Proving 0.999... Is Equal To 1

eldavojohn writes "Some of the juiciest parts of mathematics are the really simple statements that cause one to immediately pause and exclaim 'that can't be right!' But a recent 28 page paper in The Montana Mathematics Enthusiast (PDF) spends a great deal of time fielding questions by researchers who have explored this in depth and this seemingly impossibility is further explored in a brief history by Dev Gualtieri who presents the digit manipulation proof: Let a = 0.999... then we can multiply both sides by ten yielding 10a = 9.999... then subtracting a (which is 0.999...) from both sides we get 10a — a = 9.999... — 0.999... which reduces to 9a = 9 and thus a = 1. Mathematicians as far back as Euler have used various means to prove 0.999... = 1."

Comments

Re:This is second place

[Colonel Korn]

Unfortunatelly, your proof is not valid. You are trying to prove something which you postulate in your first step.
How do you know 1/9 equals 0.1111111.... ?

He begged the question! For anyone confused about the term "beg the question," this is exactly what it means: assuming the proposition to be proved in the premise.

But that begs the question: is the classical meaning already dead, replaced with the much more easily understood modern usage demonstrated here?

Re:I went one further

[Evo]

Surely the problem is that you're assuming sqrt(1) = 1 when actually it is +- 1? You're throwing away the sign change in that step :)