1=0.9999...

I always knew I didn't really understand numbers, but I thought I had a pretty good handle on basic arithmetic. So this was a bit of a surprise to me.

It's a given that we can multiply (or add, or subtract, or divide) both sides of an equality by the same factor without violating any of the laws of arithmetic operations. So, for instance:

1/2 = .50
2x(1/2)= 2 x .50
1=1

So, if

1/3 = 0.3333....
and
3 x (1/3) = 3 x 0.3333....
therefore
1 = 0.99999...

(Extracted from Strogatz, S. The Joy of X: A Guided Tour of Math, from One to Infinity (Houghton Mifflin Harcourt 2012)

There's lots of Google stuff under "1 = .9999..." if you're interested in what that really means.