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?
In: 716
If you take the diameter of a circle and try to wrap it around the circumference, it will take a little bit more than 3 times to wrap around. That’s how we got started on the mess since people wanted to know how much more than 3.
Of course, for most practical stuff they only needed to know it worked out to ~25/8
The way we got the billion whatever digits is a number of experiments in math, done over the world independently (most famously by Archimedes, but also by a number of other Greek, Chinese, Indian, and other big empires in the world at the time) where they figured out the perimeter of a Hexagon, then a 12 sided figure, then a 24, then a 48, then a 96 sided figure, and he stopped there.
In China a few hundred years later they applied the same idea but on a way bigger scale because they had better understandings of algorithms and calculated out to a 12,288 sided figure and got that pi was between 3.1425926 and 3.1415927 and that was the highest that we had for centuries.
In Persia they used 3×2^28 sides those centuries later, and really it just goes like that until we had computers do it for us
Pi is the ratio of circumference to diameter. I have my students measure these and divide to find the value of pi.
There are multiple formulas for figuring out what pi equals, just like there are multiple formulas for figuring out other answers. They seem different to us, but ultimately the different formulas for solving pi are the same, they just have a different-looking path to get there.
For instance, if we were talking about the area of a square, the most common definition is A=L^2 where L is the length of a side.
We can also use a different formula. Since this is ELI5, it’ll be easy: The other formula is A=(0.5LxL/2)x4. This formula finds the area of a triangle and multiplies it by 4, which ends up getting you to the area of a square. They’re different formulas, but both give you the same answer.
Similarly, there are formulas for pi. You could write pi=C/D. Or you could write a complex formula (that never ends) to compute pi. And since it’s a lot easier to compute this new formula than doing C/D, we use that to get a more precise answer.
Both formulas are the same, one is just easier to work with.
Pi is always an approximation. Before Newton came up with some calculus to calculate it, the Greeks did it in a way that is easier to explain.
To approximate pi, you take a circle and cut it into four pie slices – so you have four 90 degree angles forming four triangles where the length of the far side is approximately a fourth of the circles diameter. We can calculate that length by using the Pythagorean theorem. So radius squared + radius squared ~= (circumference x .25) squared. To make it a more accurate calculation, cut the circle into more pie slices. The more pie slices you use, the more digits and more accurate approximation of pie you get.
The most well known series that can be used is the sum of the reciprocals of squares of natural numbers, i.e. 1/n^2
This converges to Pi^2/6. With this you can calculate digits.
Sure, there are better series’, but I think this one is the most common one for people.
[removed]
Follow up question. It’s great some algos can calculate to the trillionth digit of Pi, but how many working digits of Pi do professionals like astronomers and rocket scientists really need?
Any formula that contains pi (this is a lot of formulas) can be rewritten into a definition of pi by making pi the subject.
You may not like this way of getting pi, but it is technically a way of getting pi.
You’d normally use one of the series in top comment, but these eventually make your computer sad.
Or you look it up. Most computers just look it up.
Latest Answers