0.9999.... = 1.0

[Jaydi]

dsgrue3 said:

ebw said:

You are deeply misguided about what a proof is.  Your calculation begins with the premise that x=2 and deduces that x = ±2, which I completely agree with.  You seem to misunderstand what the ± notation means.  The last sentence reads "x must be either 2 or -2", which of course it is, since we happen to know it is 2.  There is nothing in the sentence "x = ±2" that says that x could be -2, only that x cannot take any other value.  Likewise, Pezus's example seeks to show that if x has any value whatsoever, it must be 1.  Reading this as "-2 = 2" is utter nonsense.

Actually +/- means either one is a solution for x, which it obviously isn't.

How am I misguided about what a proof is? I'm the one suggesting this type of variable declaration manipulation is utter nonsense. There is no equation to solve, only an assertion that x is 2, or in the OP case that x = 0.999...

Soleron said:

His explanation doesn't contain any invalid steps. Yours does. Neither are formal proofs but no one here would understand the formal proof.

 

Mine does not contain invalid steps because there is no rule that describes variable declaration manipulation.

You are missing the fundamental point here, which is that it isn't an equation. X is already known, any other value you associate to x is invalid by this very reason, which is why logic goes out the window.

I'm sure many people can understand a formal proof here.

 

(x=2) implies that (x²=4). (x²=4) implies that (x=2 or x=-2). Now we "sum up" the two implications: (x=2) implies that (x=2 or x=-2) which is totally true. Logic is respected and so do the mathematics.

 

Edit: Of course you cannot "reverse" these two implications because (x=2) is not equivalent to (x²=4).