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
We have sequences that we know converge to pi, and in particular we know we have some sequences that give you a number greater than pi and a number less than pi in succession
So one of these, not the best but it illustrates the point, is the sequence
pi = 4 * (1/1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11…)
So the true value of pi must always be between 2 consecutive values in the sequence… any two values as you expand the sequence yield an upper and lower bound for pi
Expanding this for a number of places yields
4.0000
2.666667
3.466667
2.895238
3.339683
2.976046
3.283738
3.017072
3.252366
3.04184
3.232316
3.058403
3.218403
3.070255
3.208186
3.079153
3.200366
3.08608
3.194188
3.091624
3.189185
3.096162
3.18505
3.099944
3.181577
3.103145
3.178617
3.10589
3.176065
After the 8th in the sequence we know that pi is between 3.283738 and 3.017072 so we know the first digit is 3
After the 26th in the sequence we know that pi is between 3.181577 and 3.103145 so we know the first 2 digits are 3.1
There are better faster ways to calculate pi but the sequence above shows how you can KNOW what the N’th digit is
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