Why Does 0.999… Equal 1 Exactly?

It is tempting to treat 0.999… as a number that keeps approaching 1 but always remains slightly below it. That description fits 0.9, 0.99, and every decimal with only finitely many nines. It does not fit the repeating decimal itself. The ellipsis says that the nines continue without stopping at some hidden final place.

More precisely, 0.999… denotes the limit of the sequence 0.9, 0.99, 0.999, and so on. After n nines, the gap from 1 is 10 to the power of minus n, which becomes smaller as n grows. If 0.999… were still below 1, a positive real gap would have to remain between them. Yet for any positive number, however small, we can write enough nines to make the remaining gap smaller. The only possible gap is therefore zero.

The familiar algebra says the same thing. Let x equal 0.999…. Then 10x equals 9.999…. Subtracting the first equation from the second gives 9x equals 9, so x equals 1. This does not quietly remove a mythical “last nine”; it uses the definition of an infinite decimal as a limit.

So 0.999… and 1 are not two extremely close real numbers. They are two decimal representations of the same real number. Every finite truncation is below 1, but the infinite notation specifies their limit. The apparent paradox comes from assuming that because every term is below 1, the limit must be below 1 as well.

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