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Repeated decimal.

Proof that 1/7 is a repeated decimal

This blog post serves as an exercise and solution to the following question:

\frac{1}{7}=0.\overline{142857}

In plain English: is the fraction \frac{1}{7} a repeated decimal (0.142857142857142857…)?

Recurring pattern

How can we tackle such a problem? First note that we have to deal with a recurring pattern: a pattern that refers to itself. One useful concept to model such problems is the concept of sequences. We can model the repeated decimal by the following sequence:

0.\overline{142857}=x_n x_{n+1}=0.142857 + 10^{-6} x_n

With x_0=0


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How does that work? How does this sequence approximate the repeated fraction? First, take a look at x_1. x_1=0.142857+10^{-6}\cdot x_0=0.142857. Now, take a look at the following item in the sequence: x_2=0.142857+10^{-6}\cdot x_1=0.142857142857. So, it concatenates “142857” and the recurring pattern constructed so far. By that property we know that the sequence x_n equals the repeated decimal if n approaches \infty.

The proof

We would like to proof that \frac{1}{7}=0.\overline{142857}. This is equal to proving that 1=7 \cdot 0.\overline{142857}. This equals 1 - 7 \cdot 0.\overline{142857} and this is equal to proving that 1 - 7 \cdot \lim\limits_{n \rightarrow \infty} x_n = 0.

Another sequence

Lets introduce another sequence:

a_n = 1 - 7 \cdot x_n

Notice that the statement we need to proof is the same as the following:

1 - 7 \cdot \lim\limits_{n \rightarrow \infty} x_n = 0 \iff \lim\limits_{n \rightarrow \infty} a_n

Now notice that:

a_{n+1}=1-7 \cdot x_{n+1}=1 - 7 \cdot (x_n \cdot 10^{-6} + 0.142857) a_{n+1}=1 - 10^{-6} \cdot 7 \cdot x_n - 0.999999 a_{n+1}=0.000001 - 10^{-6}\cdot 7 \cdot x_n a_{n+1}=10^{-6}\cdot(1 - 7 \cdot x_n)

Thus:

a_{n+1}=10^{-6} \cdot a_n

And thus we know that:

a_{n+k}=10^{-6k} \cdot a_n

Thus (for n = 0):

a_k=10^{-6k} \cdot a_0

From this, we get that:

\lim\limits_{k \rightarrow \infty} a_k = a_0 \cdot \lim\limits_{k \rightarrow \infty} 10^{-6k} = 0

And from this we can deduce the earlier statement, thus:

\frac{1}{7}=0.\overline{142857}

Thus, \frac{1}{7} is indeed equal to the repeated fraction 0.\overline{142857}! If you have any questions or suggestions, feel free to post them below!

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Kevin Jacobs

I'm Kevin, a Data Scientist, PhD student in NLP and Law and blog writer for Data Blogger. You can reach me via Twitter (@kmjjacobs) or LinkedIn.