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Why doesn't it called like that? It seems fair, $1$ called $1$ while $0.999$... being the largest number before $1$, and not called $1$ while not look like it is. Let's say it isn't, how would that number look like?

KugBuBu
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    This has been asked many times before (e.g. here). – Ahaan S. Rungta Aug 26 '14 at 15:10
  • This isn't duplicate, I show new argument. – KugBuBu Aug 26 '14 at 15:11
  • The question is duplicate because it is based on the same fallacy : taking the symbolic representation of a number for the number itself. $2$ is the double of $1$ independently of the fact that in roman numerals $2$ is represented as $II$ while $1$ is represented as $I$; otherwise, we have to conclude that $IV$ is not the double of $II$. – Mauro ALLEGRANZA Aug 26 '14 at 15:15
  • This is like asking what is the last item in an infinite series – Some Guy Apr 03 '21 at 18:28

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$0.999999\ldots$ is exactly $1$. It is just another way of writing the number $1$, and is not less than $1$. You can prove this by subtracting $ 1 - 0.999999 \ldots $ and seeing that the answer is $0$. Alternatively, you can prove that there is no "largest number less than 1."

Proof. Note that for any fraction $\frac{p}{q}$ less than one, there is a slightly bigger fraction which is still less than one; in particular, $\frac{p+q}{2q}$ is a fraction such that $\frac{p}{q} < \frac{p+q}{2q} < 1$. $\Box$

Ahaan S. Rungta
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