0.9999.... = 1.0

[Mnementh]

Soleron said:

Troll_Whisperer said:
I know next to nothing about maths, but it makes sense.

1/3= 0.333...
3/3 = 1

So 0.333... x 3 = 0.999... = 1

I'm going to be devil's advocate here.

start with A and B such that A = B

A = B                                  First statement
A^2 = AB                           Multiply both sides by A
A^2-B^2=AB-B^2            Subtract B^2 from both sides
(A+B)(A-B)=B(A-B)          Factor both sides
A+B=B                               Divide both sides by (A-B)
2B=B                                  Since A=B we can replace A with B
2=1                                     Divide both sides by B

So with some apparently valid steps we have deduced 2=1. The contradiction is in the fifth line, where something is done that seems reasonable but is actually against the rules.

So when you write what you write "So 0.333... x 3 = 0.999... = 1", there may be something wrong you don't even realise if you aren't aware of all the relevant rules. In this case you're not wrong, but what you wrote didn't show that. This is why formal proofs are used in maths.

You cannot apply this so easily. The 'proof' you showed made the error, that the case differentiation in step 5 for divisions wasn't made. You always have to cover for the case, that you divide by zero. So step 6 and 7 are only allowed for the case A != B. But the initial statement was A=B.

That cannot be applied to the first though. The only division that is made is, that something is divided by 3. We know 3 is not equal to zero.