Let $$x=0.99999\ldots.$$
Clearly $x$ is a rational number.
I want to find $a,b$ such that $$x=\frac{a}{b}.$$
Clearly $10x-x=9$ and thus $x=1$. So $$1=0.99999\ldots$$
Where is my mistake?
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"Clearly, $x$ is a rational number." Well, it is not so clear if one does not have a clear definition for $0.99\ldots$ and "rational number". :-) – Nov 07 '18 at 14:49
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@RossMillikan: question number 11, asked on the second day that math.se was open. – GEdgar Nov 07 '18 at 14:54
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What makes you think this is wrong? – David C. Ullrich Nov 07 '18 at 15:00
1 Answers
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There is no mistake. $0{,}\overline{9} =1$ is a true statement.
Other proofs of the statement include:
$$\begin{align}0{,}\overline{9} &= 0{,}9 + 0{,}09+0{,}009+\cdots \\&=\sum_{k=1}^{\infty}9\cdot 10^{-k} \\&=9\cdot \sum_{k=1}^\infty 10^{-k} \\&=9\cdot \frac{1}{10-1} = 1\end{align}$$
or, less strictly, the thought that $\frac19=0{,}11111111\dots$, which means that $1=9\cdot \frac19 = 0{,}99999999\dots$.
