Download Lesson - week 1

Week 1
Given problems with signed real numbers, perform
basic arithmetic operations, using a hand help
calculator, including conversions of percentages,
decimals, fractions, evaluations of mathematics
expressions containing exponents and roots, and
processing complex orders of operation.
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Objectives
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Real Number System
Sets and Venn diagrams
Signed Numbers
Powers and Roots
Order of Operations
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Real Number System
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Arithmetic uses only constants like -13,
0, 2/3 which have fixed value
Algebra uses not only constants, but
also variables like a,b,x,y, which
represent different numbers
The constants and variables we use in
Algebra are called the Real numbers.
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Real Number System
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The origins of number systems date back to the
Egyptians, Babylonians, and Chinese. However,
these earliest systems were much simpler than the
real number system. For example, the number 0 was
not widely accepted before the 13th century and the
use of negative numbers (-1, -2, -3…) was not
generally accepted before the 17th century.
In this first week, you will learn about natural
numbers, whole numbers, integers, rational numbers,
irrational numbers, and real numbers. If you would
like to learn more about the historical development of
number systems you can visit:
http://mathforum.org/alejandre/numerals.html
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The Natural Numbers
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Natural numbers or counting numbers are the
most fundamental set of numbers
{1,2,3,…}
Braces, { }, are used to indicate a set of
numbers
The … after 1,2, and 3, which are read “and
so on” mean that the pattern continues
without end – there are infinitely many
natural numbers
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The Whole Numbers
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The Natural number together with Zero
{0,1,2,3,…}
Not adequate for indicating losses or
debts.
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Example: Bank statement
Previous balance: $ 50
Checks paid: $ 70
New balance: -$ 20
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The Integers
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The Whole numbers together with the
negatives of the counting numbers form
the set of integers
{…-3,-2,-1,0,1,2,3,…}
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The Rational Numbers
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Any number that can be express as a
ratio (or quotient) of two integers
{a/b, where a and b are integers, with
b≠0}
Negative and positive fractions
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Example: 3/1; 5/4; -77/3
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The Irrational Numbers
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Real number that is not rational
Can not be written as a ratio of two integers
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In Computation with irrational numbers we
use rational approximation for them
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Example: √2, √3,π
Example: √2≈1.4141;
π≈3.14 : the ratio of the circumference and
diameter of every circle
Note: not all square roots are irrational
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Example: √9 = 3
9
The Number Line
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Every real number corresponds to one and
only one point on the number line.
Every point on the number line corresponds
to one and only one real number
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Set notation
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N = {…-3,-2,-1,0,1,2,3,…}
This is infinite set
A = {0,1,2,3} is finite set
0 is a member of the set; so 0 ϵ A
Equal sets contain the same numbers
B = {0,1,2,3}; hence A=B
subsets: set C = {1,2,3} is a subset of set A
and set B
Set D = {1,2,3,4} is not a subset of A or B
Is C subset of D?
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Interval of Real numbers
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Interval of real numbers is the set of real
numbers that is between two real numbers
Interval notation is representing intervals
(4,6): all real numbers between 4 and 6, but
not 4 and 6
[4,6]: all real numbers between 4 and 6,
including 4 and 6
(-3, ∞): all real numbers between -3 and
infinity, but not -3
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Applications of Sets
Venn Diagrams
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Venn Diagrams: useful way of
visualizing the possible relationship that
can exist between various sets.
The set is represented by A and the
universal set is represented by U.
U
A
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Venn Diagrams
The set is represented by green color A’ is
the complement of set A: includes all
elements that does not belong to A.
U
A’
A
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Union of Sets
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The union of sets A and B, written as
A U B is the set that contains all the
elements of A as well as all the
elements of B.
A U B is represented by the shaded
area. In terms of set notation, suppose
A = {1, 2, 3} and that B = {d, f, g}.
Then A U B = {d, f, g, 1, 2, 3}
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Union of Sets
