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I came across this question: "Is the decimal $0.1....1$ with $1$ in every $10^i$ an irrational or a rational number?" and I am trying to figure it out, but I don't know where to start. The numbers I construct are $0.1 \cdots 1 \cdots 1$ with one in the first position (the only exception to the $10^i$th rule) , then the 10th, then the 100th, and so on... I am not sure this constitutes a "pattern", such that the number is considered rational but I am not sure I am right. Beginning with the definition if it is rational, it should be able to be written as $p \over q$. I also thought of multiplying many times with $10^i$ but I don't get anywhere with that idea. Any hints on how to approach this?

Bill Dubuque
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    Hint: decimals corresponding to rational numbers are eventually periodic. That is, there may be a (finite) initial stub block, but after that they are periodic. – lulu Jun 27 '23 at 10:19
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    Your number clearly contains arbitrarily long sequences of consecutive zeros. See the answer here to understand how that leads to irrationality. – Sarvesh Ravichandran Iyer Jun 27 '23 at 10:24
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    Your $0.1....1$ is a little misleading for a quick reader and could be closer to $\sum\limits_{i=0}^\infty 10^{-10^i} = 0.10000000010\ldots$ – Henry Jun 27 '23 at 11:02
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    @lcmpereira: Hint: Your number can be expressed as an infinite sum. --- I just got back from an extremely hard aerobic workout, so I might not be thinking very clearly, but I don't see how to apply your hint. – Dave L. Renfro Jun 27 '23 at 11:16
  • @lcmpereira there exist irrational numbers who can be expressed as an infinite sum. There also exist rational numbers who can be expressed as an infinite sum. Your comment is unhelpful. – JMoravitz Jun 27 '23 at 11:38
  • @JMoravitz: In fact, every real number can be expressed as an infinite sum (in which infinitely many terms are nonzero) because every real number has a nonterminating decimal expansion. I thought lcmpereira might have been thinking something like an infinite geometric series with rational first term and rational common ratio has a rational sum, and got confused over the hypothesis not applying (it would apply if all of the in-between sequences were the same). – Dave L. Renfro Jun 27 '23 at 11:49
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    To answer this question, you have to show that, if $n\in\mathbb{N},\ $ then $n\displaystyle\sum_{i=0}^{\infty} {10}^{ {-10}^i} \ $ is not an integer, which should be possible by elementary arguments. – Adam Rubinson Jun 27 '23 at 12:16
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    It seems to me your number is a trascendental one (read about the first study of trascendentals by the french J. Liouville). – Piquito Jun 27 '23 at 14:17
  • In fact it is a Liouville number and hence transcendental. – Peter Jun 29 '23 at 19:23

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