such as cell phones.
Be proud of the people in your life especially your mother all the time.
Be proud of all the great things you have ever done and look forward
to more and the future.
Go to cultural events periodically where tickets can be reasonably
purchased which include and are not limited to theatre productions,
symphony, band performances, ballet, operas, museums, and poetry
or literary readings.
Learn how to be studious and professional.
Know how to document any resource materials used in projects or
papers correctly and do it if you borrow from other places so that you
have no plagiarism.
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Never begin or end a sentence with a preposition.
Always act and conduct yourself like a professional, ethical, and
respectable person.
Do not be quick to act if time permits consider options otherwise
develop the ability to make swift decisions when necessary and
produce viable results and data.
Smile at least once in an interview, dress very well, and look great.
Watch your words and monitor your gestures carefully in interviews.
Be politically correct and decent in interviews.
Always introduce yourself and don’t be shy.
Keep an active portfolio, vitae (resume), and have relevant job
experience with skills in your chosen profession.
Write and keep a journal, diary, notebook, sketchbook, or log to
reflect upon yourself and your life.
Use all your talent to make a great, happy, productive beautiful life
and brilliant career for yourself all the time.
Make school your first priority and devise means to have stability in
life almost at any cost which is required for your successful
graduation and job placement.
Make sure you drop a class with paperwork if you believe you will fail
before the drop date or if unpreventable circumstances prohibit its
completion so that you receive a W instead of an F on your
transcripts.
Keep track yourself of all your grades in every class written in your
notebook so that you have a record of your works.
If you need help ask more than one person without shame until you
get what you need.
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Ask for help when you need it.
Try to have no debt or as little as possible.
Try to get a grant, scholarship, or funding to pay for school and your
necessities.
Never quit or give up.
TEST ANXIETY AND MATHE. PHOBIA
Precepts of Test Taking
1.
2.
3.
4.
There should be no talking during any test or quiz.
Always be on time for every test and prepared that day.
Submit your completed work on time or before required time.
Look over your work before you hand in any exam as it will not
be returned after submission until it is graded.
5. Teachers have at least two weeks to grade and return papers
or assignments.
6. After you submit your work exit the classroom.
7. No use of cell phones or laptops during exams.
8. Follow and satisfy the directions of the exam.
9. If you miss an exam return directly to the next class meeting.
10. Do not leave any question blank.
11. Clearly denote your answers or have neat work.
12. Put your name on your paper.
13. Do not cheat and keep your eyes on your own paper.
14. Study well in advance and know the expectations of the
exam.
15. Take any test without fear and do your best.
16.
You can and should overcome test anxiety and math phobia with an
effort if you seek help and try.
Every one does not like tests and experiences anxiety some
times.
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Every one is a little or a lot intimidated or scared by mathematics at
some time.
Most overcome anxiety and phobia by trial and error and repetition.
Realize that test taking is an acquired skill and move to gain
it.
Please learn how to take a variety of differently presented
tests, quizzes, and exams and produce great results.
Learn as much about an exam as possible and practice its required
styles and techniques well before the exam date.
Study and take exams and tests seriously and only for what they are
worth.
Learn the number of questions, question presentation format, date,
time, location of exam, type of answer format(s), time limits, required
materials you must supply (example calculator or no calculator, or
pencil), what is supplied, calculator and technology parameters if
possible.
Study well before the exam material over which it will cover and
prepare early.
Find and practice any sample test(s).
Know what it feels like in exams and practice these disciplines and
conditions.
Create your own practice test and ask advice from colleagues and
friends.
Buy a moderately priced review book (workbook) from a store where
it is available especially if it directly accompanies a standardized test
such as the Graduate Record Examination ( G.E.D.) required by
excellent graduate schools worldwide. Use the workbook please.
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Most standardized tests such as GED have very specific question
presentations and strategies to derive answers and complicated
presentations of results which make a help workbook where available
most valuable.
Most standardized tests such as GED have multiple choice answers
which require scantron input.
Multiple choice answers require you to use elimination strategies so
be familiar with them.
Practice using a scantron input bubble sheet with no. 2 pencil very
carefully.
Have a small pencil sharpener and eraser if you need one.
Taking exams requires strategy, patience, confidence, skill,
effort, and mastery over fear and anxiety.
Study well before an exam not a day or hour before it starts.
