Integers and rational numbers: how the number families fit together
Discover Mathematics P6: Term 1, Unit 2, Lessons 1–3
GoalSay which number families a number belongs to and write rational numbers as fractions a/b.
New words
integer · a whole number or its opposite: … −2, −1, 0, 1, 2 …rational number · a number that can be written as a fraction a/b with b not 0counting number · 1, 2, 3, … the numbers we use to countsubset · a smaller family that sits inside a bigger one
Explanation
Numbers belong to families that sit inside each other like boxes. The counting numbers 1, 2, 3 … sit inside the natural numbers, the natural numbers sit inside the integers, and the integers sit inside the rational numbers. Integers are the whole numbers and their opposites: … −2, −1, 0, 1, 2 …. A rational number is any number that can be written as a fraction a/b, where a is an integer and b is not 0. So 4 = 4/1, 0.75 = 3/4, −1.5 = −3/2 and −45 = −45/1. Every integer is rational, but a rational number like 0.5 is not an integer. To name a number, pick the smallest family it belongs to. In this course we treat 0 as an integer and we will not ask whether it is a natural number.
Which families? 7 belongs to all four. −3 is an integer and a rational number, but not a natural or counting number. 0.75 and −1.5 are rational only.
Number
As a/b
4
4/1
0.75
75/100 = 3/4
−1.5
−15/10 = −3/2
−45
−45/1
−0.2
−2/10 = −1/5
Writing rational numbers as a/b. Whole numbers get denominator 1. A decimal becomes a fraction with a power of 10 and is then simplified.
Plotting on a number line. A is −1.5, B is −1/2, C is 1/2 and D is 3/2. A and D are opposites, and so are B and C. Each one is the same distance from 0 as its opposite.
Class activity
Number family sort. Write ten numbers on cards, such as 12, −4, 0.5, −2.25, 100, −1/3, 0, 3/4, −9, 7. Make two piles: integers, and rational numbers that are not integers. Then tell a grown-up why each card is in its pile.
Practice
1
Which families does −8 belong to?
The integers and the rational numbers (−8 = −8/1). It is not a counting or natural number.
2
True or false: every integer is a rational number.
True. Any integer a can be written as a/1.
3
True or false: every rational number is an integer. Give an example.
False. 0.5 = 1/2 is rational but it is not an integer.
4
Write 0.6 and −2.5 as fractions a/b in simplest form.
0.6 = 6/10 = 3/5 and −2.5 = −25/10 = −5/2.
5
Point P is exactly halfway between −3 and −2. Which rational number does it show?
