Algebra
Legal Reasons: 
1 
Definition
of Set of Natural Numbers 
Counting numbers
beginning at one and continuing infinitely {1, 2, 3, 4,...} 
2 
Definition
of Set of Whole Numbers 
Set of Natural
Numbers and zero
{0, 1, 2, 3, 4,...} 
3 
Definition
of Set of Integers 
Set of Whole
Numbers and their opposites
{...3, 2, 1, 0, 1, 2, 3,...} 
4 
Definition
of Set of Rational Numbers 
All numbers
which can be written as a quotient of integers including all integers,
fractions, mixed numbers, terminating decimals, and repeating decimals. 
5 
Definition
of Set of Irrational Numbers 
All numbers
which cannot be written as a quotient of integers, including all square
roots of non perfect squares, and all non repeating, non terminating
decimals. 
6 
Definition
of Set of Real Numbers 
All Rational
and Irrational Numbers. 
7 
Addition,
Subtraction, Multiplication, and Division of Real Numbers 
All correct
use of these operations done on the set of Real Numbers will result
in an answer in the set of Real Numbers and can be used as a legal
reason in a geometric proof. Simply state for example: "Addition
fact, over the set of Real Numbers." 
8 
Definition
of Exponents 
y^n = y * y * y *
y... 'n' times.
'n' is the exponent which tells you how many times to multiply the
"base" 'y' times itself.
EX: 3^2 = 3*3=9

9 
Definition
of Absolute Value 
The absolute
value of a number is the positive distance that the number is
from zero. Zero has an absolute value of zero.
a = a if a >= 0 & a if a < 0
EX: 3 = 3
EX: 14 = 14
EX: 3 + 4 = 1 *Note: always simplify inside the
absolute value bars before making the number positive. 
10 
Commutative
Property of +, x 
For all
a,b,c elements of the Reals,
a + b + c = b + a + c
a * b * c = c * a * b
In other words, you can change the order of the numbers when multiplying
or adding, and your answer remains the same. 
11 
Associative
Property of +, x 
For all a,b,c
elements of the Reals,
(a + b) + c = a + (b + c)
(a * b) * c = a * (b * c)
Here, the only things that change are the grouping symbols. Once again,
the answers remain the same, regardless of which group gets added
or multiplied first. 
12 
Distributive
Property of Multiplication 
For all a,b,c
elements of the Reals,
a(b + c) = ab + ac
a(b  c) = ab  ac
Remember, when multiplying a number times a GROUP, always multiply
by everything INSIDE that group. 
13 
Addition
Property of Equality and Inequality 
For all a,b,c
elements of the Reals,
if a = b, then
a + c = b + c.
Also, if a < b, then a + c < b + c.
In order to maintain balance, whatever number you add to one side
of an equation must be added to the other side also.

14 
Subtraction
Property of Equality 
For
all a,b,c elements of the Reals,
if a = b, then
a  c = b  c.
In order to maintain balance, whatever number you subtract from
one side of an equation must be subtracted from the other side also.

15 
Multiplication
Property of Equality and Inequality 
For all a,b,c
elements of the Reals,
if a = b, then
a * c = b * c.
Also if a < b and c>0, then a*c < b*c.
Also if a< b and c<0, then a*c
> b*c. (Notice the inequality
sign changes direction when multiplying by a negative number.)
To maintain the balance of the equation, you must always remember
to multiply BOTH sides of the equation by the same number.

16 
Division
Property of Equality 
For
all a,b,c,elements of the Reals, where
c is NOT equal to zero,
if a = b, then
a/c = b/c.
To maintain the balance of the equation, you must always remember
to divide BOTH sides of the equation by the same number. 
17 
Property
of Opposites 
For
every a element of the Reals,
a + a = 0 
18 
Property
of Inverses 
For every
b element of the Reals,
b * 1/b = 1 
19 
Additive
Identity 
For all x
element of the Reals,
x + 0 = x
ZERO is the only number that you can add to any other number without
changing the other number's identity. That is how it gets its name
as the "additive identity". 
20 
Multiplicative
Identity 
For all x
element of the Reals,
x * 1 = x
ONE is the only number you can multiply by any other number
and not change the identity of the other number. Therefore ONE is
known as the multiplicative identity. 
21 
Substitution
Principle 
If one value
is equivalent to another value, the two can be substituted for each
other in mathematical statements. 
22 
Definition
of Slope 
When a linear
equation is written in "slope intercept
form" it is solved for y and looks like this: y
= mx + b. The number "m", in front of the x,
is the slope, or steepness of the line. The number in the "b"
spot is the y intercept, the place where the line crosses the y
axis.
HORIZONTAL
LINES have zero slope.
VERTICAL LINES have undefined
slope.
LINES WITH NEGATIVE SLOPE appear to be going "downhill"
as you look at them from left to right.
LINES WITH POSITIVE SLOPE appear to be going "uphill"
as you look at them from left to right.

23 
Pythagorean
Theorem 
Given any right triangle
with legs a and b
and hypotenuse c the
following relationship holds:

24 
Quadratic
Formula 
A quadratic equation
in x is an equation involving x to the power 2 but no higher powers
of x.
Examples:
The STANDARD FORM
of a quadratic equation is,
When a quadratic equation has solutions, and the equation is written
in STANDARD FORM, the solutions can be found using the formula below:

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