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 Property Commutative means that the order does not make any difference. a + b = b + a a b = b a Examples = = 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 1) Additive Identity What do you add to get the same #? a + 0 = a = -6 2) Multiplicative Identity What do you mult. to get the same #? a 1 = a 8 1 = 8

Inverse Properties Undo an operation 1) Additive Inverse (Opposite) a + (-a) = (-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