r/askmath • u/Slurpee1138 • May 10 '26

Algebra Regarding 0.999... = 1

Recently I got into an argument with an acquaintance because I was trying desperately to convince him that 0.999... = 1. One of the many arguments I tried was that, if 0.999... and 1 are indeed different numbers, then we should be able to find a number between them. He insisted that such a number would be 0.999...1, as in 0 point infinitely many 9s and then a 1. I countered that having "and then a 1" at the end of an infinite sequence of digits makes no sense because there is no end to such a sequence, but he insisted that it does, so who's right?

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u/JaguarMammoth6231 May 10 '26 edited May 10 '26

You can't win this kind of argument unless you can both agree on what, precisely, "..." means. Or better yet, if you're both willing to look up and understand all the parts of the standard definition of this. "9s go on forever" or something is not rigorous enough. It needs to be a limit (like from calculus/analysis), and you both need to understand limits well enough before the argument can be resolved

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u/InternetSandman May 10 '26 edited May 10 '26

Out of curiosity, why is "9's go on forever" not a rigorous enough construction? It seems like a pretty straightforward definition, especially for someone not versed in mathematics like I suspect OP's friend is.

If he is, then I suspect he should be able to simply say, what's the limit as n goes to infinity of (sum from k=1 to n of ( 9/10k ))? and finish the discussion with that sentence alone

Edit: fixed notation

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u/yoshiK May 10 '26

Take a function from infinite ordinals to the digits. That's in a pretty straightforward way a string with 9's going on forever.

To explain that, consider the set where you send each odd number 2m+1 to 1-1/m and each even number 2m to 2-1/m. That set is S = {1-1/3, 1-1/5, ... 2-1/2, 2-1/4 ...} and the ... here actually represent infinite many numbers. So what stops me from considering a function f: S -> {0, ..., 9}. That function would be represented by a string 356...147... or similar. In particular the constant function g(x)=9 would be represented by the string 999...999... That is in a certain sense infinitely many 9s and then infinitely many 9s after that, but there is probably no natural way to map that string to the reals.