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Mathematics
Basic Mathematics
Elementary Mathematics starts with Arithmetic which of course deals with numbers. The second level of
mathematics is Algebra, an extension of arithmetic, which builds upon the basics and includes symbols that
represent variables.
Trigonometry is the part of mathematics that deals with the properties of a triangle along with the study and
application of trigonometric functions.
Geometry is traditionally the science deals with relationships between lines, points, surfaces, and three
dimensional spaces.
Higher mathematics involves all of the more advanced fields of mathematics beyond Trigonometry. Those
fields include Calculus and non-Euclidean geometry. These higher orders of mathematics deal with the
abstract extensions of elementary mathematics.
Mathematics uses a special type of language that has numerals, letters and symbols. They are controlled by the rules
of logic from each of the aforementioned branches of mathematics. The mathematical symbols and abbreviations
serve as short cuts to visualize mathematical equations.
The original branches of mathematics were arithmetic and geometry.
Both of these have their roots anchored in ancient Babylonia and
Egypt.
Algebra augmented arithmetic in the middle ages through the work of
Arabic mathematicians and astronomers.
In the 10th century trigonometry became a branch of mathematics
rather than a branch of astronomy.
Figure 1
Integers
Our numbering system is called the Arabic system. This system consists of ten symbols or digits: 0,1,2,3,4,5,6,7,8,9.
The value of a number depends on its position. For example:
5000 is read as five thousand
.500 is read as five hundred
.50 is read as fifty
.5 is read as five
.05 is read as five
.005 is read as five
A zero placed before a number does not change the value of the number, a zero placed after the number does! Each
zero placed after the number increases the value of that number by a factor of ten. So two zeros multiply that same
number by one hundred, the third zero multiplies the number by one thousand; and so on.
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We count our numbers from left to right and separate them into groups of three by commas. In this way we can
facilitate the reading of each number.
Decimals
In order to express a value less than one, a decimal point is used. In the Arabic numbering system the value of a
number is multiplied by ten for each position moved from right to left. Take the following examples:
.3 equals three tenths of one.
.03 equals three hundredths of one.
.003 equals three thousandths of one.
.0003 equals three ten-thousandths of one.
With integers, a zero placed before the number does not change its value, but a zero that is placed after the number
does change its value. With decimals the opposite is true;the number is divided by ten as we move from left to right. A
zero placed after the digit does nothing to change the value of that number. The following list shows numbers of equal
value:
.7 equals seven hundred thousands.
.70 equals seven hundred thousands.
.700 equals seven hundred thousands.
.7000 equals seven hundred thousands.
Note: When reading an integer together with a decimal, inject the word “and” between integer and the decimal. For
example 15.352 is read “fifteen and three hundred fifty two thousands”
Addition
Addition is described as the sum of two or more numbers. Be they negative or positive, each number has an absolute
value. When adding numbers of like signs, we simply find the sum and use the same sign.
If the numbers are of different signs we find the difference between the numbers and use the sign of the greater
valued digit. When adding columns of mixed sign numbers you will find it easier to add like signs and then combine
the two signs.
The following are some examples of each:
5+5=8
(-2) + (-3) = -5
(-8) + 5 = 3
(-2) + 6 = 4
The addition axiom:
If equal quantities are given to different labels, the sums will always be equal.
E.g., if a = b then a + 2 = b + 2. The axiom may be expressed as the following:
If a = b and c = d then a + c = b + d.
Subtraction
Subtraction can be described as the inverse operation of addition. In the subtraction process we are removing one
absolute value from that of another.
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The subtraction axiom:
If equal quantities are subtracted from the same or equal quantities, the differences are then equal. e.g., if a = b then
a – 5 = b – 5. Therefore the axiom can be stated as the following:
If a = b and c = d then a – c = b – d.
Multiplication
Multiplication is the method by which numbers or quantities are added a given number of times. For example 2 four
times equals 2 + 2 + 2 + 2; 4 times “a” equals a + a + a + a. Both the commutative and associative properties apply to
multiplication.
Commutative property is expressed as 3 x 2 equals 2 x 3; or ab = ba. The associative property is stated as: abc =
a(bc) = (ab)c.
The Distributive property holds over addition and is expressed as 3(a + b) equals 3a + 3b.
Division
Division is the inverse of multiplication and is the process of determining the number of times that a given number
contains another number.
The division axiom:
The division axiom is an axiom of equality. If equal quantities are divided by the same or equal values, the quotients
are equal. E.g., If a = b then a / 4 = b / 4.
The axiom therefore can be expressed as:
If a = b and c = d then a / c = b / d.
Note: No number can be divided by zero; it is an undefined operation.
Order of Operation
In Algebraic operations there is a specific order that things are to be done. These rules are called the “ Rules of
Operation” and are as follows:
1) If parentheses, powers, or roots are included these operations are performed before anything else
multiplication and division.( xxx ), x2 , √ …
2) Multiplication and division are performed next.” x ” or ” / “
3) Addition and subtraction may be done in any order.” + ” or ” – ”
The Beauty of Math
The video below is the Disney classic Donald Duck in Mathmagic Land, Copyright 1959
Donald wanders into a magical land where the beauty of the laws of mathematics unfold before him. But, because of
copyright rules this film is located on YouTube at the time of publication.
