Ok, pi is probably a bit over the head of your average 5 year old. I know the definition of pi is circumference / diameter, but is that really how we get all the digits of pi? We just get a circle, measure it and calculate? Or is there some other formula or something that we use to calculate the however many known digits of pi there are?
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> I know the definition of pi is circumference / diameter, but is that really how we get all the digits of pi?
That’s the historical definition, and that’s probably how people got the approximate value of pi (slightly more than 3) thousands of years ago.
At that time, they didn’t care about the digits (they didn’t even invent decimal writing), so they often used the approximation 22/7 which was discovered to be a rather good approximation by Archimedes. (more precisely he proved that 223/71 < pi < 22/7 using a geometrical approximation of a circle by polygons)
But no we don’t use real circles to measure pi since a very very long time.
> We just get a circle, measure it and calculate?
Fun fact, if we had a perfect circle the size of the observable universe, and we were able to measure its circumference and diameter up to the atomic scale, we would only get 40 digits of the decimal expansion.
So obviously, that would not work, even with the best available equipement.
> Or is there some other formula or something that we use to calculate the however many known digits of pi there are?
Yes, there are formulas. Some formulas are easier than other. For example, a very simple formula that will get you as close to pi as you want is the following :
pi = 4 * (1- 1/3 + 1/5 – 1/7 + 1/9 – 1/11 + 1/13 + … + 1/(2n+1) + … )
Each term you add will gte you closer to pi. The problem is that this formula gets closer to pi very very slowly (You need 200 terms to get an approximation that is only as good as 22/7) .The proof of this formula is not that hard (accessible to any undergrad) but perhaps not at the ELI5 level.
Fortunately for us, we have other formulas, that are more complicated to understand, but that will get you as close to pi as you want much quicker. For example :
pi = 2 * (1 + 1/3 + (2*3)/(3*5) + (2*3*4)/(3*5*7)+ …) that will get you 10 correct digits after 30 terms
And many other formulas far more effective, but that are really ugly.
EDIT : I changed the . into * to avoid confusions.
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