I would like to understand how to "calculate" irrational numbers. By that I mean, how did an irrational number come into existence. In order to understand that, I guess I am wondering what the structure of every irrational number is (if all are an infinite series, or various other things). By knowing the structure of all irrational numbers I should be able to calculate them. I am thinking in terms of symbolically modeling them on a computer.
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$\begingroup$You "can't find seem to find an example", but there are very nice series for, say, $\sqrt{2}$ here:
Infinite series for $ \sqrt 2 $
In general, every irrational number can be written as the limit of the sum of rational numbers:
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