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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
up vote 13 down vote accepted

The real numbers do not have any infinitesimals, so there is no need to represent such a number. There are other fields that do allow them, such as the hyperreal numbers and the surreal numbers and they have ways of representing them. They give up some of the properties of the reals that many find convenient, such as completeness.

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Indeed, they give up the property of decimal representation, which pretty much all of us find convenient. – Gerry Myerson Sep 21 '12 at 23:19
@Gerry, Well, they have Gonshor's sign expansions, so that a good portion of them have something very much like binary. All dots are binary points, not decimal points: "+"=1, "+-"=.1, "+-+"=.11, "+-++"=.111, "+-+++"=.1111, ... However "+-++++++..." refers to a well defined Surreal less than 1 but greater than all of 1/2, 3/4, 7/8,... – Mark S. Feb 7 '13 at 23:41

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