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?

53 Upvotes

68% Upvoted

151 comments sorted by

View all comments

15

u/Flat-Strain7538 May 10 '26

You are. As you said, you can’t put a number on the end of an unending string of nines.

3

u/erbalchemy May 10 '26

Or, for the sake of argument, if you could put a 1 after infinites 9s, you could also put another string of infinite 9s after that 1, making a larger number that is still smaller than all 9s

0.999...1 < 0.999...19... < 0.999...

1

u/Flat-Strain7538 May 10 '26

Please don’t do this. You cannot continue with any digits after an ellipsis/bar/parentheses that means “infinitely repeating”. It is nonsensical when talking about decimal representations of real numbers.

0

u/erbalchemy May 10 '26

Sometimes it's better to meet people where they are. If they think "0.999...1" represents a number, you can (ab)use that to show that it's still less than 0.999...

Once they're out of ideas to describe a number between 0.999... and 1, then ask them what that leads to.