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Last night I could not sleep because I tried to prove that the first and the last decimal digits of infinity (whether positive or negative) must be $9$. If the biggest number is infinity, it must consist of an infinite succession of the highest valued digit, $9$. Is it possible to prove this conjecture?

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closed as unclear what you're asking by Did, Pedro Tamaroff, Maisam Hedyelloo, Ayman Hourieh, Andrey Rekalo Aug 7 '13 at 20:30

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Infinity is not a number with a decimal expansion. –  Tobias Kildetoft Aug 7 '13 at 19:49
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MSE: where dreams are crushed. –  Git Gud Aug 7 '13 at 19:54
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Why not $F$, as in hexadecimal? –  Karl Kronenfeld Aug 7 '13 at 19:55
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Yet another example of misuse of the term "philosophical". –  Did Aug 7 '13 at 19:56
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Meanwhile, I'm still wondering what color the empty set is. –  Dave L. Renfro Aug 7 '13 at 20:25

3 Answers 3

Infinity is not a real number. It has no representation as a sum of decimal digits. Infinity, as we know it from the real numbers and calculus, simply denotes "larger than any real number". It's a formal addition to the language of the real numbers which allows us to denote that a certain collection of real numbers (be it a set, a sequence, or the range of a function) is eventually larger than any number.

Note that by a similar argument, if $\infty$ had a last digit then for every base $b$, that digit has to be $b-1$. In fact, by similar arguments you can show that every digit must be $b-1$ (otherwise, it's one-before would be "non-maximal").

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Hey you stole my argument at the end. You owe me $\$\infty$. (For those who can't tell, I am kidding about both points.) –  Karl Kronenfeld Aug 7 '13 at 20:16

Infinity is not an actual number, and so doesn't have any digits. It's an abstract concept that means something without limit.

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This is essentially right, but you don't explain in what sense. –  dfeuer Aug 7 '13 at 19:58
    
Ah i understand, so my question should be: is the biggest decimal value a succession of an infinit number of the digit '9'? –  Chriss Aug 7 '13 at 19:59
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Well, there's no biggest decimal value, because it goes on forever! –  Sujaan Kunalan Aug 7 '13 at 20:01
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@Chriss your question implicitly assumes there is a "biggest decimal value", which there is not. And there is no real number whose decimal expansion has infinitely many digits -- 9 or otherwise -- before the decimal point. –  littleO Aug 7 '13 at 20:07
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@Chriss they would be just as infinite as each other –  Sujaan Kunalan Aug 7 '13 at 20:10

The decimal expansion of a non-negative real number can be viewed as an infinite series: $$\sum_{k = m}^\infty a_k 10^{-k},$$ where for each $k$, $a_k \in \{0,\dots,9\}$. A negative real number is just the additive inverse of a positive one. Extending this notion to allow the expansion to continue forever to the left as well as the right, you get $$\sum_{k = -\infty}^{\infty} a_k 10^{-k}.$$ In this case, if for each $N\in \Bbb Z$ there exists an $n\in \Bbb Z$ such that $n<N$ and $a_k\ne 0$, you end up with a series that goes to infinity.

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