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AUB
A
B
A={0,1, 2, 3}
B={d, f, g}
A U B={d, f, g, 0, 1, 2, 3}
U
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Union of Sets
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AUB
A={0,1, 2, 3}
B={d, f, g,1,2,3}
A U B={d, f, g,0,1, 2, 3}
U
A
B
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Intersection of Sets
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U
A∩B
A
B
x
A={0,1, 2, 3}
B={d, f, g,1,2,3}
Members of A are members of B: 1, 2, 3
A ∩ B = {1, 2, 3}
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Counting formula
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Strictly speaking A U B=A + B – A ∩ B
A={0,1, 2, 3}
B={d, f, g,1,2,3}
A U B={d, f, g,0,1, 2, 3}
A ∩ B={1, 2, 3}
U
A
B
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Example Venn Diagrams
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A total of 1000 students at Concorde Career College were
surveyed to determine their course-scheduling preferences. The
survey results were as follows:
620 students liked morning classes
320 students liked afternoon classes
230 students liked both morning and afternoon classes
Of those students surveyed, how many:
Like morning classes?
Like afternoon classes
Like morning classes but do not like afternoon classes?
Like afternoon classes but do not like morning classes?
Like neither morning nor afternoon classes?
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Using numbers to show
opposites
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Can you give me an examples of opposite situations
in life?
In mathematics, signed numbers are used to
represent quantities that are opposites.
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Examples:
On a temperature scale, the opposite of 10 above
zero is 10 below zero;
A profit of $100 is the opposite of losses of $100 etc.
On the real number line, +4 (read “positive 4”) is the
opposite of – 4 (read “negative 4”). Both +4 and – 4
are the same distance from zero.
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Using numbers to show
opposites
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We can say that for every positive number there is a
negative number, which is the same distance from
zero on the opposite side of the zero. Opposites have
the following properties
For any real number a,  ( a) = a
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The sum of a real number and its opposite is
zero
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Example: -1 is the opposite of 1 and vice-versa
Example: -1 + 1 = 0
Graphical representation
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Absolute Value
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The absolute value of a number is the distance between 0
and that number on a number line. Another way to put it
is that the absolute value of a number is its numerical
value, regardless of its sign.
In any pair of opposites, the positive number in the pair is
the absolute value of each of the numbers. There is a
symbol used to express the absolute value of the number.
For any number a,a is how to write an absolute value
of a. The absolute value of a (written as a) can be
found from the rule:
If a  0, then, |a|=a
If a  0, then, |a|=  a
|5 |= 5
|-5|= -(-5)=5
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Mathematical Operations
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The four basic operations used in arithmetic as well
as in algebra are:
 + addition, usually written as a + b. It means the
sum of a and b; a plus b; a added to b.
 - subtraction, usually written as a - b. It means
the difference of a and b; a minus b; b less than
a;
 * multiplication, usually written as a * b or ab. It
means the product of a and b; a times b.
 / or , division, usually written as a/b or a:b. It
means the quotient of a and b; a divided by b; a
into b.
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Fractions
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The terms of a fraction are numerator
and denominator. Thus, in the fraction
a/b (which we read as “a divided by
b”), a is the numerator and b is the
denominator.
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Type of Fractions
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Proper fraction a/b: when a<b
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Improper fraction a/b: when a>b
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Example: 1/2, 3/4, 5/7, etc.
Example: 5/3, 6/5 etc.
Mixed number c a/b: where c is a whole
number and a/b is proper fraction
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Example: 3 1/2, 5 2/3
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Equivalent fractions
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Equivalent fractions are fractions that
have the same value, but not the same
numerator and denominator. There are
two ways to find an equivalent fraction.
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Equivalent fractions
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Multiply the numerator and the
denominator by the same number. That
number cannot be zero. Why?
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Example: 3/4=3*2/4*2=6/8;
3/4=3*3/4*3=9/12. The fractions: 3/4,
6/8, 9/12 are all equivalent. Converting a
fractions to an equivalent fractions with a
larger denominator is called building up the
fraction.
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Equivalent fractions
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Divide the numerator and the denominator by