Study in groups only if it helps you to learn if not then make a good
place with good lights, quiet, and space for your books (papers)
without interruptions and a small drink or snack if possible.
Study every moment and as many others possible exceedingly weeks
or many days before an exam and/or test so that you are prepared.
Seek out the help and advice of all others who may have taken a
given test.
Be respectful of proctors and all teachers who help you and to your
fellows all in the same station as yourself.
Don’t stay up too late before an exam so that you aggravate your
normal routine and schedule.
Please make sure you are able to eat well and be rested before you
take a test. Take good care of yourself.
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Read-skim over the entire test when you first receive it to put the total
situation into your mind.
If you are stuck do not spend more than three minutes on one
problem without any advance.
Leave unmanageable problems and go to others and return to them
later.
Try a fresh approach to a difficult problem.
Stand up for a brief moment to stretch and breathe and return to your
seat.
Drink water when you can.
Draw, scribble, and use scratch paper and pictures to help yourself.
Do not proceed with many different methods until you are sure the
one you are using is not sufficient.
Try one method and if this produces answers which do not make
sense try another method until you can reach an answer. Never give
up.
Don’t try to do too much at once or too quickly.
Take a deep breath, relax, and know a test can only last for a small
portion of your life then there will be no more momentarily.
Have faith and confidence in your work and yourself.
Be polite to all people taking an exam with you please.
You are smart and every body can do and understand mathematics.
All works with math will usually end up at a task which you already
know how to do so the key is to get to that point in the work,
recognize it, and move forward to finish the problem not being
overwhelmed.
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Remember you have done mathe. your whole life and you are good
at it.
Don’t forget what you know and recognize when you start to panic
and control yourself some way.
Remember that you have multitudes of knowledge and
experiences always in yourself from all the schools and tests
you have ever attended or taken for many years and use this
now to your advantage taking great confidence from this
fact.
Remember you want to succeed and your family and friends want
you to graduate and be successful.
Remember one plus one is two.
Have humor and take life lightly when you can.
Smile sometimes even if it is hard to do it.
Check your answers and work extremely carefully before you turn in
an exam even when you wish to flee.
Make sure your name is on your paper please.
Submit the test making sure you have done everything expected of
yourself.
Get a drink of water and put some on your face and take a deep
breath.
You did it and be proud of that no matter what the results.
Relax and treat yourself to a good indulgence if you did well
otherwise don’t give up, keep studying, and prepare for the next test
and/or final.
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Study Habits and Tutoring Help
Make friends in your classes for study groups, teams and help with
notes.
Working in teams and/or groups is great for presentation or exhibition
of projects and reports promoting professional business etiquette and
poise.
Please seek out support, stability, and quiet so
you can study alone without interruptions
sometimes all the time.
Make time for the most important goals and dreams of your life.
Please seek out support, stability, and persistence from all sources to
guide and aid you in your studies in school these can include friends,
family, and co-workers.
If you need help please do not miss classes.
Please stay after class to talk to your teachers if you need help or
have questions.
Seek out help immediately and do not wait until the last minute if you
need assistance.
If you are frustrated when you study go to another problem
or take an example from the book and practice by copying it
directly as this helps with comprehension and handwriting.
Use other books, manuals, or workbooks to copy examples
or the text directly onto paper contained in this document or
otherwise written as an exercise of hand writing
transcription.
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Try different studying techniques to see how you are
strongest and most effective.
Take small periodic breaks to momentarily relax when you
study.
If you are having any difficulties with the subject matter for example
you can ask your instructor to give you information for resources that
will make your academic experience a success such as free tutoring
or other learning media.
Use the internet, computers, video tapes, friends, classmates, family,
tutors, teachers, and books to get the help you need.
Please help us create a friendly, kind, and good environment for our
campus and classes.
Mathematics, Attitudes, and
Learning
As your attitudes affect yourself and those close to you directly where
learning is concerned good attitudes towards mathematics, learning,
and sciences are extremely valuable.
Please find positive aspects of mathematics and promote its
students.
You should practice your handwriting with the homework problems
and by taking good notes in class, as mathematics is a universal
symbolic language, which makes transcription almost entirely
necessary for good results.
Why do we study mathematics? When will we ever use
what we learn again?
What purpose does mathematics serve?
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These are the most important questions you can answer to help your
mathematics education be a success.
