Types of Number

Spec reference: Number - Structure and calculation
Key idea: Recognise and use different types of number including integers, primes, squares, cubes and triangular numbers.

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▶ Types of Numbers

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Integers

An integer is any whole number - positive, negative or zero.

$$\dots -3,-2,-1,0,1,2,3\dots$$

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Square numbers

A square number is the result of multiplying an integer by itself: $n^2=n\times n$

$$1,4,9,16,25,36,49,64,81,100,121,144\dots$$

Key fact

$7^2=7\times 7=49$

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Cube numbers

A cube number is the result of $n^3=n\times n\times n$

$$1,8,27,64,125,216\dots$$

Key fact

$4^3=4\times 4\times 4=64$

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Triangular numbers

Triangular numbers are formed by adding consecutive integers starting from 1.

$$1,3,6,10,15,21,28,36,45,55\dots$$

The $n$th triangular number is: $T_n={\displaystyle \frac{n(n+1)}{2}}$

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Odd and even numbers

• Even numbers are divisible by 2: $0,2,4,6,8,10\dots$
• Odd numbers are not divisible by 2: $1,3,5,7,9,11\dots$

Operation	Result
Even + Even	Even
Odd + Odd	Even
Odd + Even	Odd
Even × Even	Even
Odd × Odd	Odd
Even × Odd	Even

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Rational and irrational numbers

• A rational number can be written as a fraction ${\displaystyle \frac{a}{b}}$ where $a$ and $b$ are integers and $b\ne 0$.
  • Examples: $3,0.5,{\displaystyle \frac{2}{3}},0.\bar{3}$
• An irrational number cannot be written as an exact fraction.
  • Examples: $\sqrt{2},\pi ,\sqrt{5}$

TIP

All terminating and recurring decimals are rational.
Square roots of non-square numbers are irrational.

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Real numbers

Real numbers include all rational and irrational numbers - essentially every number on the number line.

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Exam tips

Watch out for

• $\sqrt{4}=2$ is rational, but $\sqrt{5}$ is irrational
• 0 is an integer and an even number
• 1 is a square number and a cube number

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Test Yourself

Question 1 of 5

Which of these is a prime number?