[split] 0.999... equals 1

[CliveStaples]

But it's mathematical chicanery. The mind always wants to make numbers real. Here we go: 5.13 x 10^61. The number of the universe; well, age in Planck time. But the difference between finite and infinite is not a number, it is a perspective. One cannot write 9/10^(n) on to every n quark in the universe to add to to one, just ain't done; so whether or not it really sums to one is a metaphysical concern.

If you want to deal only with finite quantities, then just use some real analysis.

Take x = .999...

If we agree that x is indeed a real number, then by trichotomy either x > 1, x < 1, or x = 1. Clearly x > 1 is false; thus x < 1 or x = 1.

Now, suppose x < 1. Then |1 - x| must be positive; call it c. But for any positive c you give, I can show that c must be smaller. Thus c cannot be positive. Thus x < 1 is false.

Hence x = 1.