Wrong in counting real number

Question

I know that real number is uncountable,the same as the closed interval [0,1].However,I don't know what is wrong with my method of counting numbers in [0,1]:

0.1,0.2…0.9

0.01,0.02…0.99

0.01,0.02,…0.999

....

Can anyone tell me what'wrong with it?Thanks.

Answer 1

You are merely counting rational numbers, not all real numbers between $0$ and $1$. The problem with the real numbers is that from any list of real numbers, using Cantor's diagonal argument, one can show that there exists a real number that is not in that list. This is why the real numbers are uncountable.

Have fun reading this : http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

Hope that helps,

Answer 2

When does your enumeration reach pi? Or Sqrt(2)? Or even 1/13? It doesn't.