0.999999... = 1?

What the fuck. Why is that true. They got different numbers in them.

If you rewrite .9999999... as .9 + .09 + .009 + .0009 + ..., and then rewrite those numbers as fractions, you get 9/10 + 9/100 + 9/1000 + 9/10000 + ... So therefore .9999... is really just an infinite geometric series with the first term = 9/10 and the rate =1/10. Using the formula for the sum of an infinite series, which is a/(1-r), you'll get
(9/10)/(1-1/10)
= (9/10)/(9/10)
= 1

Q.E.D.

I thought this up myself at two in the morning one time. Other people have probably done it this way too, since it's the only really solid proof I've ever seen of .999999... = 1.