All arguments about .999… = 1 in my mind boil down to what do you believe 1/infinity to be. If you say 0 then they are equal. If you say no, its a really really really… Small number, then they are NOT equal. Some people don’t believe in infinity, some people don’t believe in 1/infinity (other than saying it is zero, which is just saying it doesn’t exist).

People use informal proofs like:

1/3 = .333…

And since 1/3 x 3 = 1

Then 3 x .333… = .999… = 1

The assumption here is that 1/3 in fact exactly = .333…

But if you do the math for your self you can see after every (3) in the .333… There is always a remainder. What we are supposed to believe is some how after infinitely many 3s this remainder vanishes. And why should that be? I would say it’s still there, you could say its remainder is in fact 1/(3 * infinity).

So lets look at the informal proof again:

1/3 = .333… + 1/(3 * infinty)

3 * (.333… + 1/(3 * infinity)) = .999… + 1/infinity = 1

This is not the same as saying .999… = 1 !

All algebraic proofs use this trick to make you forget about the 1/infinity.

.99 * 10 is not 9.99 but rather 9.90

Yet people will say matter of factly .999… * 10 = 9.999…(9) that extra 9/infinity just gets convenirntly thrown in there. Why not = 9.999…(0)? Okay so that number is hard to think about and so is 1/infinity. But the proof is none the less… A cheat.

There are various calculus based proofs that are supposedly more rigorous, but built into the foundation of calculus is the concept of limits which allow us to say 1/infinity is arbitrarily close to zero, so lets just call it zero. For practical purposes, this makes sense. Lets face it if .999… Is not equal to 1, it’s damn close. We all agree. But that’s different than saying it IS EXACTLY 1.

Let me ask you this. Is there a number bigger than infinity? How many rational numbers are? Infinity? Yes. How many real numbers are there? Infinity, yes, but also provably more infinite than the number of rationals. How can a number be bigger than infinity? I don’t know but clearly there can be. So when people try to prove by contadiction that 1 - .999… Can’t exist because it would be the smallest number > zero, yet certainly that number divided by 2 would be a contradiction. Its like arguing infinity can’t exist because than what’s infinity + 1, or infinity * 2.

I have never seen a proof for .999… = 1 that at its basis doesn’t assume 1/infinity = 0. And we all know 1/infinity is undefined, so at best this question is undefined.

Every time I begin to think I am the smartest person in the world, some udiot comes along to prove me even smarter. ;-p