OK, so you want to calculate pi to a jillion places. But don’t you run into a significant figures problem?

Pi is defined as a circle’s circumference over its diameter. Even if we go and say the the d. is exactly 5, say, what number do we put in the numerator? We have to actually measure C., because to calculate it we would have to use pi, which is what we want to generate, not use.

So even if we measure C to 100 places, how do we ever claim to calculate pi to 101 places, when we are limited by sig figs?

I realize someone’s probably going to say something like writing an algorithm for it, but how do you write one for a trancendental number? How would you ever know that your 287th digit was really right? You have nothing to compare it to. No patterns have ever emerged to let us simply extrapolate the next digits, so how are we so sure that we have it right to so many digits?