[vissin]

Well... here's an old trick. I was rather intrigued when I first saw the equation 0.999... = 1 somewhere around 5 years back. And I tried to find out as many mathematical proofs of it as I can and see if I can find a flaw in it. At that time I was baffled but I'm not so any more. Here I'll first reproduce the simplest proof that I remember and then dare to propose my own penny's worth on it.

0.999... = 3 * 0.333... = 3 * 1/3 = 1
ie. 0.999... = 1

Now to address the question, "how is it even possible?"

Well Raj this does not prove that Universe has limits. In fact it proves nothing. It only goes on to display the limits of human perception.

Remember, Mathematics and all Science is nothing more than merely tools that we use. Allow me to quote Neils Bohr (as I have done many times ever since I first read it) "Physics tells us not what is but what we can say to each other regarding the world." Eventhough this was said about Physics, we can very well extend this to any Science.

Let's suppose I have a tray full of randomly cut pieces of apples. I want to use these pieces to make one full apple. In analogy I'm using fractions to create 'one'. Quite easily I make an assemble that looks very much like a whole apple but I notice that they are many gaps between the pieces. Instead of '1' I may have something like 0.999 (considering that I was born lucky). Now I cut even smaller pieces from the apple in the tray to fill these gaps. I want 1 and not an approximation to it. The pieces I cut keep getting smaller and smaller until I reach a single atom and then I go further into granualities of bosons, neutrons, protons, electrons and photons. What now. Do you think I've constructed one whole apple. I doubt it. But I'm sure to be very close. So close that I'm no longer able to see the difference even with the best technology that I may have. Now this is where 0.999... = 1. As also 1.000...1 = 1.

What is infinite. It's just a concept. Something that's beyond perception. The proof is 100% right. Mathematics is flawless. It agrees completely with the definitions that we ourselves have made for our own understanding. And it only gives us what is true for us (even if it may not be the absolute truth).

The above is a outcome of my petty mind and bears no external credentials (except Bohr) and is open to criticism.