Module 8 (M8) - Algebra - Irrational / rational numbers and recurring decimals

A rational number can be written exactly in the form \(\frac{a}{b}\), where 𝑎 and 𝑏 are integers, while an irrational number cannot.

Examples of rational numbers

All recurring decimals (\(0.{\dot{n}}\)) can be written as fractions and are therefore rational.
A terminating decimal can be easily written as a fraction (as in the examples above), but a recurring decimal is not so easily converted to a fraction.

Changing a recurring decimal to a fraction

Example

Write \(0.\dot{6}\) as a recurring decimal. (We know that the answer should be \(\frac{2}{3}\)).

Let r = the recurring decimal.

r = 0.666666…

Multiply r by 10

10r = 6.666666666…

r = 0.666666666…

Subtract

9r = 6

Divide

\(r = \frac{6}{9} = \frac{2}{3} \)

Solution

1. Let r = the recurring decimal
   
   r = 0.18181818181…
   
2. Multiply r by 100 as two digits are repeating
   
   100r = 18.181818181…
   
3. Subtract to get a multiple of r
   
   100r = 18.181818181…
   r = 0.18181818181…
   99r = 18
   
4. Divide to give the required fraction
   
   \(r= \frac{18}{99} = \frac{2}{11}\)
   

Answer

\(\mathbf{\frac{2}{11}}\)

Irrational numbers cannot be written as a fraction.

The types of irrational numbers that should be known are:

• Square roots (of numbers that are not square) e. g. √2, √5, √3.65, etc.
• Multiples and powers of π e. g. \(\frac {\pi}{2}, 5\pi, 2\pi^2\) etc.

Example

Which of these numbers are irrational?

0.635 18π √400 \(\mathbf{0.\dot{1}\dot{2}}\) √28

Solution

18π, √28 are irrational. The others can be written in the form \(\frac{a}{b}\)

\(0.635 = \frac{635}{1000}\) \(\sqrt{400} = 20 = \frac{20}{1}\) \(0.\dot{12}… = \frac{4}{33}\)