1–1: Number Sets

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

represented as a/b, where both a and b are integers and b 0.

• Includes: • Common fractions • Terminating decimals • Repeating decimals • Integers

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

Irrational Numbers

Numbers that are defined as those that cannot be expressed as a ratio of two integers. These include non-terminating, non-repeating decimals.

Irrational numbers also include special numbers and ratios, such as and .

Real Numbers

• Real numbers include all rational and irrational numbers.

Rational Numbers

Integers

Whole Numbers

Counting Numbers

Irrational Numbers

Ponder the thought...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

Properties of Real Numbers • Closure Property

• Commutative Property

• Associative Property

• Identity Property

• Inverse Property

• Distributive Property

• Properties of Equality

Closure Property

The answer is part of the set. 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

Commutative PropertyCommutative 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 PropertyAssociative 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 PropertiesDo not change the value

1) Additive IdentityWhat do you add to get the same #?

a + 0 = a-6 + 0 = -6

2) Multiplicative IdentityWhat do you mult. to get the same #?

a • 1 = a8 • 1 = 8

Inverse PropertiesUndo an operation

1) Additive Inverse (Opposite)

• a + (-a) = 0

• 5 + (-5) = 0

2) Multiplicative Inverse

(Reciprocal)

The distributive property of multiplication with respect to

addition (or subtraction).

• a(b + c) = ab + bc

• 3(4 + 7) = 3(4) + 3(7)

• 3(2x + 4) = 3(2x) + 3(4) = 6x + 12

Properties of Equality

• Reflexive

a = a

• Symmetric

If a = b, then b = a

• Transitive

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