I acknowledge the existence of non-standard arithmetics in which this isn’t true. If you’re interested in those, then this post is not at your level.

The maths subreddits seem to go through phases of several people asking the same question. It’s often about fake maths. But when it isn’t, it’s usually objecting to the idea – the mathematical fact – that $0.999\dots = 1$.

Let’s examine some of the reasons it’s true, and debunk some of the reasons people think it isn’t.

The case for the truth

The fractions argument

This is by far the simplest and clearest to me: $\frac{1}{3} = 0.333\dots$. This is unambiguously true, and I find it hard to believe that anyone would object to it. You can prove it you must.

If you multiply both sides by 3, you get $\frac{3}{3} = 0.999\dots$ by just applying the basic rules of arithmetic. And three thirds – I very much hope you agree – is definitely 1.

The argument by algebra

The way I learnt it was:

  • Let $x = 0.999\dots$
  • Then $10x = 9.999\dots$
  • Subtracting the two equations gives $9x = 9$, so $x=1$.

This is morally the same as the fractions argument, but has more letters in it so it’s obviously more correct.

The geometric series argument

Let $S = 0.999\dots = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \dots$

This is a geometric series with a first term of a=$\frac{9}{10}$ and a common ratio of $r=\frac{1}{10}$. Its sum to infinity is $\frac{a}{1-r} = \frac{9/10}{1 - 1/10}$, which is 1.

(This may beg the question a little bit.)

An alternative is to say $S = \frac{9}{10} + \frac{1}{10}S$, which means $S$ has to be 1. This is also morally equivalent to the first.

The no-numbers-between argument

Slightly more sophisticatedly, two numbers are equal if there is no real number between them. There is no number greater than $0.999\dots…$ and smaller than 1 – changing any of the digits of $0.999\dots$ would make it smaller.

The case against the objections

“But it could have a 5 after it, like $0.999\dots5$!”

At first glance, this seems reasonable, but there’s a small problem: the thing about “repeating forever” is that there isn’t an after. That’s what forever means. As soon as you change one of the digits, you don’t have $0.999\dots$ any more. If you truncate it, you’ve got a number that’s smaller than $0.999\dots$, even if you put a 5 on the end.

But it’s written differently!

Yes, it is. But that’s an artifact of the decimal system – it’s perfectly possible to write the same number in several different ways (for example, 6 is the same as $3!$ or $\frac{12}{2}$ or $\sqrt{36}$, or $6.000\dots$, or any one of infinitely many others).

There’s no such thing as infinity!

Well, all I can say is that you’ve never tried to count the number of identical objections on the reddits.

Less sarcastically, you can absolutely make a finitist case that $0.999\dots$ is not properly defined. Unfortunately, that means you would need to throw out an awful lot of real analysis and probably most of the last 150 years of maths. This is a practical objection rather than a mathematical one; you can certainly develop an esoteric mathematical system that doesn’t contain infinity, but it’s necessarily incomplete.

$0.999\dots$ represents a process, not a number!

Again, this seems to be a reasonable objection – you can’t just keep sticking on nines forever, you have to stop at some point.

This, though, is conflating mathematics with the ugly constraints of a finite universe – and it’s conflating the result of a process with the process itself. Going back to $\frac{12}{2}$, that represents the result of dividing 12 by 2, not the process. The value of an expression is not the calculations you need to perform, it’s the result of those calculations.

There are more arguments, of course, on both sides. Feel free to drop me an email if you have a favourite I’ve missed out – but be warned that I’m likely to tap the sign.