Math Is Fun Forum

I don't know how to put this in a way you can understand, but it's generally accepted that 0.999... + 1/∞ = 1, people just choose to believe 1/∞ = 0. So we can use this equation to prove the value of 1/3.

0.999... + 1/∞ = 1
(0.999... + 1/∞)/3 = 1/3 <- Divide both sides by 3       
0.333... + (1/∞ * 1/3) = 1/3
0.333... + 1/(∞*3) = 1/3 <- Simplify (1/∞ * 1/3)

So to sum this up in words:
A third is equal to point three repeating plus a third of an infinitesimal, which sounds absurd, but it's true. Base 10 isn't perfect cause it's a round number. There's always a flaw. But it's too late to change number systems easily, so we just have to deal with these flaws.

0.999... is only equal to 1 because we humans made rules stating it is. I'd like to see you prove 0.999... = 1 without those rules.

For example. Take a common non-calculus proof.

1/3 = 0.333...
0.333... * 3 = 0.999...
1/3 * 3 = 1
∴ 0.999... = 1

That explodes on the first line. 0.333... is not exactly equal to 1/3. It is the closest Base 10 number to 1/3.  1/3 is in fact: 0.333... + 1/∞*3. This proof was made to trick people into believing 0.999... = 1 without calculus proofs.
Now lets rewrite that proof.

1/3 = 0.333... + 1/∞*3
0.333... * 3 = 0.999...
1/3 * 3 = 3(0.333... + 1/∞*3) = 0.999... + 1/∞
∴ 0.999... + 1/∞ = 1

So the big question is: How much is an infinitesimal?

Calculus tells us that the limit of 1/x as x goes to infinity is 0, but that's calculus. It was practically made for answering this. Can you prove that  "1/∞ = 0" without those rules?

Almost all people say 0.000...1 (1/∞) is 0 because you'd never write that 1 down. Why would this be? There are infinite numbers between 0 and 1, but 1 still exists. This is a concept, an idea, not a physical object. To apply physical-object ideas to a concept would be reification.

I will say this again.
0.999... is equal to 1 ONLY BECAUSE we humans made extra rules saying it is. It is not equal to 1 without those rules.