0.9 Repeating is Equal to 1

9.9(repeating) = 10x
0.9(repeating) = 1x
Subtracting gives you 9 = 9x, or 1=x.

This seems odd because we think the real numbers are repre-
sented by decimals in some exact fashion, when in fact that is not true at
all (this is the most obvious manifestation). Figuring out exactly what
real numbers were took an awfully long time for mathematicians of previous
centuries. One way to think about this particular problem is, 0.9 = 9/10,
0.99 = 99/100 etc., and so what is the number you might expect to get as you
continue that sequence? Mathematically, the limiting value cannot be
anything other than 1, because there is no real number smaller than 1 which
does not eventually also become smaller than one of these fractions
99 . . . 9/100 . . . 0

Edit:

Wrong forum, please move this.