the same number- common factor. That
number cannot be zero. Why?
 Example: 6/8=6÷2/8÷2=3/4. In this case
2 is the common factor of 6 and 8. We
have to continue dividing until the
numerator and denominator have no
common factors greater then 1.
 3/4 is in lowest terms or written in simplest
form.
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Greatest Common Factor
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To find the simplest form of a fraction is
to look for the greatest common factor
(GCF) of the numerator and
denominator.
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Example: Let’s find the simplest form of
6/24. The factors of 6 are: 1,2,3,6; the
factors of 24 are: 1,2,3,4,6,8,12,24. The
GCF of 6 and 24 is 6. So, 6÷6=24÷ 6=1/4;
So ¼ is the simplest form of 6/24.
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Addition and subtraction of
fractions with same denominator
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When the fractions have same
denominators
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Example: 2/5+1/5=3/5; Add the
numerators and keep the same
denominator
Example: 2/5-1/5=1/5; Subtract the
numerators and keep the same
denominator
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Addition of fractions with
different denominator
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5/9 + 3/2=?
Find least common denominator: LCD
 Make a list of all multiples of both denominators:
 For 9: 9,18;
For 2: 2,4,6,8,10,12,14,16,18
 LCD is the lowest number that both of these
denominators would be able to divide into evenly ; For
9 and 2 this number is 18; so the common
denominator of the resulted fraction should be 18.
 Multiply the numerator and denominator of the 5/9
with 2 and numerator and denominator of 3/2 with 9:
 5*2/9*2 + 3*9/2*9 = 10/18+27/18= 37/18; this is
improper fraction and should be converted to mixed
number ( explanation of how to do this later in the
lecture).
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Subtraction of fractions with
different denominator
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1/3-1/12=?
Find least common denominator: LCD
For 3: 3,6,9,12;
For 12: 12,
 LCD is the lowest number that both of these
denominators would be able to divide into evenly ;
For 3 and 12 this number is 12; so the common
denominator of the resulted fraction should be 12.
 Multiply the numerator and denominator of the
1/3 with 4 and numerator and there is no need to
multiply second fraction because it already has
denominator is 12.
 1*4/3*4 – 1/12= 4/12 – 1/12 = 3/12
 Reduce to lowest term:1/4 is in its simplest form
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Multiplication of fractions
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Multiplication of fraction is the easiest
of all fraction operations. All you have
to do is to multiply straight acrossmultiply the numerators and the
denominators.
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Example: 4/5 * 2/3 = 4*2/5*3=8/15
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Division of fractions
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Let’s consider following example:
2/5÷1/3=?
Division is a opposite of multiplication;
so we change ÷ to * sign and 1/3 with
its reciprocal 3/1;
So 2/5÷1/3=2/5*3/1 = 6/5
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How to convert mixed number
into improper fraction
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3 1/2=?
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Multiply denominator 2 by the whole
number 3, so 2*3=6;
Add the numerator 1 to the product of the
whole number and the denominator-this
will be your new numerator: 1+6=7
Keep the same denominator; So, the new
improper fraction is 7/2
3 1/2=7/2
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How to convert improper
fraction into mixed number
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Convert 45/4 to a mixed number.
First, do the long division to find the
"regular" number part:11 and the
remainder:1
Since the remainder is 1 and you're
dividing by 4, the fraction part will be
1/4.
so 45/4 = 11 1/4
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How to multiply/divide mixed
numbers
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(4 1/2) * (3 1/3)=?
Convert mixed numbers to improper
fractions
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4 1/2=9/2
3 1/3=10/3
Perform multiplication/division of the two
improper fractions following the rules
9/2*10/3=90/6=15
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How to subtract/add mixed
numbers
 16 1/8 + 5 5/8 = 21 6/8 = 21 3/4
 129/8 + 45/8 = 174/8 = 21 3/4
 8 5/8 – 3 1/12
 LCD = 24
 8 15/24 – 3 2/24 = 5 13/24
 207/24 – 74/24 = 133/24
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Carrying when adding mixed
numbers
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9 5/8 + 13 7/8 = 22 12/8
12/8 can be written in its lowest terms
as 3/2 = 1 ½
So, 22 12/8 = 22 3/2 = 22 + 3/2
22 + 1 ½ = 23 1/2
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Borrowing when subtracting
mixed numbers
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8 1/3 – 4 3/5
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LCD = 15
8 5/15 – 4 9/15
You can not subtract 9/15 from 5/15
We borrow from the whole number 8
8 5/15 = 7+1+5/15 = 7+15/15+5/15
7 + 20/15 = 7 20/15
7 20/15 - 4 9/15= 3 11/15
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Examples
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7 – 2 5/6
