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In other words, there are (probably) infinite combination of numbers/operations which leads to irrational numbers. So I wonder, if there is one which gives exact number representation of P(π)? Do we need to measure the ratio of diameter/circumference or we can actually get infinitely accurate representation by just doing simple math? Thanks.

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That depends on what you consider simple math. The most basic sequence you can get (i.e. looks simple) is $\pi/4 = 1 -\frac13+\frac15-\frac17+\frac19+...$.

However, as this one converges painfully slow, no one in their right mind uses it because more effective formulas have been known for ages (for example, I once used $4\arctan {\frac15} - \arctan {\frac1{239}}$). Currently rapidly converging series are used. Read the Wikipedia article on this, as suggested in the comments :)

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  • $\begingroup$ Sorry, well... $arctan(x)$ itself is calculated using taylor series $x - \frac{x^3}3 +\frac{x^5}5 -...$, which, I guess, is simple enough math, even if proving the fact that it actually equals this is beyond what you can consider "elementary math" that anyone knows. $\endgroup$
    – Shady_arc
    May 21, 2014 at 15:32

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