<s>[INST] How can the decimal expansion of this rational number not be periodic? I just noticed that dividing $1 \div 998$ gives me the apparently non-periodic
$$0.001002004008016032064\ldots ,$$
which is $$10^{-3} + 2\times 10^{-6} + 4\times10^{-9} + 8\times 10^{-12} + \cdots = \sum_{i=0}^\infty 2^i 10^{-3(i+1)}.$$
Since every rational number expands into a finite or periodic decimal expansion, does that become periodic somehow? If so, how?
 [/INST] Emphasis on "apparently".  The same thing happens with $1/7$: $$\begin{array}{crrrrrrrrrl} & 0 & . & 14 \\
+ & & & & 28 & & & & & & =2\times 14 \\
+ & & & & &  56 & & & & & = 4\times 14 \\
+ & & & & & 1 & 12 & & & & =8\times 14 \\
+ & & & & & & 2 & 24 & & & =16\times 14 \\
+ & & & & & & & 4 & 48 & & = 32\times 14 \\
+ & & & & & & & & & & \cdots \cdots \cdots \\
= & 0 & . & 14 & 28 & 57 & 14 & 28 & \cdots & & \cdots\cdots\cdots \end{array}$$
and $14\ 28\ 57$ keeps repeating, although the pattern identified above also continues.
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