I want to prove that $1,999\dots$ Is an element of $\mathbb{Z}$. Here is my try :
$x = 1,9999\dots \\ 10x = 19,9999\dots \\ 10x - x = 18 \\ 9x = 18 \\ x = 18/9 \\ x = 2 $
So $x \in \mathbb{Z}$
I know something is wrong but where ?
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How do you define $1.999 \ldots$ ? – Martin R Mar 01 '18 at 20:19
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1 with a infinity of 9999 after the comma – Amine HANINI Mar 01 '18 at 20:22
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Why do you think something is wrong with what you wrote? – Mathematician 42 Mar 01 '18 at 20:29
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@Mathematician42 because $1,9\dots \neq 2$... – Amine HANINI Mar 01 '18 at 20:32
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Actually you have just proved that they are indeed equal. Well done. And no, there is nothing wrong with that. – Joffan Mar 01 '18 at 20:41
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That's where you are wrong. They are equal! Many people just don't realize that the decimal representation of real numbers is not necessarily unique. – Mathematician 42 Mar 02 '18 at 09:02
1 Answers
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$$x=1+9\sum_{i=1}^{+\infty}10^{-i} $$
$$=1+9\sum_{i=1}^{+\infty}(\frac {1}{10})^i$$
$$=1+9\frac {1}{10}\frac {1}{1-\frac {1}{10}}$$
$$=1+9\frac{1}{10}\frac {10}{9}=2$$
hamam_Abdallah
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