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14 6/7+15 ½
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19 2/3 – 11 ¾
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12 8/15 + 18 3/5
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From Fractions to Decimals
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Fraction with a denominator of 10, 100,
1000 etc. can be written as decimals
numbers:
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Examples: 3/10=0.3; 25/100=0.25;
5/1000=0.005
Fractions with a denominator of 100 are
often written as percents
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Examples: 25/100=25%; 3/100=3%;
300/100=300%
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From Decimals to Fractions
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Count the positions to the right of decimals, i.e., tenth’s,
hundredth’s, thousandth’s, ten thousandth’s, etc.
Whatever that number ends up being, that number
becomes your denominator.
The number in the numerator now loses its decimal, any
zeros that remain to the left of any natural number are
eliminated.
 Example: 0.56=? Fraction; the position to the right of
decimal is hundredth’s; hence the denominator of the
fraction will be 100
 So, 0.56 = 56/100
 Example: 1.56=?; the denominator is 100, the
numerator is 56 and the whole number is 1; the whole
number of original decimal now is the whole number of
the mixed number
44
 Hence: 1.56 = 1 56/100=1 14/25
From Decimals to Percents
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Fractions Decimals
Percents
Move decimal point two places
From left to right
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Example: 5.45=?%
Move decimal point two places to the right
5.45=545%
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From Percents to Decimals
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Fractions
Decimals
Percents
Move the decimal point two places
From right to left
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Examples: 3.67% = 0.0367
The process converting decimals to percents and viceversa calls for a two space movement in either
direction. When the spaces are not there, add Zeros
(0’s), as many as necessary, to make that 2 position
movement and proceed.
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Ratio
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Previously, we defined a rational number as
the ratio of two integers.
More general definition of a ratio:
If a and b are any two real numbers, with
b≠0, then the expression a/b is called the
ratio of a and b, or the ratio of a to b.
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Example: During soccer game, 240 tickets were
sold. There were 500 tickets available. Find the
ratio of tickets sold to the total number of tickets.
240/500 = 12/25; after reducing the initial fraction
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Proportion
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A proportion is any statement of equality of
two ratios. The statement a/b=c/d , (where
b, d ≠0) is a proportion;
When two pairs of numbers have the same
ratio we say the they are proportional
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Example: 4/8=1/2
Cross-multiplying: the cross products of a
proportion are equal
If a/b=c/d then a*d=b*c
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Example: 4/8=1/2 , so 4*2=8*1
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Proportion
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Cross products can be used to find
missing numbers in a proportion
2/3 = n/12
2*12 = 3*n
24 = 3n
n=8
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Operations with signed
numbers-Addition
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Adding numbers with like signs: add their
absolute values; the sum has the same sign
as the given numbers
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Example: 23+56=79; (-23)+(-56)=-79
Adding numbers with unlike signs: subtract
the absolute value of the smaller number
from the absolute value of the larger number;
the sum has the sign of the larger number
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Example: -23+56=33; -7 + 5 = -2
50
Operations with signed
numbers-Subtraction
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For any real numbers a and b:
a – b = a + (-b)
To do any subtraction, we can change
to the addition of the opposite
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Example: -5 -3 = -5 +(-3)= -8
Example: -5 - (-3) = -5+ (3) = -2
Example: 5 - (-3) = 5+ (3) = 8
51
Operations with signed
numbers-Multiplication
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To find the product of two real
numbers, multiply their absolute values
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The product is positive if the numbers have
like signs, i.e. -*-=+; +*+=+
Example: (-2)*(-3)=2*3=6
The product is negative if the numbers
have unlike signs, i.e.-*+=Example: (-2)*(3)=2*3=- 6
52
Operations with signed
numbers-Division
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To find the quotient of two nonzero real
numbers, divide their absolute values
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The quotient is positive if the numbers
have like signs, i.e. -÷-=+; +÷+=+
Example: (-8)÷(- 4)=8÷4=2
The quotient is negative if the numbers
have unlike signs, i.e.-÷+=Example: (-8)÷(4)=8÷4=- 2
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Exponents
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For any counting number a,
an =a*a*a*a…*a
n factors