There are good reasons why we study the areas of mathematics and
logic.
As part of important and essential data for humankind’s welfare
mathematics is required in all degree programs of respectable
institutions of higher learning as culture.
Culture is the perpetuation and maintenance of knowledge.
As part of logic and technology mathematics is essential for deductive
analytic reasoning and symbolic manipulation using comprehension
in abstraction.
The study of Mathematics strengthens our abstract reasoning facility
and allows us to understand symbolism and abstraction which can be
used in meaningful and powerful ways.
Understanding mathematics allows one to interpret legal literature
and use computer programs more easily than if we do not
understand.
As mathematics enables the viable transcription of abstract
symbols and language in forms where technology may not
be present or fails for this reason we wish to learn its
applications.
As mathematics allows one to use proper variable substitution into
formulas of mathematics for their relation to all sciences where
calculation through order of operations agreement produces accurate
results we wish to learn these applications of mathematics.
Mathematics allows learning of analysis for complex abstract
situations.
Mathematics allows accurate and efficient organizational skills for
reasoning of complex events.
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Mathematics helps us understand and interpret technical graphs,
charts, diagrams, and figures.
Mathematics teaches us to collect, arrange, and analyze data and
samples.
Mathematics teaches us to use creativity and imagination in
productive and critical directions.
Mathematics teaches us to use thought experimentation.
Mathematics allows skills beyond the direct applications we will study.
Methods and skills gained by learning mathematics are useful beyond
their vehicles of presentation.
Mathematics helps us understand chemistry, physics, biology, and
medicine.
Analysis and creativity through imagination are enhanced by being
able to apply mathematics.
Using mathematics we can utilize money, business, accounting,
sweepstakes, odds and games.
Complex charts and diagrams which are used for weather, sports,
and money markets are understood by the study of mathematics.
Proficiency with mathematics and logic allows us to understand and
create forms and documents and interpret them.
Proficiency with mathematics helps us utilize algorithmic procedures
and schemes to satisfy goals with strategies.
Mathematics helps us understand units and measures and their
conversions with applications.
EXPERIENCE
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Please join professional organizations such as www.ams.org
national organization.
it is a
There are several branches of the ISA. They include controls,
robotics, automotive, wastewater, and power. They also host trade
shows.
Membership in ISA – be a part of a mentorship. Membership can
help when students are working on a class project.
Resume writing tips also can be learned. Several different courses
are offered.
Try to maintain work experience related to the field of work you
choose
for your professional resume.
Develop an electronic portfolio and start to keep a working vitae or
resume.
Learn how to go through interviews and have business etiquette.
Make sure you are familiar with modern marketing, sales, strategies,
and customer services.
Get an internship in your chosen career area and gain much related
work experience so that you have at least three current letters of
recommendation always.
Most importantly research the relevant skills and knowledge which
will be required of you in applications for the career field you enter.
Use the Library resources such as books, journals, and computer
resources available to you.
Read books, journals, and magazines relevant to the career field you
wish to enter.
Abstract
Any equations, laws or formulae which are fundamental to physics have many mathematical implications beyond their
known or accepted interpretative meanings and these cannot be discarded in reality as their conditional truth is verified by
experimental evidence and the laws and axioms of mathematics exist beyond the realms of physical sciences and
computers.
What cannot be determined (experimentally) cannot be discarded and most often yields important information concerning
the truth.
The equation e=mc2 is demonstrated in physics where its variable assignments are given the properties of units of energy
and mass with the speed of light held constant. Graphs of the equation e=mc2 in the real Euclidean plane when the
164
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c. t( x )2012
freedom of the individual variables is constrained or given representations other than those of standard physics (yet
validated by mathematics) have interesting real limits.
These limits give insight into the nature of matter and time where relation to the absence of energy or mass or the vectors
of electromagnetic radiation are negative and exist.
Conservation of energy and mass implies that where there is an absence or deficit of some units of mass or energy
equilibrium occurs where the surplus of these units of energy or mass are taken to make up the difference and not
necessarily at the same locations. Black holes may be considered as the absence or deficit of energy and mass.
Electromagnetic radiation considered as a vector with direction and magnitude exists in lines of force and otherwise it is
reasonable to assume that the negative of such vector unit amounts may apply to physical theories as the opposite of a
vector is well defined in real analysis. Light has that its speed is not constant near or in a black hole or when transmitted
through super cold mediums and only constant in a vacuum. Light is also non linear in its paths due to gravity. Mass and
length are not constant at large velocities.
