# 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 $8/17$:

Using this result and the Euler totient function, we concluded that the repeating block had length $16$. So we multiply twice by $10^8$ (since $10^8 \times 10^8 = 10^{16}$) to deduce the decimal representation, concluding that

$\displaystyle \frac{8}{17} = 0.\overline{4705882352941176}$

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

$8 \times 10^8 = 17 \times 47058823 + 9$ and $9 \times 10^8 = 17 \times 52941176 + 8$

Therefore,

$8 \times 10^{16} = 17 \times (47058823 \times 10^8) + 9 \times 10^8$ and $9 \times 10^8 = 17 \times 52941176 + 8$

so that

$8 \times 10^{16} = 17 \times (47058823 \times 10^8) + 17 \times 52941176 + 8$

$8 \times 10^{16} = 17 \times 4705882352941176 + 8$

$8 \times 10^{16} - 8 = 17 \times 4705882352941176$

$8 (10^{16}-1) = 17 \times 4705882352941176$

$\displaystyle \frac{8}{17} = \displaystyle \frac{4705882352941176}{10^{16}-1}$

Using the rule for dividing by $10^k -1$, we conclude that

$\displaystyle \frac{8}{17} = 0.\overline{4705882352941176}$

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