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# Decimal representation of rational numbers

Every number has a unique decimal representation.

• If the prime factorisation of the denominator of a rational number only has $2$s and $5$s, the decimal has finite length. It is called a terminating decimal. $$\frac{1}{2} = 0.\Tblue{5},\quad \frac{1}{5} = 0.\Tblue{2},\quad \frac{1}{8} = 0.\Tblue{125}, \quad\frac{231}{125} = 1.\Tblue{848}.$$
• All other rational numbers have recurring decimals. This means that they have an infinite decimal pattern that repeats itself. The recurring part is often written with a dot or a bar on top. \begin{align*} \frac{1}{3} &= 0.\Tgreen{3}\Tblue{3}\Tviolet{3}\dots = 0.\dot{\Tgreen{3}},\quad \frac{1}{7} = 0.\overline{\Tgreen{142857}},\\ &\frac{231}{162} = 1.4\Tgreen{259}\Tblue{259}\Tviolet{259}\ldots = 1.4\overline{\Tgreen{259}} \end{align*}
• Irrational numbers have infinite non-recurring decimals. $$\sqrt{2}= 1.\Torange{414\;213\;562\dots}, \quad \pi = 3.\Torange{141\;592\;653\dots}$$
Timings in races are given as decimals.