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 General Math Legal Reasons: 1 Definition of Prime Number A prime number is an integer, other than zero or one, that has only two factors, itself and one. 2 Definition of Even Number An even number is an integer which is divisible by two. All even numbers can be written in the form 2n, where n is an integer. 3 Definition of Odd Number An odd number is an integer which is not divisible by two. All odd numbers can be weritten in the form 2n + 1 or 2n - 1, where n is an integer. 4 Definition of Radius of a Circle The radius of a circle is the distance from the center of the circle to any point on the circle. 5 Definition of the Diameter of a Circle The diameter of a circle is equal to two times the radius, (d=2r).

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:

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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