Bailey–Borwein–Plouffe formula

The Bailey–Borwein–Plouffe formula, often called the BBP formula, is a special way to calculate the number π, which 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 allows computers to find a specific digit of π in the hexadecimal (base-16) system without needing to calculate all the digits that come before it. This was a big surprise because before this, people thought finding a single digit of π far out in its sequence would be just as hard as calculating all the digits up to that point.

The BBP formula works by using a special kind of mathematical series, and 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 the door to many new ways of studying π and other numbers. Because of this, the formula inspired many similar methods for calculating other irrational numbers, which are numbers that cannot be expressed as simple fractions. These related formulas are called BBP-type formulas and have helped scientists and mathematicians explore the properties of numbers in new and exciting 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 (a special number system) without needing to calculate all the digits before it.

This formula uses a special math idea called the P function. It can show many math patterns, like how to find parts of the natural logarithm for certain numbers. The BBP formula for π is one example of this, showing how π can be broken into smaller, easier pieces to work with.

BBP compared to other methods of computing π

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

Even though BBP is smart 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 created a version of the BBP algorithm that helps us calculate many special numbers quickly. This includes things like Catalan's constant, π³, π⁴, Apéry's constant, and the Riemann zeta function at different points, as well as logs of numbers like 2, 3, 4, and 5. These calculations use a clever method called polylogarithm ladders.