Types of Numbers Worksheets | Questions and Revision | MME

# Types of Numbers Worksheets, Questions and Revision

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## 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. See the list below to help you get started.

### What is an Integer?

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

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

### What is a rational number?

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: $\dfrac{3}{8},\,\,\dfrac{1}{6},\,\,\dfrac{19}{5},\,\,\dfrac{12}{1}$

### What is an irrational number?

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: $\pi,\,\,\sqrt{2},\,\,\sqrt{7}$

### Different types of integer:

Square Numbers

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

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

Cube Numbers

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

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: $2,\,\,3,\,\,11,\,\,19,\,\,37$

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

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: $8,\,\,24,\,\,64,\,\,112,\,\,8,008$

### Example 1: Types of Numbers

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 2: Rational Numbers

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

a) $\sqrt{5}$

b) $0.66\dot{6}$

c) $\pi$

d) $-\sqrt{8}$

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

$0.66\dot{6}=\dfrac{2}{3}$

### 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

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

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) $0.33\dot{3}$ is the only rational number as the recurring decimal can be written as a fraction,

$0.33\dot{3}=\dfrac{1}{3}$

### Worksheets and Exam Questions

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