Eli5:How do you get a new digit of pi?

Question

Eli5:How do you get a new digit of pi?

In: Mathematics

Answer 1

There are various mathematical series that converge to π, so keep adding more terms, and your sums will be closer and closer approximations. Warning: it’ll take you a while to get to a “new” digit.

> [one formula] has been used for several record-setting π calculations, including the first to surpass 1 billion (10^(9)) digits in 1989 by the Chudnovsky brothers, 10 trillion (10^(13)) digits in 2011 by Alexander Yee and Shigeru Kondo,[136] over 22 trillion digits in 2016 by Peter Trueb[137][138] and 50 trillion digits by Timothy Mullican in 2020.[139]

https://en.wikipedia.org/wiki/Pi#Modern_quest_for_more_digits

Answer 2

There are formulas for calculating the successive digits of pi. By computing more and more terms of these formulas, you get more digits of pi.

Answer 3

There are no “new” digits of pi, can you perhaps rephrase your question? It might help if you look at the Wikipedia entry for pi and if you have questions after that you can ask about it more specifically

Answer 4

Archimedes’ method: start with an inscribing and circumscribing triangles or squares, calculate their area, then repeat, keep doubling the number of sides. The smaller one give lower bound on pi, bigger one upper bound. With enough size, you get enough bound to gain a new digits. Very intuitive method to understand, but very slow.

Newton and Leibniz’s method: obtain what’s called a Taylor’s series of a inverse trigonometric function. Taylor’s series is an infinite number of terms to add up that allow you to compute the value of the function at a certain point, using values of the functions and its derivatives at another point. Since there are an infinite number of terms, you never finish adding them, but the more you add the closer it is to the final result, and this gives you more digits. The nice thing about inverse trigonometric functions is that they have really simple formula for the Taylor’s series at 0, and you can compute pi or something easily related to pi by evaluating the function at a different point.

Ramanujan’s method: the modern method, obtained from number theory, with the current version made by Chudnovsky brothers. Sorry but I can’t ELI5 this one.