Thoughts on 1/7 and other rational numbers (Part 4)

In Part 3 of this series, I considered the conversion of a repeating decimal expansion into a fraction. This was accomplished by an indirect technique which was pulled out of the patented Bag of Tricks. For example, if x = 0.\overline{432} = 0.432432432\dots, we start by computing 1000x and then subtracting.

1000x = 432.432432\dots

x = 0.432432\dots

999x = 432

x = \displaystyle \frac{432}{999} = \displaystyle \frac{16}{37}

As mentioned in Part 3, most students are a little bit skeptical that this actually works, and often need to type the final fraction into a calculator to be reassured that the method actually works. Most students are also a little frustrated with this technique because it does come from the Bag of Tricks. After all, the first two steps (setting the decimal equal to x and then multiplying x by 1000) are hardly the most intuitive things to do first… unless you’re clairvoyant and know what’s going to happen next.

In this post, I’d like to discuss a more direct way of converting a repeating decimal into a fraction. In my experience, this approach presents a different conceptual barrier to students. This is a more direct approach, and so students are more immediately willing to accept its validity. However, the technique uses the formula for an infinite geometric series, which (unfortunately) most senior math majors cannot instantly recall. They’ve surely seen the formula before, but they’ve probably forgotten it because a few years have passed since they’ve had to extensively use the formula.

Anyway, here’s the method applied to 0.\overline{432}. To begin, we recall the meaning of a decimal representation in the first place:

0.432432432 \dots = \displaystyle \frac{4}{10} + \frac{3}{100} + \frac{2}{1000} + \displaystyle \frac{4}{10^4} + \frac{3}{10^5} + \frac{2}{10^6} + \displaystyle \frac{4}{10^7} + \frac{3}{10^8} + \frac{2}{10^9} + \dots

Combining fractions three at a time (matching the length of the repeating block), we get

0.432432432 \dots = \displaystyle \frac{432}{10^3} + \displaystyle \frac{432}{10^6} + \frac{432}{10^9} + \dots

This is an infinite geometric series whose first term is \displaystyle \frac{432}{10^3}, and the common ratio that’s multiplied to go from one term to the next is \displaystyle \frac{1}{10^3}. Using the formula for an infinite geometric series and simplifying, we conclude

0.432432432 \dots = \displaystyle \frac{ \quad \displaystyle \frac{432}{1000} \quad }{\quad 1 - \displaystyle \frac{1}{1000} \quad}

0.432432432 \dots = \displaystyle \frac{ \displaystyle \quad \frac{432}{1000} \quad}{ \displaystyle \quad \frac{999}{1000} \quad}

0.432432432 \dots = \displaystyle \frac{ 432}{ 999}

0.432432432 \dots = \displaystyle \frac{ 16}{ 37}

green line

For what it’s worth, the decimal representation could have been simplified by using three separate geometric series. Some students find this to be more intuitive, combining the unlike fractions at the final step as opposed to the initial step.

0.432432432 \dots = \left( \displaystyle \frac{4}{10} + \frac{4}{10^4} + \displaystyle \frac{4}{10^7} + \dots \right)

\quad \quad \quad \quad + \left( \displaystyle \frac{3}{100} + \frac{3}{10^5} + \displaystyle \frac{3}{10^8} + \dots \right)

+ \left( \displaystyle \frac{2}{1000} + \frac{2}{10^6} + \displaystyle \frac{2}{10^9} + \dots \right)

0.432432432 \dots = \displaystyle \frac{ \quad \displaystyle \frac{4}{10} \quad }{\quad 1 - \displaystyle \frac{1}{1000} \quad} + \frac{ \quad \displaystyle \frac{3}{100} \quad }{\quad 1 - \displaystyle \frac{1}{1000} \quad} + \frac{ \quad \displaystyle \frac{2}{1000} \quad }{\quad 1 - \displaystyle \frac{1}{1000} \quad}

0.432432432 \dots = \displaystyle \frac{ \quad \displaystyle \frac{4}{10} \quad }{\quad \displaystyle \frac{999}{1000} \quad} + \frac{ \quad \displaystyle \frac{3}{100} \quad }{\quad \displaystyle \frac{999}{1000} \quad} + \frac{ \quad \displaystyle \frac{2}{1000} \quad }{\quad \displaystyle \frac{999}{1000} \quad}

0.432432432 \dots = \displaystyle \frac{ 400}{ 999} + \frac{30}{999} + \frac{2}{999}

0.432432432 \dots = \displaystyle \frac{ 432}{ 999}

0.432432432 \dots = \displaystyle \frac{ 16}{ 37}

green lineFinally, this direct technique also works for repeating decimals with a delay, like 0.41\overline{6}.

0.41666\dots = \displaystyle \frac{4}{10} + \frac{1}{100} + \left( \frac{6}{1000} + \frac{6}{10^4} + \frac{6}{10^5} + \dots \right)

0.41666\dots = \displaystyle \frac{4}{10} + \frac{1}{100} + \displaystyle \frac{ \quad \displaystyle \frac{6}{1000} \quad }{\quad 1 - \displaystyle \frac{1}{10} \quad}

0.41666\dots = \displaystyle \frac{4}{10} + \frac{1}{100} +\displaystyle \frac{ \quad \displaystyle \frac{6}{1000} \quad }{\quad \displaystyle \frac{9}{10} \quad}

0.41666\dots = \displaystyle \frac{4}{10} + \frac{1}{100} +\frac{6}{900}

0.41666\dots = \displaystyle \frac{375}{900}

0.41666\dots = \displaystyle \frac{5}{12}

2 thoughts on “Thoughts on 1/7 and other rational numbers (Part 4)

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