The Real Number System

Numbers that are the same distance away from zero, but on opposite sides of the number line. Opposites are also referred to as ADDITIVE INVERSES. Opposites always combine (add) to be 0 (They cancel each other out!). Ex.: -3 and 3 x and -x 3 + (-3) = 0 -x + x = 0

Opposites

A rational number that is equal to the square of another rational number. Examples: 9 is a perfect square, because 3² (or 3 squared...3×3) = 9 36 is a perfect square, because 6² (or 6 squared...6×6) = 36 Non-Examples: 2 is not a perfect square, because you can't multiply a number by itself to get 2. 24 is not a perfect square, because you can't multiply a number by itself to get 24.

Perfect Square

The set of both rational and irrational numbers.

Real Numbers

This set of numbers contains whole numbers and their opposites. {..., -3, -2, -1, 0, 1, 2, 3, ...}

Integers

Numbers that cannot be expressed as the ratio of two integers (as a simple fraction or ratio). These numbers can be represented by a non-terminating, non-repeating decimal. And, these numbers can also be the square roots of non-perfect squares. Pi is the most famous of this type of number. Examples: π, 3.1415926..., √7, 0.131131113... Non-Examples: 0.5, 0, -2, 0.29, -0.375, 1/42, -1/3, 2_1/8, 0.2727..., 25/3, √49, -1

Irrational Numbers

This set of numbers contains the basic counting numbers. {1, 2, 3, ...}

Natural Numbers

An expression that has a square root, cube root, etc. The symbol is √.

Radical

Any number that can be written as a quotient of two integers in the form a/b (b cannot be equal to 0). These numbers include: natural numbers, whole numbers, integers, simple fractions (and mixed numbers), ratios, and certain decimals. In decimal form , they can be either a terminating or a repeating decimal. Finally, these numbers can also be the square roots of perfect squares. Examples: 0.25, 0, -3, 0.23, -0.675, 1/2, -2/3, 1_2/5, 0.3737..., 9/3, √16, -1 Non-Examples: π, 3.1415926..., √3

Rational Numbers

This set of numbers contains the natural numbers and zero. {0, 1, 2, 3, ...}

Whole Numbers

The distance of a number on the number line from 0 without considering which direction from zero the number lies. This value is NEVER negative. The symbol for this is two straight lines surrounding the number or expression. Examples: |-3| = 3 and |3| = 3 * The OPPOSITE of the absolute value is always negative (or 0). Remember, the opposite of symbol (-) just means you are multiplying by a HIDDEN -1. Anytime you multiply something by -1, it turns that term into its opposite. Examples: -|-6| = -6 and -|6| = -6

Absolute Value

A decimal that never repeats and never terminates. It cannot be rewritten as a simple fraction, so it is not a rational number. These types of decimals are irrational numbers. π is the most commonly referred to non-terminated, non-repeating decimal. Examples: 3.1415926..., -√11, 0.121121113..., √2 Non-Examples: 0.35, 0.6, 0.111..., 0.2727..., -2.333...

Non-Terminating, Non-Repeating Decimal

A decimal number that has digits that repeat forever. The part that repeats is usually shown by placing a line (bar notation) over the repeating digit or digits. This decimal can be rewritten as a simple fraction (rational number). We also say that repeating decimals are non-terminating, because they do not stop. Examples: 0.333... or 1/3, 0.666... or 2/3, 0.8333... or 5/6 0.1666... or 1/6, -1.1428571428571428571... or -1/7 Non-Examples: 0.75 or 3/4, 0.8 or 4/5, 3.1415926... or π, √5 or 2.2360679...

Repeating Decimal

A number that, when multiplied by itself, will result in a given number. Examples: √4 (or the principal square root of 4) is equal to 2, because 2 squared or 2² is 4. *A number can have more than one square root if you don't use the √ (radical symbol). For example, the square roots of 25 are ±5. If you square positive 5, or 5², you will get 25. In addition, if you square -5, or (-5)², you will also get 25.

Square Root

A decimal that ends. It's a decimal with a finite number of digits. This decimal can be rewritten as a simple fraction (rational number). We also say that terminating decimals are non-repeating, because they do not repeat forever. Examples: 0.25, 0.3, 0.675 Non-Examples: 0.333..., 0.8333..., 3.1415926...

Terminating Decimal