The role of 1

1.2.5 The role of 1

Description

Quick Overview

This section discusses the significance of the number 1 in rational numbers as the multiplicative identity.

Standard

The section illustrates how multiplying any rational number by 1 yields the original number, establishing 1 as the multiplicative identity. It prompts learners to verify this property across different sets of numbers, including integers and whole numbers.

Detailed

The Role of 1

In this section, we explore the foundational role of the number 1 in the context of rational numbers. When any rational number is multiplied by 1, the product is that same rational number. For example:

  • For whole numbers:

5 × 1 = 5

1 × 5 = 5

  • For integers:

−2 × 1 = −2

  • For rational numbers, demonstrating this can be shown with:

3/8 × 1 = 3/8

This property indicates that 1 functions as the multiplicative identity for all numbers in the set of rational numbers. By investigating the multiplicative effects of 1 on integers and whole numbers, students realize that this property holds true across these sets as well. The overarching goal is for students to recognize the importance of the identity element in multiplication for these numerical types and to apply this understanding in broader mathematical contexts.

Key Concepts

  • Multiplicative Identity: The number 1 is the multiplicative identity, meaning any number multiplied by 1 remains unchanged.

  • Rational Numbers: These can include fractions and mixed numbers that represent a part of a whole.

  • Closure Property: Rational numbers maintain equality when multiplied by 1.

Memory Aids

🎵 Rhymes Time

  • One is fun, it makes numbers run; multiply with me to stay the same, that’s my game!

📖 Fascinating Stories

  • Once in a land of numbers, there lived a number named One. Whenever a number was feeling lost, One would come by, and the number would feel just like itself again, proving it was the identity everyone loves.

🧠 Other Memory Gems

  • Remember 'One keeps it won!', meaning that multiplying by one keeps the value the same.

🎯 Super Acronyms

I = Identity, means you keep what you got, when multiplied by 1, your value’s not lost.

Examples

  • 5 × 1 = 5 (demonstrating the property with a whole number)

  • −2 × 1 = −2 (showing how this applies to integers)

  • 3/8 × 1 = 3/8 (illustrating with a rational number)

Glossary of Terms

  • Term: Multiplicative Identity

    Definition:

    A number that, when multiplied by another number, does not change its value, which is 1 in multiplication.

  • Term: Rational Numbers

    Definition:

    Numbers that can be expressed as the quotient of two integers, where the denominator is not zero.

  • Term: Whole Numbers

    Definition:

    The set of non-negative integers including zero.

  • Term: Integers

    Definition:

    Whole numbers that can be positive, negative, or zero.