## What you need to know

Numbers come in all different shapes and sizes. We have to know the words used to describe different types of numbers in different scenarios.

“Integer”

An integer is a whole number. This means no decimals or fractions.

Examples of integers: $7,\,\,23,\,\,1,\,\,3,583$

“Rational”

A rational number is any number that we can write as a fraction. Specifically, a fraction that has an integer on the top and the bottom.

Note: just because some number isn’t written as a fraction doesn’t mean it can’t be. For example, any integer is also a rational number, because we can just write it “over 1” as a fraction, see:

$5=\dfrac{5}{1}$

Lots of decimals (but not all!) can also be written as fractions. See:

$0.3=\dfrac{3}{10}$

Examples of rational numbers: $\dfrac{3}{8},\,\,\dfrac{1}{6},\,\,\dfrac{19}{5},\,\,\dfrac{12}{1}$

“Irrational”

An irrational number is any number that we can’t write as a fraction. In other words, it is the opposite of rational. Another way to see irrational numbers is decimals that go on forever and never repeat.

Examples of irrational numbers: $\pi,\,\,\sqrt{2},\,\,\sqrt{7}$

“Real”

A real number is just about any number you’ll come across in GCSE Maths. Integers, rational numbers, and irrational numbers are all real numbers.

Examples of real numbers: $\dfrac{4}{11},\,\,12,\,\,0.3299,\,\,\pi$

*** The rest of the definitions are all different ways of describing certain kinds of integers. ***

“Square”

A square number is the result of multiplying any integer by itself. We denote square numbers like

$\text{“4 squared” }=4\times 4=4^2=16$

You are expected to know the first 15 square numbers. They are shown below.

Examples of square numbers:

$1,\,\,4,\,\,9,\,\,16,\,\,25,\,\,36,\,\,49,\,\,64,\,\,81,\,\,100,\,\,121,\,\,144,\,\,169,\,\,196,\,\,225$

“Cube”

A cube number is the result of multiplying any integer by itself twice. We denote cube numbers like

$\text{“2 cubed” }=2\times 2\times 2=2^3=8$

You should be familiar with the first few cube numbers.

Examples of cube numbers: $1,\,\,8,\,\,27,\,\,64,\,\,125$

“Prime”

A prime number is only divisible by 1 and itself. Every whole number is made up of prime numbers.

Note: the number 1 is not a prime number.

Examples of prime numbers: $2,\,\,3,\,\,11,\,\,19,\,\,37$

“Factor”

A factor is a number that goes into another number. For example, we say that “2 is a factor of 8” because 2 goes into 8 with no remainder:

$8\div 2 = 4$

Most integers have multiple factors.

All the factors of 12: $1,\,\,2,\,\,3,\,\,4,\,\,6,\,\,12$

“Multiple”

A multiple of a number is any value that appears in the times tables for that number. For example, we say that “30 is a multiple of 6” because

$6 \times 5 = 30$

Every number has an infinite number of multiples.

Some multiples of 8: $8,\,\,24,\,\,64,\,\,112,\,\,8,008$

Example: State which of the words below correctly describe the number 3.5.

rational   prime square real

3.5 is rational – it can be written as the fraction $\frac{7}{2}$.

3.5 is not prime – only whole numbers can be prime.

3.5 is not a square number – only whole numbers are square numbers.

3.5 is real – since it is a rational number, we know it is also real.

## Example Questions

The easiest way to consider factors is in pairs: two number that, when multiplied, make 45. We get

$1 \times 45 = 45$

$3 \times 15=45$

$5\times 9 = 45$

There are no more factor pairs, so we’re done. Therefore, the complete list of factors is

$1,\,\,3,\,\,5,\,\,9,\,\,15,\,\,45$

$1^3=1$, so 1 is a cube number.

$3^3=27$, so 27 is a cube number.

$4^3=64$, so 64 is a cube number.

Therefore, 100 must not be. Indeed, the next cube number after 64 is $5^3=125$, which is bigger than 100. As it happens, 100 is a square number, but it is not a cube number.

0.89 is a rational number, because we can write it as a fraction, as shown:

$0.89=\dfrac{89}{100}$