0.999... = 1

Maybe this mathematical curiosity is more widely appreciated than I realize. However, I only recently became aware of it, so I will share with you the madness that I've discovered.

It turns out that 0.999... (the “...” means that the 9s go on forever) is equal to 1. And I don't just mean that the two numbers are extremely close to one another or that they are equal by convention. Rather, I'm telling you that the symbols 0.999... and 1 represent the exact same real number and that this equality can be proven.

If you're learning of this for the very first time, I can guess how you are feeling:

I was feeling the exact same way, and I knew what had to be done. Research mode engaged.

After about 20 minutes of intense research (e.g. Wikipedia, various math blogs, etc.), I was entirely satisfied that 0.999… = 1.

Preliminaries

First off, Wikipedia has an entire page on 0.999… = 1. If that doesn’t settle the matter, then I don’t know what will.

Next stop WolframAlpha, a “computational knowledge engine” that knows pretty much everything. For example, WolframAlpha knows the estimated number of atoms in the sun (9 x 10^56), the average undergraduate tuition at Harvard ($33,700 US), the population of Bahrain in 1962 (161,000 people), and even the phase of the moon on the night that Leonardo da Vinci was born (waning crescent). WolframAlpha also knows that 0.999… is equal to 1.

Evidence

Of course, the above arguments are really only appeals to authority, not actual evidence. So let’s get down to business.

Hopefully we can all agree that

1/9 = 0.111…

If so, we can also agree that

9 * 1/9 = 9 * 0.111...

We’ve simply multiplied both sides of the equation by 9, so the equality still holds true.

If we now simplify both sides of the equation, we are left with

1 = 0.999…

Crazy, right?! Another way to think about this is to ask: is there any number that can fit between 0.999… and 1? If two numbers are truly different, then at least one other number should fit in between them. But nothing fits between 0.999... and 1. For example, if we subtract 0.999... from 1, we get

1 - 0.999... = 0.000...1

The “...” on the right side of the equation represents an infinite number of 0s, so the 1 at the end is irrelevant. Subtracting 0.999... from 1 leaves us with nothing – a big fat goose egg.

You might have guessed that this strange phenomenon is not limited to 0.999… and 1. For the same reasons described above

1.999... = 2
0.42999... = 0.43
99.999... = 100, etc.

In fact, every nonzero number that ends in an infinite number of 0s (e.g. 1 can also be written as 1.000…) has a counterpart that ends in an infinite number of 9s.

See here and here for more details on this mathemagical weirdness.