I'm a novice here, but have always wondered this... I can divide 1 by 3 to get the recurring decimal 0.3333... But how would I find pi? I could take the circumference of a giant circle and compare it to its diameter, or use the pi button on a calculator, but when someone figures out pi to the 10,000th digit or so, how do they know they've got it right? Is there a simple algorithm (such as using long division to compute 1/3) that laymen can use to explore the digits of pi?
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See Simple numerical methods for calculating the digits of pi as well. The breakthrough is that $\pi$ is not just literally the ratio of the circumference to the diameter, but also the summation of an infinite series. For example, since $\tan \frac{\pi}{4} = 1$, $\pi = 4 \arctan 1$, and the series for $\arctan x = x - x^3/3 + x^5/5 + \cdots$ which can be found using Taylor series (calculus), which gives $\pi/4 = \arctan 1 = 1 - 1/3 + 1/5 \cdots$ – Toby Mak Sep 19 '21 at 11:24
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Please also read how to ask a good question. Without an idea of your background, we can't give you an appropriate answer (only after going to your profile did I see you also were on Electrical Engineering SE). – Toby Mak Sep 19 '21 at 11:27
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So, perhaps the answer to my question is, no, there is not a simple algorithm like with 1/3, but several other methods do exist that require a little more elbow grease. – nuggethead Sep 19 '21 at 11:31