In mathematics, 0.999... is exactly equal to 1. This seems to be so mysterious. In other words, the symbols '0.999...' and '1' represent the same number. Yet, a number of proofs of this identity have been formulated.
A related mystery is "The mystery of 0.33333".
We all know that 1/3 = 0.3333... repeating. Multiplying both sides of the equation by 3, we find that 1 = 0.9999.... How can this be?
Answer from the book A Passion for Mathematics: The reason that we find 1 = 0.9999... is because it is true. There are numerous mathematical ways to show this involving the sum of an infinite series, but my favorite way doesn't involve too much math. Consider that any two distinct (different) real numbers must have another number in between them. However, there is no number between 1 and 0.9999.... Thus, 1 and 0.9999... are not different numbers. (Of course, this question becomes slightly more complicated when considering alternative number systems, including those that consider an "Absolute Continuum.")
Additional web sites devoted to the beauty and wonder of 0.9999999999999999.... include: