Types of Number (Edexcel GCSE Maths): Revision Note

Types of Number

You will come across vocabulary such as

  • Integers and natural numbers

  • Rational and irrational numbers

  • Multiples

  • Factors

  • Prime numbers

  • Squares, cubes and roots

  • Reciprocals

Knowing what each of these terms mean is essential.

What are integers and natural numbers?

  • Integers are whole numbers;

    • They can be positive, negative and zero

    • For example, -3, -2, -1, 0, 1, 2, 3  are all integers

  • Natural numbers are the positive integers

    • They can be thought of as counting numbers

    • 1, 2, 3, 4, … are the natural numbers

      • Notice that 0 is not included

What are rational and irrational numbers?

  • A rational number is a number that can be written as a fraction in its simplest form

    • A rational number can be written in the form a over b

      • where a and b are both integers, and b is not zero

    • All terminating decimals are rational numbers

      • e.g. 0.32 equals 32 over 100

    • All recurring decimals are rational numbers

      • e.g. 0.4444... space equals fraction numerator space 4 over denominator 9 end fraction 

      • Recurring digits are indicated by dots or bars
        e.g.  0.4 5 with dot on top space equals space 0.4 5 with bar on top equals 0.455 space 555 space 555 space... 
        e.g.  0.5 with dot on top 82 4 with dot on top equals 0.5824 with bar on top equals 0.5824 space 5824 space 5824 space...

  • An irrational number cannot be written as a fraction

    • All non-terminating, non-recurring decimals are irrational numbers

    • pi (pi) is an irrational number

      • It has an endless amount of decimal places and there is no pattern or recurring digits

      • Any (simplified) fraction involving pi, such aspi over 2, will also be irrational (as a in a over b is not an integer)

Examiner Tips and Tricks

  • In the calculator paper, use your calculator to your advantage!

    • To determine whether a number is rational or irrational, type it into your calculator and see if it can be displayed as a fraction (with integers)

Worked Example

Explain why 0.3 with dot on top 49 2 with dot on top is a rational number.

The dots indicate the first and last of the recurring digits

0.3 with dot on top 49 2 with dot on top equals 0.3492 space 3492 space 3492 space...

bold 0 bold. bold 3 with bold dot on top bold 49 bold 2 with bold dot on top is a rational number as it is a recurring decimal

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Jamie Wood

Author: Jamie Wood

Expertise: Maths

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.