Distributivity of multiplication over addition for rational numbers

1.2.6 Distributivity of multiplication over addition for rational numbers

Description

Quick Overview

The section introduces the distributive property of multiplication over addition for rational numbers, establishing how it works through various examples.

Standard

In this section, the distributive property of multiplication over addition for rational numbers is discussed. Examples illustrate how multiplying a rational number by a sum can be expressed as the sum of products, and this property is crucial for simplifying expressions in algebra.

Detailed

Distributivity of Multiplication over Addition for Rational Numbers

The distributive property states that for all rational numbers a, b, and c, the following holds:

  • Distributivity Over Addition: a(b + c) = ab + ac
  • Distributivity Over Subtraction: a(b - c) = ab - ac

This means that if you have a rational number multiplied by the sum (or difference) of two other rational numbers, you can distribute the multiplication across the addition (or subtraction). This is a fundamental property in mathematics that enables the simplification and restructuring of expressions to make calculations easier and more manageable. For instance:

Given:
- a = -3/4, b = 2/3, c = -5/6

We can show the distributive property:

  • Example:
  • Calculate (-3/4) × (2/3 + -5/6).
  • By distributive property: (-3/4) × (2/3 + -5/6) = (-3/4) × (2/3) + (-3/4) × (-5/6).
  • Solve the two products separately and then add them together.

In this section, we explore various examples to make this property evident and reinforce its significance in mathematical operations involving rational numbers.

Key Concepts

  • Distributive Property: It allows multiplication to distribute over addition, making calculations easier.

  • Rational Numbers: Fractions that can be expressed as a ratio of two integers.

Memory Aids

🎵 Rhymes Time

  • If you see a sum, and multiplication is near, distribute it well and have no fear!

📖 Fascinating Stories

  • Once there was a number, let's call it a, who had two friends b and c. A wanted to share its gifts equally with b and c, so it multiplied by their total; together, they became an even better sum.

🧠 Other Memory Gems

  • Remember: D for Distributive, A for Addition, S for Subtraction, helps us keep the numbers in the right function!

🎯 Super Acronyms

DAS

  • Distributive add then solve! An easy way to recall how to apply it.

Examples

  • Example 1: a = -3/4, b = 2/3, c = -5/6. Calculate (-3/4) × (2/3 + -5/6)

  • Example 2: Using the distributive property, calculate -2 × (5 + 1) as -2 × 5 + -2 × 1.

Glossary of Terms

  • Term: Distributive Property

    Definition:

    A property that states a(b + c) = ab + ac, allowing multiplication to be distributed over addition or subtraction.

  • Term: Rational Number

    Definition:

    A number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero.