Saturday, September 17, 2011

What does .9999999999999... really mean?

Haven't we all heard people say that there is a 99.999 "repeating" percent chance that something will happen. They think that this means they are almost sure it will happen. Though I don't want to make things complicated, this is actually equal to 100 percent.

To make things more simple, let's look at if 0.9999... = 1. First off, realize that 1/3 x 3 = 1, but .33333... x 3 = .99999... Same with 1/7, or other repeating fractions.

However, this won't convince people. So, I like to show them my favorite way to look at it, the algebraic proof, the one that proves this fact, or all of algebra and other mathematics incorrect. Say that .9999... is equal to S.

S = .9999...

Try multiplying that all by ten.

10S = 9.9999...
     S =    .9999...

What if we subtract both of these equations from each other.

10S = 9.9999...
   -S =    .9999...
  9S = 9
     S = 1

By dividing both sides by nine, it shows that S is both of these numbers.

This is another thing that is absolutely shocking. Before I saw this, I always thought that it was too close to call, but not exactly one. This is probably one of my favorite proofs in number theory, and all of mathematics.

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