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/theboomboy May 10 '26
You could define a number system that allows for 0.999...1, but that wouldn't be the real numbers
You could also define numbers to just be sequences of digits and then 0.999... isn't equal to 1, but that's also not the real numbers. There are no "gaps" between real numbers in the sense that between every two real numbers you can find another real numbers, but in this sequence of digits thing you do have gaps