Irrational Numbers (AQA GCSE Maths) : Revision Note

Irrational Numbers

What is a rational number?

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

  • It must be possible to write in the form , where and are both integers

    • cannot be zero

  • This includes all terminating and recurring decimals

    • E.g. 0.15 (which is ) and 0.151515151515... (which is )

  • This also includes all integers

    • can be written as

    • can be written as or

    • can be written as (it is ok to have 0 in the numerator)

What is an irrational number?

• An irrational number is a number that cannot be written in the form , where and  are integers and is in its simplest form

  • A decimal which is non-terminating and non-recurring is an irrational number

  • The number , where is not a square number, is an irrational number

    • This is also known as a surd

What irrational numbers should I know?

• You may be asked to identify an irrational number from a list

• Irrational numbers that you should recognise are π, ,  

  • Any multiple of these is also irrational

    • For example  are all irrational

  • Be careful: is not irrational as it equals

• Most calculators will show irrational numbers in their exact form rather than as a decimal