0.999 is just a geometric series of fractions: 9/10 + 9/100 + 9/1000 etc. Do you remember how to calculate the sum of a convergent geometric series? The formula is a/(1-r), where a is the coefficient and r is the ratio. In this case, a = 9/10 and r = 1/10, so we have

a/(1-r)= (9/10)/(1-(1/10))= (9/10)/(9/10)= 1

If you don't believe that infinite series can be summed, then what is your solution to Zeno's Paradox?[0].

Also, conveniently enough, there's a pretty thorough Wikipedia article on the whole .999 question. [1]

[0] https://en.wikipedia.org/wiki/Infinity#Zeno:_Achilles_and_th... [1] https://en.wikipedia.org/wiki/0.999...