I’m not sure if I worded this correctly but I’ll try to elaborate more. I know that there are formulas that can calculate pi to the Nth digit. But my question is how can we be certain that the formulas accurately calculate the Nth digit of pi when we have nothing to compare it to since pi is an irrational number that keeps on going?
For example if I discovered a new formula than can calculate pi to 1 higher digit than what we currently know and the value I got was 4. How do I confirm that it is indeed 4 and not any other number? I have nothing else to compare it to?
I hope this makes sense
In: Mathematics
A method could work like this:
We have a circle with radius 1, we know its surface is pi.
We start by drawing a square inside of it, and then a square that’s completely outside of it. We know that pi is smaller than the big square, and bigger than the small square.
Now, for the smaller square we start adding smaller squares in the leftover areas but still within the circle, and for the bigger square we start removing squares that are just outside the circle.
Step by step, as we add/remove smaller squares, we approach pi closer and closer.
At some point, we can say that pi is bigger than one bunch of squares, and smaller than another bunch of squares.
If we have calculated the smaller squares to have an area of 3.13, and the larger squares an area of 3.15, we can be confident that the first to digits of pi are 3.1
If we continue adding/removing smaller squares, and we end up finding out that pi lies between 3.14152 and 3.14163, we can be sure that pi has digits 3.141.
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There are formulas for describing these smaller and smaller squares, allowing us to calculate pi to arbitrary precission.
(There are also other methods used, and formulas for e.g. calculating the 10000th digit of pi are often slightly different, but they still use similar tricks, where we can be sure that the value is correct.
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