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Prove that any number, with zeros standing in all decimal places numbered $10^n$ and only in these places, is irrational.

This question I got from the book Problems in Calculus of One Variable by I.A.Maron. I can't get what the question means even. Any insight will be helpful

V.G
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Anwesh Sarkar
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    Can you try to clarify the question a bit? What do you mean by "standing in all decimal places"? As I understand it, this means your number has zeroes in the first, tenth, hundredth, etc.. decimal places and nowhere else. Is this correct? – SescoMath Aug 26 '20 at 02:59
  • see, I myself have no clue what does this question means. I mentioned the source though. – Anwesh Sarkar Sep 11 '20 at 02:51

1 Answers1

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Rational numbers have eventually periodic decimal expansions. Suppose the period is of length $d$, the periodic sequence starts on place $q$ and $10^n>q$. Then the digit on place $10^n+kd$ for every positive integer $k\ge 1$ must be 0. So $ 10^n+kd$ should be a power of $10$ for every $k$, a contradiction.

markvs
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