Question

I need to know a formula, that if I plug a number in to a variable (let's say N) it will give me that digit of pi. For example, plugging in 5 would give me the fifth digit of pi, 9.

Answer 1

There is no closed-form algebraic formula to compute the n-th digit of pi for an arbitrary n. However, there are various algorithms that can be used to compute pi to a desired number of digits, and it is possible to extract individual digits from the resulting decimal representation.

One such algorithm is the Bailey-Borwein-Plouffe (BBP) formula, which can be used to compute the n-th hexadecimal (base-16) digit of pi without needing to compute any of the preceding digits. The formula is:

pi = SUM[k=0 to infinity] 16^-k {4/(8k+1) - 2/(8k+4) - 1/(8k+5) - 1/(8k+6)}

Using this formula, the n-th hexadecimal digit of pi can be computed using the following expression:

floor(16^(n-1) * {4/(8n+1) - 2/(8n+4) - 1/(8n+5) - 1/(8n+6)})

Note that this formula only gives the n-th hexadecimal digit of pi, not the n-th decimal digit. To convert a hexadecimal digit to a decimal digit, you can multiply it by 0.0625 (i.e., 1/16), which is the value of one hexadecimal place.