Pi Day 2021: Approximating Pi with Infinite Series

The number π is irrational and transcendental, which means it goes on forever without repetition. For millenia, people have tried to come close to π's true value by using rational fractions:

π ≈ ²²⁄₇ = 3.142857142857142 (3rd millenium BCE Egypt)
π ≈ ³⁵⁵⁄₁₁₃ = 3.142857142857142 (3rd century China)
π ≈ ⁶²⁸³²⁄₂₀₀₀₀ = 3.1416 (6th century India)

Eventually, mathematicians tried adding a series of fractions over and over to get even better approximations of pi. For example, in 1676, Gottfried Lebniz discovered: $π = 4 Σ_{i = 0}^{\infty} \frac{(- 1)^{i}}{2 i + 1} = 4 (\frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} . . .)$

Here are nine different methods from different time periods and cultures. You can choose a method, see information about it, and also calculate as many terms of each as you like to get better and better guesses at the value of pi.

Each time you click the Go button, the program will generate a table of each succesive term. The terms are shown both as fractions and as decimals numbers. The last column is a running total of each term, getting closer and closer to the correct answer.

Learn more:

Estimate π using random numbers (Monte Carlo).
Read Wikipedia on approximating π

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