Lauris

16

Chapter P

Prerequisites

P.2

Properties of Real Numbers
Mathematical Systems
In this section, you will review the properties of real numbers. These properties make up the third component of what is called a mathematical system. These three components are a set of numbers, operations with the set of numbers, and properties of the numbers (and operations). Figure P.8 is a diagramthat represents different mathematical systems. Note that the set of numbers for the system can vary. The set can consist of whole numbers, integers, rational numbers, real numbers, or algebraic expressions.
Whole numbers Input set Integers

What you should learn:
• Identify and use the basic properties of real numbers • Develop and use additional properties of real numbers

Why you shouldlearn it:
Understanding properties of real numbers will help you to understand and use the properties of algebra.

Defined operation Addition Multiplication Subtraction Division Exponentiation Etc.

Rational numbers

Real numbers

Properties Commutative Associative Identity Distributive Etc.

Algebraic expressions

Algorithms Vertical multiplication Long division Etc.

Figure P.8Basic Properties of Real Numbers
For the mathematical system that consists of the set of real numbers together with the operations of addition, subtraction, multiplication, and division, the resulting properties are called the properties of real numbers. In the list on page 17, a verbal description of each property is given, as well as one or two examples.

Section P.2 Properties of Real NumbersLet a, b, and c represent real numbers. Property Closure Property of Addition a b is a real number.

Properties of Real Numbers

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Verbal Description The sum of two real numbers is a real number. Example: 1 5 6, and 6 is a real number.

Closure Property of Multiplication ab is a real number. Commutative Property of Addition a b b a

The product of two real numbers is a real number.Example: 7 3 21, and 21 is a real number.

Two real numbers can be added in either order. Example: 2 6 6 2

Commutative Property of Multiplication a b b a

Two real numbers can be multiplied in either order. Example: 3 5 5 3

Associative Property of Addition a b c a b c

When three real numbers are added, it makes no difference which two are added first. Example: 1 7 4 1 7 4

AssociativeProperty of Multiplication ab c a bc

When three real numbers are multiplied, it makes no difference which two are multiplied first. Example: 4 3 9 4 3 9

Distributive Properties ab a c bc ab ac ac bc

Multiplication distributes over addition. Examples: 2 3 3 4 42 2 3 3 2 2 4 4 2

Additive Identity Property a 0 0 a a

The sum of zero and a real number equals the number itself. Example:4 0 0 4 4

Multiplicative Identity Property a 1 1 a a

The product of 1 and a real number equals the number itself. Example: 5 1 1 5 5

Additive Inverse Property a a 0

The sum of a real number and its opposite is zero. Example: 5 5 0

Multiplicative Inverse Property a 1 a 1, a 0

The product of a nonzero real number and its reciprocal is 1. Example: 7 1 7 1

The operations ofsubtraction and division are not listed above because they fail to possess many of the properties described in the list. For instance, subtraction and division are not commutative. To see this, consider 4 3 3 4 and 15 5 5 15. Similarly, the examples 8 6 2 8 6 2 and 20 4 2 20 4 2 illustrate the fact that subtraction and division are not associative.

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Chapter P

Prerequisites

Example 1Identifying Properties of Real Numbers

Name the property of real numbers that justifies each statement. (Note: a and b are real numbers.) (a) 9 (b) 4 a (c) 6 (d) (e) b 3 8
1 6

5 3

5

9 4 a 4 3

1 2 0 b b 3 8 2 b

Solution
(a) This statement is justified by the Commutative Property of Multiplication. (b) This statement is justified by the Distributive Property. (c) This statement...