Thoughts on 1/7 and other rational numbers (Part 10)

In the previous post, I showed a quick way of obtaining a full decimal representation using a calculator that only displays ten digits at a time. To review: here’s what a TI-83 Plus returns as the (approximate) value of :

Using this result and the Euler totient function, we concluded that the repeating block had length . So we multiply twice by (since ) to deduce the decimal representation, concluding that

Though this is essentially multi-digit long division, most students are still a little suspicious of this result on first exposure. So here’s a second way of confirming that we did indeed get the right answer. The calculators show that

and

Therefore,

and

so that

Using the rule for dividing by , we conclude that