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I was reading Elements of Set Theory by Herbert B. Enderton, and I saw there is written: $$0.1000...=0.0111...=\frac{1}{2}$$

I don't understand how $0.0111...$ is a binary expansion of $\frac{1}{2}$. I'm giving you the text of the book to see if I'm getting it wrong.

Page 149, Academic Press, 2009

  • $0.01111... = \sum_{n\ge 2} 2^{-n} = 2^{-1}$. Maybe the confusion is that this is the binary expansion, not the decimal expansion. – Manlio Jun 06 '23 at 11:06
  • Take $a_n= 0.0\underbrace{1\cdots 1}_n$ so $a_1 = 0.01, a_2=0.011$ and $b_n=0.\underbrace{0\cdots 0}_n1$ so $b_1 = 0.01, b_2=0.001$ then $a_n+b_n = 0.1$ so letting $n$ go to infinity we have $b_n\to 0$ so we get that $0.0111\cdots = 0.1$ – kingW3 Jun 06 '23 at 11:06
  • It is similar to how in decimal notation you have $1/2=0.5000... = 0.49999...$. – Jaap Scherphuis Jun 06 '23 at 11:21

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$0.01111\dots$ means the value of the series $$ \sum_{k=2}^\infty 2^{-k} $$ Do you know how to evaluate that series?

GEdgar
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