Real number system

Graph of an inequality

One variable is represented by either a ray or a line segment on the real number line. The endpoint is not a solution if the variable is strictly less than or greater than a particular value. In those cases, the endpoint is indicated by an open circle. The endpoint is a solution if the variable is either (1) less than or equal to or (2) greater than or equal to a particular value. In those cases, the endpoint is indicated by a closed circle.

Rational numbers

Q = Integer / Integer. The denominator can't be zero. (1/2 = .5)

Whole Numbers

W = Start from zero. 0,1,2,3,4,.......

Adding odd & even numbers

even + even = even odd + odd = even even + odd = odd

Multiplying odd & even numbers

even x even = even even x odd = even odd x odd = odd

Absolute Value

A number is represented by two vertical lines around the number and is equal to the positive value, regardless of sign. How far away are you from zero.

Inequality

A statement in which the value of one quantity or expression is greater than (>), less than (<), greater than or equal to, less than or equal to, or not equal to that of another.

Odd Integers

Not divisible by 2: ( ..., -3, -1, 1, 3, 5, 7, ...).

Identity Property

Adding 0 or multiplying by 1 doesn't change the original value: Addition: 3 + 0 = 3 Multiplication: 3 x 1 = 3

Real numbers

All of the numbers on the number line.

Improper fractions

All other rational numbers; the numerator is greater than or equal to the denominator. Also called mixed numbers d/t they can be written as a whole number with a fractional part. 4/3, 19/17. The mixed numbers are 11/3 and 1 2/17, respectively.

Fractions

All rational numbers can be displayed this way.

Factor

Any counting number that divides into another number with no remainder. Example: 20 --- 1,2,4,5,10, and 20.

Multiple

Any number that can be divided by another number with no remainder. Example: 20 --- 20,40,60,80,etc.

Proper fractions

Are numbers between -1 and +1; the numerator is less than the denominator. 1/2, 3/4, 17/19

Consecutive Integers

Differ by 1: ( n, n + 1, n + 2, ...) n = an integer.

Even Integers

Divisible by 2: ( ..., -4, -2, 0, 2, 4, 6, ...).

Open interval

Does not include any endpoints. Intervals on the number line represent sets of points that satisfy the conditions of an inequality.

Integers

I = Positive and negative numbers. (..., -3,-2,-1,0,1,2,3,...)

Addition property of inequality

If a, b, and c are real numbers and a > b, then a + c > b + c and a - c > b - c.

Transitive property of inequalities

If a, b, and c are real numbers, the following statements are true: If a < b and b < c, then a < c. If a > b and b > c, then a > c.

Order property of real numbers

If x and y are real numbers, then one and only one of the following statements is true: x > y, x = y, or x < y.

Half-open interval

Includes one endpoint.

Closed interval

Includes two endpoints. Example {x | x is any real number}. The graph would be a solid line.

Natural Numbers

N = Numbers you can count. 1,2,3,4,....

Irrational numbers

S = Has a decimal that is nonterminating and does not have a repeating block. Can't be integer / integer. ( .101001000...., pi, square root of 2.) Every real number is either rational or irrational.

Negative Integers

Starting with -1 and decreasing: (-1, -2, -3, ...)

Distributive Property

The first number gets distributed to the ones in parentheses: 2 x (3x4) = (2x3) + (2x4)

Inverse Property

The inverse of addition is subtraction and the inverse of multiplication is division: Additive Inverse: 3 + (-3) = 0 Multiplication Inverse: 3 x 1/3 = 1 The multiplication inverse doesn't work for 0 because division by 0 is not defined.

Associative Property

The numbers can be grouped, or associated, in any order. Addition: 2+ (3+4) = (2+3) + 4 Multiplication: 2 x (3x4) = (2x3) x 4 You can move the parentheses and it won't make a difference.

Commutative Property

The numbers commute, or move. Addition 2+3 = 3+2 Multiplication 2X3 = 3X2

Composite Numbers

The set of integers that are not prime.

Prime numbers

The set of positive integers greater than 1 that are divisible only by 1 and themselves: (2,3,5,7,11,...).