Time is altered at large velocities.
Perception and interpretation are not confined by all existing physical theories as some of them may be incorrect and they
are all relative to human observation.
Introduction
Mathematics used by physicists and if truth is discerned this is certain for all such conditions as the laws of those such as
Maxwell, Newton, and Einstein must hold [phy1] in their cases.
Vast areas of mathematics exist not utilized or known to most scientists who are not also mathematicians so that it is
incorrect to say the mathematics does not exit to satisfy the unification of the physical forces. The theorems of the great
mathematicians Paul Erdos [e] or A. Lelek [top] contain many statements yet unexplored and of interest for scientists who
are willing to understand them for example concerning Ramsey theory and dimension theory.
It may be determined from existing formulas of physical sciences many statements that seem strange yet are valid
mathematically as that which is currently known or understood by men under the empirical process’ determination is not
total and can never be complete [godel].
Mathematics is the Mother of all sciences and indiscriminant, incorrect or invalid applications and uses of her axioms
and/or laws are never sanctioned by any mathematician or computer over all time. Order of operations agreement is held
across the world by everyone so that there is well definition [royden] in computation for example.
Logic is the Father of the sciences’ empirical process and technology its mandates cannot be violated by nature or
engineering yet only in human imagination.
If the mathematics for some given existing formula of physics yields statements or implications that seem nonsensical to
physicists these may not be discarded as they are most likely applicable somehow based on the validity of the instances
where the given formula is true and note that nothing is wasted in mathematics as the remainder is given the greatest
meaning in the fundamental theorem of Algebra [Hungerford] and the freedom of variables taken as axiomatic.
Any mathematics or technology used by physicists, financiers, or scientist should never be used incorrectly, irresponsibly,
partially or to the detriment of humankind, freedom or life in the future or on the Earth.
Standard models used to understand the propagation of energy such as light or electromagnetic radiation use linear rays
and where rays are non linear, directed, and terminated they represent lines of force [physics] such as magnetism or
directed paths in graphs [graph].
Energy propagation from a point on the boundary of an object may or may not follow a linear path like a ray [Royden] but
may also propagate in a non linear way such as radiant heat energy or solar flares [so].
All rays ending at a fixed point in a Euclidean space can represent the possible paths a boundary point may take over
time due to some force placed upon the point at a source.
Propagation of properties, energies, or forces directed from a point as a source at an existing boundary point for a given
object through itself, its complement [set1] space or other objects located properly in its complement may be modeled with
linear rays.
Existential influence from an object’s geometry and topology while embedded in spaces effects the object’s complement
space[set] and any other object contained in the given space, intersecting the given object itself or while placed in one of
many possibly connected or disjoint other spaces for almost every standard observer at any time in almost every universal
space or set axiomatically [foundations] in mathematics.
Propagation of energy and/or force from existing boundary points of a given object effects the object’s complement
space[set] and any other existing object.
It can be assumed that no object has existential influence on any other object or its complement space unless it has at
least one boundary point and if the collection of boundary points is empty that the object does not incur nor produce
influence from the existence of any other object and almost every force.
The above statement can be modified yet its acceptance can produce a viable theory for energy propagation,
transmission, absorption, and reflectance [color2] using sets, Euclidean product space, topology, and linear real rays
[ana].
Propagation of forces from an object’s boundary points may occur in a linear, non linear, or discrete way and possibly into
other existing spaces which may not contain the given object or its boundary or be disjoint from any other space.
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Granting properties to sets in a Euclidean topological space [top] or Euclidean topological product spaces which mimic the
attributes of an objects propagation for its influences, properties, and energies into the space itself and with relation to
some other objects in the space yield new models of physical phenomena involving the use of rays and facts of
topologically normal space.
Ideas that spaces, objects, and/or forces can be composed as unions of other spaces of lower dimension or intersections
of spaces with higher dimensions or some combination or permutation thereof originates from the topologist Andrej Lelek
[lelek] and teachers such as Knaster [knaster], R.Sinkhorn, and S. Fajtlowicz.
Properties used for this model include transmission, propagation, and emission of energy (such as light), reflectance
[color], absorption, and luminescence [spie].
