Is there any combination of numbers which upon division gives the exact number of P?

Question

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.

Answer 1

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 :)