Donald Duck - Donald in Mathmagic Land
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Cartesian measuring coordinates
The Cartesian measuring coordinate system is made up of
“rectangular coordinates”.
This system of measurement was named for the French
Mathematician René Descartes. Descartes was the originator of most
of the basic ideas behind the analytical geometry, what we refer to as
coordinate geometry.
It is a system that is used to find a location of a point in reference to
two or three perpendicular lines.
A point is found in terms of its distance from two of the perpendicular
axis (lines). The horizontal axis is termed the X axis and the vertical
axis is the Y axis on a two dimensional plane.
Figure 2
The center where the two axes intersect is termed the “ origin“. The coordinates of any given location or point in this
system is in ordered pairs (x, y).
The “abscissa” is the name given to the first component “x” axis. The “y” axis component is called the “ ordinate” and it
too represents a value plus or minus along the y axis from the center or origin, Figure 2.
Numbers to the right of the abscissa “x” are considered to be positive or plus and those to the left of origin are in a
negative or minus position. Above origin for the ordinate “y” is the positive or plus direction, and any coordinate below
the intersection with “x” would be considered a minus or negative position.
The intersection of x and y effectively divide the two dimensional plane
into four quadrants, figure 2.
To locate a point at (5,4), first count 5 units in the positive (to the right)
direction along the x axis and then 4 units up the y axis in a positive (up)
direction. Should the coordinates include a negative number, for example
a point “B” at (-5,4) you would count 5 in the negative (left) direction and
then 4 units up the y axis in a positive (up) direction. Also shown in figure
3.
In a three dimensional space, a coordinate would be located in terms of its
relationships to three separate axis X, Y, and Z. The planes created by this
intersection are xy, xz, and yz, figure 4. The intersection of the three axes
is still called the origin.
Figure 3
A point in a three dimensional position is an ordered triple (x,y,z) and position would first be located along the x
axis, second along the y axis and lastly along the z axis.
To plot a point “C” (-5,4,3) first move to the left along x 5 units, up the y axis 4 units and toward you 3 units.
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Prime and Composite Numbers
A prime number is any number that is only divisible by the number one and the number itself. The following numbers
are examples of prime numbers:
2,3,5,7,11,13,17,19,23…etc.
The numbers 9, 10 and 15 are not prime because they can be divided more
than two different numbers; that makes these composite numbers.
9
18
=
=
(3 x 3)
(9 x 2)
=
=
(3 x 3)
(9 x 2)
=
(3 x 3)
= (3 x 3 x 2)
Fractions
A fraction has both a Numerator and a Denominator. The expression written
below the line in a common fraction and indicates the number of parts into
which one whole is divided.
Figure 4
2 is the numerator of the fraction 2 / 3 and 3 the denominator
Reducing Fractions
To simplify or reduce a fraction to its lowest term we need to use prime numbers.
9
18
=
=
(3 x 3)
(9 x 2)
=
=
(3 x 3)
(9 x 2)
= (3 x 3)
=
= (3 x 3 x 2) =
(3 x 3)
(3 x 3) x 2
=
(1) = 1
= (1) x 2 = 2
The fraction 9/18 can be expressed as (3×3) / (2x3x3) and (3×3) / (3×3) = 1
Therefore 9/18 = 1/2
Converting Fractions to Decimals
To convert a decimal number to a fraction, again we look to division, 9/18 is simply 9 divided by 18 or .500.
Converting Decimals to Factions
Converting decimals back into fractions is a little more difficult. The first step is to write the decimal number divided by
one. Take .750 as an example. It would look like this: .75/1
For each number position to the right of the decimal point, multiply both the Numerator (top number) and the
Denominator (the lower number) by a factor of ten. If there are two number positions, you would multiply by a factor
of 100. If you have three positions to the right, then you would multiply by 1000, and so on.
(.75/1) x 100 (there are two decimal places)
After multiplying both the Numerator and Denominator by 100, we have 75/100.
Now that you have converted a decimal into a fraction, you must reduce it to its simplest form. Find the largest
number that will go into both the Numerator and Denominator. In this example, that value is 25.
(75/100) / 25 = 3/4
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The Circle
A Circumference is the distance around a circle. Divide the circle created by the circumference along the center line
and you have the Diameter; divide the diameter if half, and you have found the Radius, figure 5.
Mathematically speaking, the circle is defined as the set of all
points that are the same distance from a given point, called the
center. That distance is called the radius “r”.
The word radius comes from the Latin word for a rod; for
example, the spoke of a wheel can defined as a rod, a spoke or
radius. Using the definition of a circle as: “the set of points all at
the same distance from a given point called the center,” we can
graph a circle, figure 6.
Figure 5
The equation of the circle in figure 6, with its center at the origin, and its radius equal to 3 (the square root of 9) is:
x2 + y2 = 9
The equation comes directly from Pythagorean Theorem: the sum of the squares of the legs of a right triangle is equal
to the square of the hypotenuse; the “x-distance” being one leg and the “y-distance” being the other leg of a right
triangle with hypotenuse “r”.