Perceptive and reporting abilities of standard observers can be similar and communicated similarly at times and more
than one such standard observer exists with these abilities is axiomatic.
Assume that the conditions and positions of objects and standard observers can be held constant for time intervals of
observations and replicated by other standard observers at similar of different times.
Given a real object in physical real space the object’s complementary space is the collection of all the space that the
object does not occupy at a certain given time as perceived by a standard observer which does not intersect the object or
contain it.
Given an object in physical real space its property space is the collection of all properties possessed by the object at a
certain given time as perceived by a certain standard observer which does not intersect the object.
Given an object in physical real space its complement property space is the collection of all properties not possessed by
the object at a certain given time as perceived by the same standard observer given above.
Given an object in physical real space the object’s indeterminant space is the collection of all properties not able to be
determined for the object at a certain given time as perceived by any standard observer which does not intersect the
object.
The indeterminant space of an object includes all properties not able to be determined in our real physical universe with
its laws yet also all properties not able to be determined in any universe with any set of laws as the location of the
standard observer is not fixed other than they do not contain or intersect the given object and conditions can be
considered as constant for a given interval of time.
What is possible intersects that which is indeterminant and contains that which is determined for a given object over any
time, location, and condition for every standard observer.
If the standard observer’s and object’s locations are real or in our physical universe then what is indeterminant is
controlled by our universal physical laws as an axiom.
What can be determined absolutely is connected with questions of computation [c] and completeness [g] (Does p=np?)
yet these definitions yield a means to consider relativity and models and gravity.
More information concerning properties, standard observers, and conditions can be found in [zprop].
An object’s existence propagates properties to its complement space in a way that gravity, light and matter must obey with
respect to a given standard observer’s perceptions due to Einstein’s General theory of relativity.
Over intervals of finite time with acceleration placed on the object this relationship between the object, its complement
space and the standard observer must yield to Einstein’s special theory of relativity [eee] .
It is known that perception of properties effect the instances of objects and observers at the quantum level by
Heisenberg’s uncertainty principle [h].
Key Words
Source, electromagnetic, energy propagation, complement space, topology, real analysis, foundations of physics,
standard observer, segmentation, graphs, paths, product space, continua,
References
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c. t( x )2012
Pollock, Nomic Probability and the Foundations of Induction, Oxford, Oxford
University Press, 1990, isbn 0-19-506013-x
[l] Luminous and Reflective Sets, s. Zoch, Lumino.doc
[n] Negative Light Vectors and the Deficit of Energy and Mass, A. Zoch, Energia.doc
Probable Categories
S. Zoch t(2007)
Primordial Integers
Stephen Zoch
Primordial.doc
C. t(2007)
Devlin, K., The Millennium Problems, Basic Books, N. Y., N. Y., 2002.
Royden, H. L. , Real Analysis, Macmillan Publishing Company, New York, 1988.
Buck, R. C., Advanced Calculus, McGraw-Hill, Inc., U.S.A., 1978.
Kramer, Mathematics for Electricity and Electronics, Second Edition, Delmar
Thomson Learning: Albany, New York (2002).
Devlin, K., The Millennium Problems, Basic Books, N. Y., N. Y., 2002.
Royden, H. L. , Real Analysis, Macmillan Publishing Company, New York, 1988.
Buck, R. C., Advanced Calculus, McGraw-Hill, Inc., U.S.A., 1978.
Kramer, Mathematics for Electricity and Electronics, Second Edition, Delmar
Thomson Learning: Albany, New York (2002).
D. Boneh, “Twenty years of attacks on the RSA cryptosystem”, Notices of the
American Mathematical Society 46(2):203—213 (1999).
[J. Yepez, “Quantum computation for physical modeling”, Computer Physics
Communications, 146(3):277-279 (2002).
M. Luskin, “Computational Modeling of Microstructure”, Proceedings of the
International Congress of Mathematicians, Higher Education Press:
Beijing, China 3:707-716 (2002).
R. H. Bing, “A Translation of the Normal Moore Space Conjecture”, Proceedings
of the American Mathematical Society 16: 612-619 (1965).
F. H. Croom, Principles of Topology, Saunders College Publishing through Holt,
Rinehart, Winston: Orlando, Florida (1989).
