Mathematics can be crazy at times!

One very interesting article I came across recently is the proof of the fact that 0.999.... = 1. Like most students, I immediately questioned the validity of the steps in the proof, but it turns out that this is actually true. Three proofs are shown below

Proof 1:
Let S = 0.999......
Therefore 10S = 9.999......
Subtracting the above two equations, 9S = 9.000....
Therefore S = 1

Proof 2:
We know that 1/9 = 0.111....... (by long division)
Therefore 1 = 0.999......

Proof 3: (probably the most correct one)
Let S = 0.999....
Therefore S = 0.9 + 0.09 + 0.009 + ......
The RHS of the above equation is an infinite geometric progression with first term = 0.9 and common ratio = 0.1
Therefore S = 0.9 / ( 1 - 0.1) = 1

Read all about it on http://en.wikipedia.org/wiki/0.999...