# Types of Numbers

GCSEKS3Level 1-3Level 4-5AQACambridge iGCSEEdexcelEdexcel iGCSEOCRWJEC

## Types of Numbers

Understanding the different types of numbers is key to other areas of maths. Types of numbers is all about terminology and knowing what each number actually is.

Level 1-3GCSEKS3AQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

## Type 1: Integers vs Non-Integers

The word integer is just another way of saying whole number. This is a number with no decimals or fractions.

Examples of integers: $7,\,\,23,\,\,-11,\,\,3,\,0,\, 583$

Non-integer numbers is just a way of referring to all numbers that are not whole numbers.

Examples of non-integers: $0.25,\,\,-5.5,\,\,\pi,\,\,\dfrac{1}{3},\,\, \sqrt{2}$

Level 1-3GCSEKS3AQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

## Type 2: Special Integers

There are some notable integers that you should be able to recognise, these include…

• Square Numbers

A square number is the result of multiplying any integer by itself.

Examples of square numbers: $1,\,\,4,\,\,9,\,\,16,\,\,25,\,\,36,\,\,49,\,\,64,\,\,81,\,\,100$

• Cube Numbers

A cube number is the result of multiplying any integer by itself twice.

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

• Prime Numbers

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

Examples of prime numbers: $2,\,\,3,\,\,5,\,\,7,\,\,11,\,\,13,\,\,17,\,\,19,\,\,23$

Level 1-3GCSEKS3AQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE
Level 1-3GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

## Type 3: Rational Numbers

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.

Numbers that are rational include:

• Integers – all integers are rational numbers as they can be written as a fraction over $1$ e.g. $6=\dfrac{6}{1}$
• Decimals – decimals that are recurring e.g. $0.16\dot{6}$ or terminate e.g. $0.375$ are rational.
• Fractions – all fractions that are in the form $\dfrac{a}{b}$ where $a$ and $b$ are integers e.g. $\dfrac{2}{3}$

Remember: Just because a number isn’t written as a fraction doesn’t mean it can’t be.

Level 4-5GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

## Type 4: Irrational Numbers

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.

• Square roots – if the square root of a positive whole number is not an integer then it is irrational, i.e. $\sqrt{9}=3$ is an integer whereas $\sqrt{3}=1.732050808...$  is a non-terminating and non-repeating decimal so it is irrational. Such numbers containing irrational roots are called surds.

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

Level 4-5GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE
Level 1-3GCSEKS3AQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

## Factors

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 $12$ are: $1,\,\,2,\,\,3,\,\,4,\,\,6,\,\,12$

## Multiples

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 $8$ are: $8,\,\,24,\,\,64,\,\,112,\,\,888, \,2008$

Level 1-3GCSEKS3AQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

## Example 1: Types of Numbers

State which of the words below correctly describe the number $3.5$

rational, prime, square

[2 marks]

• $3.5$ is rational as it can be written as the fraction $\dfrac{7}{2}$.
• $3.5$ is not prime as only whole numbers can be prime.
• $3.5$ is not a square number as only whole numbers are square numbers.
Level 1-3GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

## Example 2: Rational Numbers

State which of the following numbers is rational giving a reason for your answer.

$\sqrt{5}, \quad 0.\dot{6}, \quad \pi, \quad -\sqrt{8}$

[2 marks]

• $0.66\dot{6}$ is the only rational number as the recurring decimal can be written as a fraction (as shown below) and fractions are rational numbers.

$0.\dot{6}=\dfrac{2}{3}$

Level 4-5GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

## Types of Numbers Example Questions

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

\begin{aligned} 1 \times 45 &= 45 \\ 3 \times 15 &= 45 \\ 5\times 9 &= 45 \end{aligned}

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

$1,\,\,3,\,\,5,\,\,9,\,\,15,\,\,45$  Gold Standard Education

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

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

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

d) The next cube number after $64$ is $5^3=125$. Therefore, $100$ must not be a cube number.  Gold Standard Education

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

$0.89=\dfrac{89}{100}$  Gold Standard Education

b) $2\sqrt{4}$ is an integer as,

$2\sqrt{4}=\sqrt{4\times4}=\sqrt{16}=4$  Gold Standard Education

c) $0.\dot{3}$ is the only rational number as the recurring decimal can be written as a fraction,

$0.\dot{3}=\dfrac{1}{3}$  Gold Standard Education

## Types of Numbers Worksheet and Example Questions

### (NEW) Types of Numbers - Exam Style Questions - MME

Level 1-3GCSEKS3NewOfficial MME Product