P. Alexandroff and H. Hopf, Topologie, Springer: Berlin (1935).
s. Zoch, The Jordan Polygonal Theorem, a, Houston, TX (1994).
Washington, Basic Technical Mathematics with Calculus, Seventh Edition,
Addison Wesley:New York, (2000).
J. Nagata, Modern Dimension Theory, Vol. 2, Heldermann Verlag: Berlin (1983).
167
UNIVERSITY MATHEMATICS t(1994)
S. Zoch
c. t( x )2012
F. Harary, Graph Theory, Addison-Wesley: Manila, Philippines (1969).
A. Lelek, “Properties of Mappings and Continua Theory”, Rocky Mountain
Journal of Mathematics 6(1):47-59 (1976).
K. Menger, Dimensionstheorie, B. G. Teubner: Leipzig (1928).
s. Zoch, Polygonal Cellular Continua, a, Houston, TX (1995).
s. Zoch, Power Numbers, a, Houston, TX (2003).
A. Zoch, Geometric Language, A, Houston, TX (2004).
G. Cantor, “De la puissance des ensembles parfait de points”, Institut MittagLeffler, Djursholm, Sweden, Acta Mathematica 4:381-392; in
Gesammelte Abhandlungen, pp. 252-260 (1884).
s. Zoch, Harmonic Cellular Continua, A, Houston, TX (1994).
s. Zoch, Total Rational Partitive Color Mixture Continua and Triangles, A,
Houston, TX (1996).
S. MacLane, G. Birkhoff, Algebra, Macmillan: New York (1967).
E. DeLaVina, Ramseyan Properties and Conjectures of Graffiti, Ph.D.
Dissertation, University of Houston (1997).
Graphing websites, resources, engines, and internet tools.
sample problem only
Graph
2 x 3 y cos z  e
x
in three dimensions using any computer
software programs, engines, tools, websites, or resources located on a computer
or the internet.
HARMONIC CELLULAR CONTINUA
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I feel there is another type of mathematics that will be able to produce a unified theory of physics.
This new ‘meta-algebraic’ mathematics is not probably based on the horizontal vertical first order
predicate calculus that we normally have used to simplify equations but will encompass it. To
state ‘horizontal vertical first order predicate calculus’ I mean the usual properties we used to
formulate theory such as De Morgan’s laws, laws of exponents ,distributions , integrals, and what
is mostly now considered the properties of spaces in modern mathematics. This ‘meta-algebraic’
mathematics is possibly based on properties of circles and sets and geometry. I do not claim
these things without reason as I have produced interesting results among which include a
revealing situation between algebraic simplification of complex fractions and the Cantor sets.
This pointing to a higher geometry for unknown but applicable non typical equations with new and
encompassing rules for their meta’-algebraic simplifications.
Some laws of the physical universe are not affected by observation and some laws of the
universe yield variations based on physical observation.
Yo me siento hay otro tipo de las matemáticas que podrán producir una teoría unificada de la
física. Este nuevo ‘meta-algebraic’mathematics no es probablemente se basó en el primer
cálculo vertical horizontal del predicado de la orden que hemos utilizado normalmente simplificar
las ecuaciones pero lo abarcará. Para indicar 'horizontal vertical primero cálculo de predicado de
orden' significo las propiedades usuales nosotros formulábamos la teoría tal como De las leyes
de Morgan, las leyes de exponentes, las distribuciones, integrante, y lo que en su mayor parte
ahora es considerado las propiedades de espacios en matemáticas modernas. Esto matemáticas
meta-algebraicos son basadas posiblemente en propiedades de círculos y conjuntos y de la
geometría. Yo no reclamo estas cosas sin la razón como yo he producido los resultados
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c. t( x )2012
interesantes entre que incluyen una situación reveladora entre simplificación algebraica de
fracciones complejas y los conjuntos de Solista. Este señalar a una geometría más alta para
ecuaciones desconocidas pero aplicables no típicas con nuevo y las reglas que abarcan para su
meta' -simplificaciones algebraicas.
Algunas leyes del universo físico no son afectadas por la observación y algunas leyes de las
variaciones del rendimiento del universo se basó en la observación física.
