A Story of Numbers

Overview

This chapter traces how number systems developed across cultures and extends our idea of numbers from natural numbers to integers and rational numbers. It examines place value, different bases and the special properties that make our decimal system powerful.

Key concepts

• Number sets: natural numbers, whole numbers, integers and rational numbers, each larger than the last.
• A rational number is any number written as p ÷ q where q is not 0.
• The decimal (base-10) place value system uses powers of 10 and the symbol 0.
• Rational numbers obey closure, commutative and associative properties for addition and multiplication.

Important formulae

• Commutative: a + b = b + a and a × b = b × a
• Distributive: a × (b + c) = a × b + a × c
• Additive identity 0; multiplicative identity 1; additive inverse of a is −a.

Solved example

1. Find a rational number between 1 ÷ 3 and 1 ÷ 2.
2. Make like denominators: 1 ÷ 3 = 2 ÷ 6 and 1 ÷ 2 = 3 ÷ 6.
3. Take the average: (2 ÷ 6 + 3 ÷ 6) ÷ 2 = (5 ÷ 6) ÷ 2 = 5 ÷ 12.

Important questions

1. Insert three rational numbers between −1 and 0.
2. Verify the distributive property using a = 2, b = 3, c = 4.
3. Is the set of integers closed under division? Justify with an example.
4. Write the additive and multiplicative inverse of −5 ÷ 7.

Quick revision

Numbers grew historically to meet new needs; rational numbers fill the gaps between integers. Remember closure, commutative, associative and distributive properties, and that between any two rationals there is always another rational.