**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}

**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

**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.

**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}

**Type 5: Multiples and Factors **

**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

**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.

**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}

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### Example Questions

**Question 1:** List all the factors of 45.

**[2 marks]**

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

**Question 2:** Which one of the following is not a cube number?

a) 1

b) 27

c) 64

d) 100

**[2 marks]**

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.

**Question 3:** State whether or not 0.89 is a rational number. Explain your reasoning.

**[2 marks]**

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

0.89=\dfrac{89}{100}

**Question 4:** Show that one of the following numbers is an integer.

a) 2\sqrt{7}

b) 2\sqrt{4}

c) 0.90

d) \sqrt{6}

**[2 marks]**

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

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

**Question 5:** State which of the following numbers is rational.

a) 2\sqrt{2}

b) \pi

c) 0.\dot{3}

d) -\sqrt{5}

**[2 marks]**

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

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