Irrational Numbers (Cambridge (CIE) O Level Maths): Revision Note

Rational and 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 $\frac{a}{b}$, where $a$ and $b$ are both integers

    • $b$ cannot be zero

  • This includes all terminating and recurring decimals

    • E.g. 0.15 (which is $\frac{15}{100}$) and 0.151515151515... (which is $\frac{15}{99}$)

  • This also includes all integers

    • $5$ can be written as $\frac{5}{1}$

    • $-3$ can be written as $\frac{-3}{1}$ or $\frac{3}{-1}$

    • $0$ can be written as $\frac{0}{1}$ (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 $\frac{a}{b}$, where $a$ and $b$ are integers and $\frac{a}{b}$ is in its simplest form

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

  • The number $\sqrt{n}$, where $n$ 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, $\sqrt{2},\sqrt{3},\sqrt{5}$,  

  • If you multiply an irrational number by a non-zero rational number, you get an irrational number

    • For example $2\mathrm{\pi },3\sqrt{2},\frac{3}{4}\sqrt{5}$ are all irrational

  • Be careful: $\sqrt{2}\times \sqrt{2}$ is not irrational as it equals $2$

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