0.999... = 1?? sir said it in class and my brain refuses. Is there a way to actually see it?

Bhai, sir said today that $0.999\ldots = 1$ (the 9s never stop) and I swear I almost laughed. To me it's obviously a bit less than 1, like 0.999999 and then still some tiny gap left.

Sir did the usual trick: let $x = 0.999\ldots$ , then $10x = 9.999\ldots$ , subtract, you get $9x = 9$ , so $x = 1$ . I can follow each line but it feels like a magic trick, like we're quietly "subtracting infinity". Then he did $\frac{1}{3} = 0.333\ldots$ , so $3 \times \frac{1}{3} = 0.999\ldots = 1$ . That felt like an even bigger cheat, honestly.

My real doubt: if two numbers are different there should be a gap between them. And I can see the gap, it's 0.000...1. Why isn't that a real number?

I don't want to just believe it because the algebra says so. I want to see why. Any picture, number line idea, or other way to think about it? Also, is this only a school maths thing or do actual mathematicians also say this?

10 answers & replies

[+]Kabir Joshi11thSample· 13d ago

Number line argument. If two different numbers a < b, then their average (a+b)/2 sits strictly between them. So any two different numbers have another number between them. Now try 0.999... and 1. Is there any number strictly between them? Try to write one, it has to be bigger than 0.999... so it would need to start 0.999... and then beat it somewhere, which it can't, and smaller than 1. There's nothing to put there. If no number fits between them, they are the same number.

[+]Meera Patel11thSample· 13d ago

About your "0.000...1": that gap needs a last digit, a place where the 1 sits. But the 9s in 0.999... never end, so the zeros in the gap would also have to go on forever, and there is no "end" at which to put the 1. You can't write it down, not because it's hard to write but because it doesn't exist. The gap isn't tiny, it's zero.

[+][–]Yash Tiwari11thSample· 13d agoAuthor
ooh the "no place to put the 1" bit actually hit. okay

[+]Dev Agarwal12thSample· 13d ago

Isn't it more like 0.999... is infinitely close to 1 but never actually reaches it? Like limits, you approach but don't get there.

[+][–]Arjun Nair12thSample· 13d ago
That describes the sequence 0.9, 0.99, 0.999, ... Each of those is less than 1 and they keep getting closer, yes. But "0.999..." with the dots is not any one of those terms. It means the limit of that sequence, and the limit is exactly 1. The "approaches but never reaches" language is about the list of numbers, not about the value the dots stand for. Also, in the real numbers we use at school there are no positive numbers smaller than every $1/10^n$ , so the "infinitely small gap" doesn't exist there.

[+]Aman YadavDropperSample· 13d ago

Try this. Suppose the gap between 0.999... and 1 is some positive number g. Pick n big enough that $1/10^n < g$ . But 0.99...9 with just n nines is already within $1/10^n$ of 1, and 0.999... is at least that close. So the actual gap is smaller than g. Contradiction. So g can't be positive. It's zero.

[+]Arjun Nair12thSample· 13d ago

Also the series view, for completeness: $0.999\ldots = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots = \frac{9/10}{1 - 1/10} = 1$ It's a geometric series with first term 9/10 and ratio 1/10. And your teacher's subtraction trick is only legitimate because that sum has a well-defined value, we are allowed to do arithmetic on it. Good that you're suspicious of the trick, it is an informal version of this.

[+]Rohan Sharma11thSample· 13d ago

tbh the 1/3 one convinced me first and I don't feel bad about it. Also fun fact my brain: 0.5 can be written 0.4999... too then?

[+][–]Kabir Joshi11thSample· 13d ago
Yes! Any number that ends in a terminating decimal has a second representation ending in infinite 9s. 0.5 = 0.4999..., 2 = 1.999.... The decimal system just isn't a one-to-one naming system. It's not weird, it's a quirk of how we write numbers, not of the numbers.
- [+][–]Yash Tiwari11thSample· 12d agoAuthor
  ok so what I'm taking: (1) no number fits between them, (2) 0.999... means the limit, and the limit is 1, (3) my 0.000...1 has no last place. Thanks everyone, sir is right and I'm sad about it 😂

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