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 = = 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 number a + 0 = a = -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) = (-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) Real Numbers Properties of Real Numbers (regentsprep) Properties of Real Numbers Math Properties (purplemath) Math Properties Properties of Equality (hotmath) Properties of Equality Glossary of Properties (dr.math/mathforum) Glossary of Properties