Re: Why is pi an irrational number?

That you can never know pi "exactly" is like saying that pi is irrational, 
that is that pi cannot be expressed in the form of the ratio of two entire 
numers.

About the "as much accurate estimation as you like" you may consider the 
following well known equality expressing pi as the sum of of a numerical 
series:
                                 n
pi = 4(1-1/3+1/5-1/7 ......+ (-1) 1/(2n+1) +.........)

and if you take for the estimation :
                            n 
4(1-1/3+1/5-1/7 ......+ (-1) 1/(2n+1)  

you make an error less than 1/(2n+1).

This seems to answer your question, even if the above calculation, how 
simple it can be, may not be the best way to evaluate pi with a high 
precision !