This blew my mind.  If you asked me a few days ago, I’d have said that 0.999999 (with 9’s repeating to infinity) could never exactly equal 1, since it would always be that little bit less than one.  Sure, it would tend towards one asymptotically, but it would never quite get there.  But here’s a proof someone told me to show that it does actually equal precisely one:

1/9 = 0.111111 (with 1s repeating to infinity)

Multiplying both sides by 9 gives:

1 = 0.999999 (with 9s repeating to infinity)

This seems to make much and little sense, simultaneously!  I guess infinity means an awful lot of nines.  My brain hurts.