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

Let’s take another look at the decimal expansion of 1/7:


This result from a calculator should convince most students that \displaystyle \frac{1}{7} = 0.\overline{142857}. After all, there’s a second 142 after the first 7, and the ending 9 is consistent with rounding up the 857.

So the evidence that \displaystyle \frac{1}{7} = 0.\overline{142857} is persuasive.

But does this prove beyond a shadow of a doubt that this decimal representation is correct?

Sadly, no. Taken by itself, the result of the calculator is also consistent with, to give just one example, \displaystyle \frac{1}{7} = 0.\overline{142857142910235}, which also would truncate after 10 decimal places to the result shown above.

In short, the calculator gives evidence that the decimal expansion is correct, but does not prove that it’s correct.

Which leads to the obvious question: how do we prove it?


One method, which used to be taught in elementary school (I honestly don’t know if this is taught anymore), is by traditional long division:longdivision17

After six steps, we finally get to a remainder that was previously seen (in this case, on the first step). Therefore, we tell students, the subsequent digits have to repeat.

By the way, this is the essence of the proof for why every rational number has either a repeating decimal representation (possibly with a delay, like 0.1\overline{6}) or else a terminating decimal representation. Though a more formal proof would be preferred by professional mathematicians, the idea is simple: in the algorithm for long division for k/n, there are only n possible remainders: 0, 1, \dots, n-1. So we eventually have to arrive at a remainder that was seen before. If that remainder is 0, then the decimal representation terminates. Otherwise, the decimal representation repeats itself.

In my experience, every math major that I’ve ever met intuitively knows that the above theorem is true. After all, they’ve worked intensively with decimals since 5th grade and have seen decimals in the lower elementary grades. However, very few can articulate why it’s true.

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

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