Bailey–Borwein–Plouffe formula

The Bailey–Borwein–Plouffe formula, often called the BBP formula, is a special way to calculate the number π. π is the ratio of a circle’s circumference to its diameter. It was discovered in 1995 by Simon Plouffe and is named after the three mathematicians who published it: David H. Bailey, Peter Borwein, and Plouffe.

This formula is interesting because it lets computers find a specific digit of π in the hexadecimal (base-16) system without needing to calculate all the digits before it. This was a surprise because before this, people thought finding a single digit of π would be as hard as calculating all the digits up to that point.

The BBP formula works by using a special kind of mathematical series. It can be used to find the nth digit of π in base-16. While it does not directly find digits in the more common decimal (base-10) system, it opened new ways to study π and other numbers. Because of this, the formula inspired many similar methods for calculating other irrational numbers. These related formulas are called BBP-type formulas and have helped scientists and mathematicians explore the properties of numbers in new ways.

Specializations

The Bailey–Borwein–Plouffe formula is a special way to calculate the number π. It was discovered in 1995 and helps find each digit of π in hexadecimal without needing to calculate all the digits before it.

This formula uses a special math idea. It can show many math patterns. The BBP formula for π is one example, showing how π can be broken into smaller, easier pieces.

BBP compared to other methods of computing π

The BBP formula can find any single digit of π without needing to calculate the digits that come before it. This makes it faster and easier because it doesn’t need special tools for very big numbers. Another version of this method, called Bellard's formula, was found by Fabrice Bellard and works even quicker.

Even though BBP is clever and saves time, it still takes longer to find digits that are farther along in the number π. As the digits get farther away, it needs more time to calculate them, just like other common ways of finding digits of π.

Generalizations

Mathematician D. J. Broadhurst made a version of the BBP algorithm. It helps us find many special numbers quickly. These numbers include Catalan's constant, π³, π⁴, Apéry's constant, and the Riemann zeta function at different points. It can also find logs of numbers like 2, 3, 4, and 5. These calculations use a smart method called polylogarithm ladders.