Q: How accurately do we need to know π? Is there a reason to know it out to billions of digits?
So if you know π out to twenty digits, that’s not 20 times more accurate than just “3”, that’s 100,000,000,000,000,000,000 times more accurate. For example, in parts of ancient India they used π ≈ 62832/20000 (5 digit accuracy), for a while in ancient Egypt they used π ≈ 22/7 (3 digits), and π = 3.2 (1 digit) was nearly used in Indiana embarrassingly recently. So how do we know that π is irrational? In practice (when building physical things) it couldn’t matter less that π is irrational. But ultimately there are two reasons to know π out to more than a handful of digits: studying π for purely mathematical pursuits, and winning competitions.