1–2: Properties of Real Numbers

Counting (Natural) Numbers

{1, 2, 3, 4, 5, …}

Whole Numbers

{0, 1, 2, 3, 4, 5, …}

Integers

{…–3, –2, –1, 0, 1, 2, 3 …}

Rational Numbers

• All numbers that can be expressed as a/b, where both a and b are integers and b 0.

• Includes common fractions, terminating decimals, repeating decimals, and integers.

• They do not include non-repeating decimals, such as .

Irrational Numbers

• Those numbers that cannot be expressed as a ratio of two integers

• Includes non-terminating, non-repeating decimals and special numbers, such as π and

Real Numbers•Real numbers include all rational and irrational numbers.

Rational Numbers

Integers

Whole Numbers

Counting Numbers

Irrational Numbers

Ponder the statements...True or False?

• All whole numbers are integers.

• All integers are whole numbers.

• All natural numbers are real numbers.

• All irrational numbers are real numbers.

Classify each of the following numbers using all the terms that apply: natural (counting), whole, integer, rational, irrational, and real.

A) B) 3 C) D) –7

Classifying Numbers

Properties of Real Numbers • Closure Property • Commutative Property • Associative Property • Identity Property • Inverse Property • Distributive Property • Properties of Equality

Closure Property• When you combine any two numbers in a

set, the answer is part of the set.• For example, when you add or multiply real

numbers, the result is also a real number.

a + b is a real number

a x b is a real number • Learn more

Commutative Property

• Commutative means that the order does not make any difference.

a + b = b + a a • b = b • a

Examples

4 + 5 = 5 + 4 2 • 3 = 3 • 2

• The commutative property does not work for subtraction or division.

Associative Property

• Associative means that the grouping does not make any difference.

(a + b) + c = a + (b + c) (ab) c = a (bc)

Examples

(1 + 2) + 3 = 1 + (2 + 3) (2 • 3) • 4 = 2 • (3 • 4)

• The associative property does not work for subtraction or division.

Identity Properties

• Do not change the value!• Additive Identity – When you add zero to

any number, the result is the same numbera + 0 = a -6 + 0 = -6

• Multiplicative Identity – When you multiply a number by one, the result is the same number

a • 1 = a 8 • 1 = 8

Inverse Properties

• Undo an operation• Additive Inverse – when you add a number

and its opposite, the result is 0

a + (-a) = 0 5 + (-5) = 0• Multiplicative Inverse – when you multiply

a number and its reciprocal, the result is 1

Distributive property

Distributive property of multiplication with respect to either addition or subtraction.• a(b + c) = ab + bc

• 3(4 - 7) = 3(4) - 3(7)

• 3(2x + 4) = 3(2x) + 3(4) = 6x + 12

• Reflexive

a = a • Symmetric

If a = b, then b = a • Transitive

If a = b and b = c, then a = c

Properties of Equality

More info…• Real Numbers (mathisfun)• Properties of Real Numbers (regentsprep)• Math Properties (purplemath)• Properties of Equality (hotmath)• Glossary of Properties (dr.math/mathforum)