Converting Recurring Decimals Into Rational Numbers
Converting Recurring Decimals Into Rational Numbers
When the decimal is a recurring decimal, a bit more work is needed to write the fractional part of the decimal number as a fraction.
Example
Question
Write \(\text{0.}\dot{3}\) in the form \(\cfrac{a}{b}\) (where \(a\) and \(b\) are integers).
Define an equation
\[\text{Let } x = \text{0.33333...}\]Multiply by \(\text{10}\) on both sides
\[10x = \text{3.33333...}\]Subtract the first equation from the second equation
\[9x = 3\]Simplify
\[x = \cfrac{3}{9} = \cfrac{1}{3}\]
Example
Question
Write \(\text{5.}\dot{4}\dot{3}\dot{2}\) as a rational fraction.
Define an equation
\[x=\text{5.432432432...}\]Multiply by \(\text{1 000}\) on both sides
\[\text{1 000}x=\text{5 432.432432432...}\]Subtract the first equation from the second equation
\[\text{999}x = \text{5 427}\]Simplify
\[x = \cfrac{\text{5 427}}{\text{999}} = \cfrac{\text{201}}{\text{37}} = \text{5}\cfrac{\text{16}}{\text{37}}\]
In the first example, the decimal was multiplied by \(\text{10}\) and in the second example, the decimal was multiplied by \(\text{1 000}\). This is because there was only one digit recurring (i.e. \(\text{3}\)) in the first example, while there were three digits recurring (i.e. \(\text{432}\)) in the second example.
In general, if you have one digit recurring, then multiply by \(\text{10}\). If you have two digits recurring, then multiply by \(\text{100}\). If you have three digits recurring, then multiply by \(\text{1 000}\) and so on.
Not all decimal numbers can be written as rational numbers. Why? Irrational decimal numbers like \(\sqrt{2}=\text{1.4142135...}\) cannot be written with an integer numerator and denominator, because they do not have a pattern of recurring digits and they do not terminate.
This lesson is part of:
Algebraic Expressions Overview