The beauty of recurring decimals

Jul 21, 2014

One of the coolest parts about a geometric series is the ability to rationalize and control patterns that emerge from it. Althought these recurring patterns can often seem arbitrary, they can actually be represented in a concise way using basic formulae of geometric series.

For example, the pattern . Without the dot notation, this works out to be ad infinitum, which is annoying when we need a form to work with. How can we express this recurring decimal in a fraction? Well, when we think about it, a recurring decimal is just a geometric series:

Each term is related to its predecessor by a common ratio, . To find this ratio, we divide a term by its predecessor:

As the series progresses, the next term is equated by dividing the current term by 100. We know that the first term, , is 0.50. To find the total sum of a geometric series to terms:

But we want to find the sum of the series to an infinite amount of terms. How do we do this?

We say that tends towards a limiting value as approaches infinity. So, in our case, each term will approximate to zero for large values of , i.e. as . Thus:

Which works out as:

$$ S_{\infty} = \frac{a}{(1 - r)}$$

However, only exists for infinite geometric series in which . This is called a convergent series. So when we apply this equation to our number:

Cool! The recurring decimal can also be represented as .

Let’s try another: ():