Does there exist an algebraic or arithmatic means, completely devoid of any need for graphical or trigonometric methodologies, and, you just might guess, similarly requiring no prior understanding of integrals or derivatives, such that you can produce a fraction/ratio by means of some formulaic procedure that, applied over and over, ad infinitum, which, by infinite application thereof results in a quotient that is "equal to" pi in that the more such operation is applied/calculated (longhand) the more infinitely accurate it is in answering C/D?

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There are Monte-Carlo methods for calculating pi that require nothing but iterating a certain procedure, recording the ratio or successes to total attempts, and then using some algebraic formula to convert that ratio to pi.

See this user's articles for a humorous take on calculating pi with such techniques:

https://hubpages.com/education/How-to-Estimate-Pi-...

There are also infinite series and infinite products that converge to algebraic functions of pi. For example, the alternating series

1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15 ...

converges to pi/4. Here is a mathworld reference for similar pi formulas

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You could always do what the ancient Babylonians did, which sufficed for centuries:

1) Get some circular objects and measure their diameter

2) Measure those same objects' circumference (which can be tricky!)

3) Compare the two measurements for each circle and observe that they are about 3 (a tad bit more than 3, to be precise) times bigger around than they are across, which seems to happen for every circle, all the time

And there you have the essence of pi! Nowadays with our fancy mathematics and computing machines, we have far more information about pi and all its infinite decimals, but back in the day, "about 3" was good enough.

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Thanks for the reminder of the seeds of concrete and manipulable numeracy, and the great thinkers and how they started. I am, however, ahem...infinitely more interested in the formulae and mathematics used to refine it to its nonexistant end! :)