Je me sens il y a un autre type de mathématiques qui pourront
produire une théorie unifiée de physique. Ce nouveau ‘metaalgebraic’mathematics n'est pas probablement a basé sur le premier
calcul de prédicat d'ordre vertical horizontal que nous avons utilisé
normalement pour simplifier des équations mais l'entourerons. Pour
déclarer ‘le premier calcul de prédicat d'ordre vertical horizontal’ je
signifie les propriétés normales que nous avons utilisées formuler la
théorie telle que De Morgan’les lois de s, les lois d'exposants, les
distributions, intégrales, et ce que maintenant est surtout considéré
les propriétés d'espaces dans les mathématiques modernes. Ces
mathématiques de ‘meta-algebraic’ sont probablement basées sur les
propriétés de cercles et de séries et la géométrie. Je ne réclame pas
ces choses sans la raison comme j'ai produit des résultats
intéressants parmi qui incluent une situation révélant entre la
simplification algébrique de fractions complexes et le Chantre règle.
Cet indiquer à une plus haute géométrie pour inconnu mais
applicable non les équations typiques avec nouveau règles et les
règles entourant pour leurs simplifications meta’-algébriques.
Excerpt from [ave] ‘Property and Theory’ A. Zoch t(2000)C .
I would like to thank many students for all your valuable
questions, friendliness, and graduating. I hope this book will
help new students of sciences and mathematics and
promote learning.
Note: Technology tools, such as calculator and/or computer software, used are
campus specific.
under revisions
s. Zoch copyright t(2004) all rights reserved
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ALGEBRA FOR EVERYBODY
Revised t(9/1/2007)
Revised t(9/11/2007)
Revised t(8/28/2010)
COMMENTS OF THE AUTHOR
Mathematics and art are connected and can be used to understand
many abstract situations and thoughts were other media are
insufficient .
Diagrams, Graphs, Continua, and
Fractals
h
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Figure
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c. t( x )2012
173
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c. t( x )2012
174
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All Rights reserved by Stephen Zoch t(12/22/04)
All national and international copyrights reserved by Stephen Zoch t(12/22/04)
Stephen P Zoch
U. S. A.
Cc arc e2
Sample writing unedited version
Topics for great searches or research
papers.
Topics
Science Technology
Engineering
Mathematics and Accounting
Education
1. Women Engineers, Scientist, or Mathematicians
2. Concerning theorems of the computer program Graffiti or the program itself created by Dr.Siemion
Fajtlowicz in 1986
3. Graffiti.pc by Dr. Ermalinda DeLaVina
4. HILBERT’S DECISION PROBLEM
5. What is Autosophy?
6. Gödel’s INCOMPLETENESS PROBLEM
7. Turing Test
8. Sapir-Worf Hypothesis
9. Heisenberg’s Uncertainty Principle
10. Does P=NP?
11. Mixed states or phase changes.
12. When is an arbitrarily large natural number prime?
13. Given a random natural number how do we know when it is divisible by seven or any prime number
larger than seven in general?
14. STUDY SKILLS, LEARNING METHODS, AND STRATEGY FOR STUDENTS of MATHEMATICS
15. TEST ANXIETY AND MATH PHOBIA
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c. t( x )2012
16. Mathematics learning, education, tutoring, and help
17. UNIFICATION OF THE FORCES IN PHYSICS
18. GENERAL AND SPECIAL RELATIVITY
19. Psychology and Statistics of Mathematics Education in U.S.A.
20. Geometries of Proteins and Enzymes
21. Graphs of Cyber Spaces, Systems, and/or Networks
22. Cardinal and Ordinal Numbers
23. Non Standard Analysis
24. Universal Algebras
25. Mathematics of Ancient Societies and Communities
26. Dimension Theory
27. Ramsey Theory
28. When is a collection of line segments, polygons and/or circles a continuum or fractal?
29. Color Theory and color order systems used in industries (ex. Munsell color system)
30. Nano and/or Micro Technologies
31. Given a circle C of positive radius what is the maximum number of non overlapping circles contained in
the interior and/or boundary of C each of a similar fixed radius less than the radius of C?
32. Patents of N. Tesla
33. Encoding and Encryption.
34. Misapplication of formulae concerning economics and finance.
35. Discontinuity of functional rules in economics.
36. Loss or gain due to rounding interest computations.
37. Manipulations of unequal time change interval (daylight savings time) with respect to interest
computations.
38. Least Integer Function.
39. relationships between communities and mathematics with respect to technology and Industry
40. Learning Styles
41. relationships between the arts, design and fashion with mathematics and/or sciences
Achieving the dream and writing across the curriculum components.
