Why Does 0.999… = 1?
Consider the real number that is represented by a zero and a decimal point, followed by a never-ending string of nines:
0.99999…
It may come as a surprise when you first learn the fact that this real number is actually EQUAL to the integer 1. A common argument that is often given to show this is as follows. If S = 0.999…, then 10*S = 9.999… so by subtracting the first equation from the second, we get
9*S = 9.000…
and therefore S=1. Here’s another argument. The number 0.1111… = 1/9, so if we multiply both sides by 9, we obtain 0.9999…=1.
