Lesson 1.9

Properties of Real Numbers

Addition and multiplication follow invisible rules that you've used your whole life without naming them. The Commutative, Associative, and Distributive properties are the backbone of algebra.

Introduction

You already know that . That fact has a name: the Commutative Property. Naming these properties lets us use them deliberately to simplify complex expressions.

Past Knowledge

You can perform addition, subtraction, multiplication, and division with real numbers.

Today's Goal

Identify and apply the Commutative, Associative, Distributive, and Identity properties.

Future Success

The Distributive Property is used every time you solve an equation or simplify an expression.

Key Concepts

The Big Five Properties

PropertyAdditionMultiplication
Commutative
(order swaps)
Associative
(grouping shifts)
Identity
Inverse

The Distributive Property (The Bridge)

This is the single most important property in algebra. It connects multiplication and addition.

Worked Examples

Example 1: Name the Property

Basic

Identify the property:

1

What Changed?

The order of the two numbers swapped. The grouping and values are the same.

Commutative Property of Addition

Example 2: Apply the Distributive Property

Intermediate

Use the distributive property to expand .

1

Distribute the 4

2

Simplify

Example 3: Using Properties to Compute Mentally

Advanced

Compute mentally.

1

Commutative Property — Reorder

2

Associative Property — Regroup

Common Pitfalls

Subtraction Is Commutative

. The commutative property only applies to addition and multiplication, not subtraction or division.

Confusing Commutative & Associative

Commutative changes the order. Associative changes the grouping (parentheses move). The numbers stay in the same order.

Real-Life Applications

Cashiers use the commutative property every time they rearrange coins to count change faster. Software compilers use the associative property to reorder calculations for speed. Every time you "distribute" a discount across items in a shopping cart, you"re using .

Practice Quiz

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