I always love the discussion on this one...

Is .9999~ (repeating to infinity)  exactly equal to 1?

No, it's off by 0.000...1!

No, it is an irrational number and 1 is a rational number/integer

Unfortunately for you both, they are EXACTLY equal. Proof: Let x = 0.99999~ 10x = 9.99999~ (just move the decimal) 10x - x = 9.99999~ - 0.99999~ = 9 (All the nines after the decimal fall off) 9x = 9 x = 1 But we started with x = 0.99999~, ended with x = 1. Thus, 0.9999~ = 1

I'm pretty sure it doesn't work that way txags911.

edit: I guess I should extrapolate a little.

 Take x to equal 0.9999

x = 0.9999
10x = 9.999
10x - x = 9.999 - 0.9999 = 8.9991
9x = 8.9991
x = 0.9999

You can then continue adding 9's towards infinity and it never changes.  What you are implying is as x nears infinity, the number tends towards 1 which is true, however it never reaches 1, just gets really really close.

 Tell me where I'm off, then.

Hey Alex, you mean its off by 0.000...0001, not 0.1111...1111

Please don't make me post my insanely large post again to show you where you're off.

Woops. Let me edit that.

As the number of 9's nears infinity, it gets closer to 1, yes. And if you ever stopped adding 9's, you would not quite be to 1. But when the number of 9's reaches the theoretical point of infinity, the number would be exactly 1.