a is called the base; n is exponent

Example: 24=2*2*2*2=16

Common mistake: 24≠2*4=8
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Exponents
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Example: - 42=-(4)2 = -(4)*(4)=-16
The base is 4 and than form the opposite
of the product
Example: (- 4)2 = (- 4)*(- 4) = 16
Example: - (- 4)2 = -(-4)*(- 4) = -16
Example: (-1/5) 3 =(-1) 3 /(5) 3 =-1/125
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Odd and even powers
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Another aspect of exponents is the determination of
the sign of the number we develop from a number
raised to a particular power.
2
 Example: (-1) states that (-1) * (-1) which
results in a positive
The questions becomes then what sign results if we
have (-1)175 ? The rule states
When a negative number is raised to an odd
power, the resulting answer will be negative
and when a negative number is raised to an
even power, the answer will be positive.
175 = -1
 Example: (-1)
174 = 1
 Example: (-1)
56
Square Roots
Finding the square root of a number is the
inverse operation of squaring that number.
So, 52 = 25 and (-5)2 =25
The principle root of √25 = 5
√11+25 = √36 = 6
It is not as: √11+√25 = √11+5
57
Arithmetic expression

The result of writing numbers in a meaningful
combination with the ordinary operations is
called arithmetic expression or simply an
expression.

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Example: (3+2)*5
The parentheses are used as grouping
symbols and indicate which operation to
perform first. Because of the parentheses, the
expressions below have different values:


Example: (3+2)*5=25
Example: 3+(2*5)=13
58
Order of Operations
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
Often there will be more than one operation
or set of grouping symbols in a problem.
TIP: Your may find it helpful in remembering
the order of operations to memorize the
expression: P. E. M. D. A. S, or Please Excuse
My Dear Aunt Sally, which stands for
Parentheses, Exponents, Multiplication,
Division, Addition, Subtraction.
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Order of Operations
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1. Evaluate each exponential expression in order
from left to right
2. Perform multiplication and division in order from
left to right
3. Perform addition and subtraction in order from left
to right
Note: Multiplication and Division have equal priority
in order of operations. If both appear in an
expression, they are performed in order from left to
right.
The same holds for Addition and Subtraction.
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Order of Operations
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Example: 8÷4*3=2*3=6
Example: 9-3+5=6+5=11
Example: 2*(6-1) - 3*4 +42
 Evaluate the parentheses first (6-1)=5
2
 2*5 – 3*4 + 4
2
 do the exponent second 4 =16
 2*5 – 3*4 + 16
 Multiplication from left to right:
 10 – 12+ 16
 Addition and subtraction from left to right:
 -2 + 16 = 14
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Examples:
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
Consider the following example.
Simplify:
4+62+3*5–4
Consider the following example.
Simplify:
6 + 7 + 8  2(3 + 2) * 32 - 6 + 3
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Algebraic expression

Evaluate the following expressions for
x=3, y=-1, z=5
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x-y+z; we substitute x, y, and z with their
values
3-(-1)+5=3+1+5=9
x*(y+z); substitute x, y, and z
3*(-1+5)=3*4=12
63
Distributive Property
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For any real numbers a, b, and c,
a*(b + c)=a*b + a*c

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
a*(b - c) = a*b - a*c



Example: 4*(5 +2) = 4*5 + 4*2
Example: 4(x+2) = 4x+4*2
Example: 4*(5 -2) = 4*5 - 4*2
Example: 4(x-2) = 4x-4*2
Remember: to distribute the negative
sign across the parentheses.
64
Commutative Property


For any real numbers a and b
a+b=b+a


a*b= b*a


Example: 5 + 4= 4 + 5
Example: 5*4=4*5
Valid only for addition and multiplication
65
Associative property
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
For any real numbers a, b, and c
(a + b) + c = a +(b + c)
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(a*b)*c=a*(b*c)
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Example: (3+8) + 5 = 3 + (8+5)
Example: (6*7)*8=6*(7*8)
Valid only for addition and multiplication
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Identity Property
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The numbers 0 and 1 have special
properties
For any real number a,
a*1=1*a=a: Multiplication Identity
a+0=0+a=a: Additive Identity
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Inverse Property
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For any real number a, there is a
number –a, such that:
a + (-a) = 0
For any nonzero real number a there is
a number 1/a such that:
a*1/a=1
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