Pi
The constant pi, is a real number defined as the ratio of a circle’s circumference
“C” to its diameter d = 2r
Pi = C / d = C / 2r
It is also sometimes called Archimedes’ constant and is equal to:
3.1415926535997932384626433832….
In sheet metal calculating the circumference is required to compute the flat size of
cylinder.
Taking these examples
Diameter of 5.750-inches x Pi = 1.8064
Figure 6
and
Diameter of 6.250-inches.6.250 x Pi = 19.643
Negative Numbers
A negative number is any number value less than zero. To work with negative numbers, you should review few rules
of signs.
Addition Rules
Positive + Positive = Positive: 5 + 4 = 9 Negative + Negative = Negative: (- 7) + (- 2) = – 9
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Sum of a negative and a positive number: use the sign of the larger number and subtract
(- 7) + 4 = -3 6 + (-9) = – 3 (- 3) + 7 = 4 5 + ( -3) = 2
Subtraction Rules
Negative – Positive = Negative: (- 5) – 3 = -5 + (-3) = -8
Positive – Negative = Positive + Positive = Positive: 5 – (-3) = 5 + 3 = 8
Negative – Negative = Negative + Positive =
Use the sign of the larger number and subtract (Change double negatives to a
positive) (-5) – (-3) = ( -5) + 3 = -2 (-3) – ( -5) = (-3) + 5 = 2
Multiplication Rules:
Positive x Positive = Positive: 3 x 2 = 6 Negative x Negative = Positive: (-2) x (-8) = 16
Negative x Positive = Negative: (-3) x 4 = -12 Positive x Negative = Negative: 3 x (-4) = -12
Division Rules
Figure 7
Positive ÷ Positive = Positive: 12 ÷ 3 = 4 Negative ÷ Negative = Positive: (-12) ÷ (-3) = 4
Negative ÷ Positive = Negative: (-12) ÷ 3 = -4 Positive ÷ Negative = Negative: 12 ÷ (-3) = -4
Squaring
Squaring a number simply means to multiply any number by itself. Three
times three equals nine, figure 8.
When a number is to be squared it is written like this: X²
Using the example in figure 8 we see:
3² = 3 x 3 = 9
If there is a raised three, it means to cube the value:
3 x 3 x 3 = 27
Figure 8
You can also square negative numbers.
Example: What happens when you square (-5) ?
Answer: (-5) × (-5) = 25
Remember, a negative times a negative gives a positive . When you square a negative number you get a positive
result, just as if you had squared a positive number.
Square Roots
A square root goes the other way: 3 squared is 9, so a square root of 9 is 3. A square root of a number is a value that
can be multiplied by itself to give the original number.
A square root of 9 is 3, because when 3 is multiplied by itself you get 9.
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Algebra
Algebra is traditionally the branch of mathematics in which the basic operations of arithmetic are generalized through
the use of letters (alpha) to represent quantities in formulas and equations.
Algebra was invented by Mohammed al-Khowarizmi in 825 AD in Baghdad, Arabia (Iraq). Al-jabr is the Arabic word for
equation and was so important to European and Asian mathematicians that the name algebra was adopted in various
spellings.
Algebraic Expressions
An algebraic expression is made up of numerals, letters and algebraic symbols (alpha-numeric). The term of an
expression comes from a plus or minus that connects different parts of the expression. If a sign precedes the term it
would be considered the sign of the term. That term would be a negative number as no sign is used to represent a
positive number. If no sign is seen it is automatically assumed to be a positive number. Whether an expression is
3x3yz, x/y, or 2b they each represent a single term. An expression 4x – 3y is said to have two terms, Therefore
(3x3yz) – (x/y) + (2d) would be considered to have three terms.
If an algebraic expression has more than one term it is defined as a polynomial. An expression with one term is a
monomial, an expression with two terms is a binomial; and if it contains three terms, it is considered to be a trinomial.
Algebraic symbols
Assuming the value n as certain number that can be expressed algebraically; using only conventional mathematical
expressions we are able to abbreviate the following mathematical expressions.
1) 6 n = The product of 6 and the number n
2) n/4 = The number divided by 4
3) 8/2 n = 8 divided by 2 times the number.
4) 7(n + 7) = 7 times the sum of 7 plus the number.
5) √( n + 5) = The square root of the sum of the number and 5.
6) n + 4 = The sum of the number and 4.
7) x – y = The difference between x and y.
8) b-(8+a) = The sum of 8 plus a subtracted from b.
9) 2ab = The product of 2 times a times b.
10) a/b = The quotient of a divided by b.
Algebraic Order of Operations
In an equation (a series of algebraic operations) any operation that is a parentheses, roots, or powers are always
done first, followed by multiplications and divisions.
Third the additions and subtractions are performed in any order.
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The expression:
3+9/2
9 is divided by 2 and then added to 3
Here the 3 is the first term and 9 / 2 is the second term.
(3 + 10) / 2
3 is added to 10 and then divided by 2
This expression (3+10)/2 is a single term to be treated as a single division problem.
The theorem that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two
sides.
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