1 = .9999.....

[Raistlin]

Let ".999..." equal a decimal followed by an infinite series of nines.

Ok, all of you who are claiming that .999... does not equal one are wrong. The number between them is not "infinitely small," because there is no number between them. They are not "basically equal with rounding," they are in fact exactly equal.

Three thirds together make a whole, so the number .999... is a human construct that doesn't exist in the practical world.

Yeah it does. I'm pretty sure you can identify a single object in the "practical world". ".999..." is just a different representation of it, but just as valid as "1."

It's from this misunderstanding that the equation 1/3 + 2/3 = 3/3 / 0.333... + 0.666... = 0.999.... can arise, because you're essentially adding together decimals that aren't a correct representation of the fraction being shown. You can't add infinite series of decimals together even if logically they make sense because they are infinitely long.

This makes no sense. It's not a "misunderstanding," it's adding together rational numbers.

This is a falsity because it only applies to finite decimal values. If a given value is an infinite series then it becomes impossible to locate it within a number line because of this as the inherent value of that number is impossible to place.

This is also wrong, and I'm not even sure where you got this. It is a basic rule of algebra that any two, distinct real numbers have an infinite number of other distinct, real numbers between them. 0.999... is a real number.

Someone who remembers more calculus than me will have to post the actual proof. It was posted a couple of years ago in the last thread but I'm too lazy to look it up. But yes, it is actually, definitively proven that .999... = 1.

EDIT: Ok, previous threads on this: First one. Second one. Third one.