 Types of Numbers Worksheets | Questions and Revision | MME

# Types of Numbers Worksheets, Questions and Revision

Level 1-3

## 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.

## 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 1-3

## 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 1-3

## 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 4-5

## 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 decimal so it is irrational. Such numbers containing irrational roots are called surds.

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

Level 4-5

## 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 1-3

## 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 1-3

## 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 1-3

### 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

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

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.

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

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

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

$2\sqrt{4}=\sqrt{4\times4}=\sqrt{16}=4$

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

$0.\dot{3}=\dfrac{1}{3}$

### Worksheets and Exam Questions

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

Level 1-3 New Official MME

Level 6-7

Level 1-3

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