# What is the representation for a number that is not quite one?

If: $$0.\overline{9999999} \equiv 1$$

Then how would you represent a value that is infinitesimally close to one, but not quite one?

i would have thought: $$1-\frac 1 \infty$$

But i would take that to be: $$0.\overline{9999999} = 1$$

Or do i have to subtract an infinitesimal amount from one?

$$1 - 0.\overline{000000}1$$

$$1 - 1 \times 10 ^{-\infty}$$

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If $x$ is a real number, it's either $1$ or it isn't. If it isn't, then $|1-x|$ is a real number greater than $0$, and that's how exactly far $x$ is from $1$. Not an infinitesimal. If you want a different answer, then you need to work outside the real numbers. –  mjqxxxx Sep 21 '12 at 23:19
@mjqxxxx What is the smallest value x can take satisfying 1-x < 1? –  Ian Boyd Sep 22 '12 at 3:34
@IanBoyd: There isn't one. For any $x$ that satisfies $1-x < 1$, there is always a smaller $x$ (just divide by two!). –  jwodder Sep 22 '12 at 4:21