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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1What do you mean by "calculate"? Do you mean, by a series of rational numbers, like $\pi=4(1-\frac{1}{3}+\frac{1}{5}-+\cdots)$? – Dietrich Burde Jun 09 '18 at 18:47
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Do you mean how to find the value of an irrational number? – The Integrator Jun 09 '18 at 18:48
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Yeah to find the value, yes, by series if that's what it is. I don't know if all irrational numbers are series, for example. – Lance Jun 09 '18 at 18:48
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you can usually find them with the help of an infinte series . for example $e = \displaystyle\sum_{n=0}^\infty \frac{1}{n!}$. This is usually done by setting specific values in a functions Taylor series – The Integrator Jun 09 '18 at 18:52
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Any real number between $0$ and $1$ has a series expansion of the form $\sum_{n=1}^\infty a_n/10^n$ where each $a_n\in{0,1,2,3,4,5,6,7,8,9}$. – Angina Seng Jun 09 '18 at 18:53
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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:
Dietrich Burde
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Perfect, thank you. That's exactly what I was wondering but didn't know how to put it. – Lance Jun 09 '18 at 18:53