Is there mathmatical proof that the decimal expansion of Pi never ends or repeats?

Question

I picked Pi as a specific example, but my question is really about irrational numbers in general. How do we know for sure that irrational numbers actually exist? How do we know that they don't eventually have a last decimal place, or turn into a repeating decimal sequence? How do we know that they can't be expressed as a fraction?

And if you happen to know of any philosophical ideas concerning the meaning of irrational numbers, I would like to hear ideas about this as well. What does the existence of irrational numbers tell us about the nature of reality? Why should there be such numbers?

Theta40 · Accepted Answer

That's a proof fro Pi as being irrational:

http://www.lrz-muenchen.de/~hr/numb/pi-irr.html

Also Pi was proven to be transcendental by Lindemann in 1832, which is a stronger condition.
"How do we know for sure that irrational numbers actually exist?"
You can prove they are not rational. You assume ad absurdum and you get a contradiction. That is a way to prove it.

 Here are some proofs to show sqrt(2) is irrational:
http://www.cut-the-knot.org/proofs/sq_root.shtml

"Why should there be such numbers?"
I think those numbers appear in nature. The perimeter of a circle of radius 1/2 is Pi. Now, I think you always can measure, supposing that you have tools for measuring. But by finite nature of our capabilities, you get only an approximation of the number.
Other examples: the golden radius number( related to Fibonacci series) appears in artistic proportions, the form of a flower( much like your beautiful avatar), the cycle of appearance of cicadas, 
http://en.wikipedia.org/wiki/Golden_ratio

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Is there mathmatical proof that the decimal expansion of Pi never ends or repeats?

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