TITLE: Why is $\frac{10}{8.1}$ so weird?
QUESTION [24 upvotes]: Here's something I came across last night:
$$\frac{10}{8.1} = 1.23456790123456790123456790123456...$$
Notice there is no "8" repeating in the decimal. But all the other digits are there, in order: $012345679...$
Why is this?
Also I noticed if I took the square root of it, more weirdness appeared:
$$\sqrt{\frac{10}{8.1}} = 1.111111111... = \frac{10}{9}$$
What in the world is going on here?

REPLY [2 votes]: One more remark on this problem: to compute how many repeating decimals a fraction $\frac{p}{q}$ has, one has to compute the order of $10$ modulo $q$ (because then $\left(10^n-1\right)\frac{p}{q} \in \mathbb{Z}$ for the first time).
If we use the "Lifting the exponent lemma", we get that:
$$v_3(10^n-1) = v_3(10-1)+v_3(n) = 2+v_3(n)  $$
And this should equal $4$ because $81 = 3^4$. So $n = 9$.