Topic 17.-Unification of the Forces in Physics
Any equations, laws or formulae which are fundamental to physics have many mathematical implications beyond their
known or accepted interpretative meanings and these cannot be discarded in reality as their conditional truth is verified by
experimental evidence and the laws and axioms of mathematics exist beyond the realms of physical sciences and
computers [1].
What cannot be determined (experimentally) cannot be discarded and most often yields important information concerning
the truth.
The equation e=mc2 is demonstrated in physics where its variable assignments are given the properties of units of energy
and mass with the speed of light held constant. Graphs of the equation e=mc2 in the real Euclidean plane when the
freedom of the individual variables is constrained or given representations other than those of standard physics (yet
validated by mathematics) have interesting real limits [2].
These limits give insight into the nature of matter and time where relation to the absence of energy or mass or the vectors
of electromagnetic radiation are negative and exist.
Conservation of energy and mass implies that where there is an absence or deficit of some units of mass or energy
equilibrium occurs where the surplus of these units of energy or mass are taken to make up the difference and not
necessarily at the same locations. Black holes may be considered as the absence or deficit of energy and mass.
Electromagnetic radiation considered as a vector with direction and magnitude exists in lines of force and otherwise it is
reasonable to assume that the negative of such vector unit amounts may apply to physical theories as the opposite of a
vector is well defined in real analysis. Light has that its speed is not constant near or in a black hole or when transmitted
through super cold mediums and only constant in a vacuum. Light is also non linear in its paths due to gravity. Mass and
length are not constant at large velocities.
Time is altered at large velocities.
Perception and interpretation are not confined by all existing physical theories as some of them may be incorrect and they
are all relative to human observation.
Mathematics used by physicists and if truth is discerned this is certain for all such conditions as the laws of those such as
Maxwell, Newton, and Einstein must hold [3] in their cases.
Vast areas of mathematics exist not utilized or known to most scientists who are not also mathematicians so that it is
incorrect to say the mathematics does not exit to satisfy the unification of the physical forces. The theorems of the great
mathematicians Paul Erdos or A. Lelek contain many statements yet unexplored and of interest for scientists who are
willing to understand them for example concerning Ramsey theory and dimension theory.
176
UNIVERSITY MATHEMATICS t(1994)
S. Zoch
c. t( x )2012
It may be determined from existing formulas of physical sciences many statements that seem strange yet are valid
mathematically as that which is currently known or understood by men under the empirical process’ determination is not
total and can never be complete.
Mathematics is the Mother of all sciences and indiscriminant, incorrect or invalid applications and uses of her axioms
and/or laws are never sanctioned by any mathematician or computer over all time. Order of operations agreement is held
across the world by everyone so that there is well definition in computation for example.
Logic is the Father of the sciences’ empirical process and technology its mandates cannot be violated by nature or
engineering yet only in human imagination.
If the mathematics for some given existing formula of physics yields statements or implications that seem nonsensical to
physicists these may not be discarded as they are most likely applicable somehow [4] based on the validity of the
instances where the given formula is true and note that nothing is wasted in mathematics as the remainder is given the
greatest meaning in the fundamental theorem of Algebra [5] and the freedom of variables taken as axiomatic.
Any mathematics or technology used by physicists, financiers, or scientist should never be used incorrectly, irresponsibly,
partially or to the detriment of humankind, freedom or life in the future or on the Earth.
References
[1] Spanier, Edwin H. Algebraic Topology, New York, NY: McGraw-Hill, 1966.
[2] Buck, R. C., Advanced Calculus, McGraw-Hill, Inc., U.S.A., 1978.
[3] Kramer, Mathematics for Electricity and Electronics, Second Edition, Delmar Thomson Learning: Albany, New York
(2002).
[4] www.
[5] Hungerford, T.W. Algebra , Springer-Verlag New York Inc., New York,1974.
[6] Zoch, S., Luminous and Reflective Sets, x, Houston, Texas, USA, 2008, Lumino.doc
Excerpt from Negative Light Vectors and the Deficit of Energy and Mass by S Zoch.
Comments on Mathematics and Mathematics Education
All Rights reserved by Stephen Zoch t(12/22/04)
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Stephen Zoch
U. S. A.
2/4/2011 10:22 AM
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