Pi Calculator - Free Online Tool
Calculate Pi using multiple algorithms with animated convergence visualization.
Compare Leibniz, Monte Carlo, Nilakantha, and Machin's formula in real-time.
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What is Pi (ฯ)?
Pi is the ratio of a circle's circumference to its diameter. It's an irrational number, meaning its decimal representation never ends and never repeats. Despite being defined by a simple geometric relationship, Pi appears throughout mathematics, physics, and engineering.
Leibniz Series
ฯ/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...
Discovered by Gottfried Wilhelm Leibniz in 1674, this is one of the simplest formulas for Pi. However, it converges extremely slowly - you need about 5 billion terms to get 10 correct decimal digits! This makes it impractical for actual computation, but it's beautiful in its simplicity.
Monte Carlo Method
Imagine throwing darts randomly at a square board with a quarter circle inscribed in it. The ratio of darts landing inside the quarter circle to total darts approximates ฯ/4.
This probabilistic approach converges as O(1/โn), meaning you need 4x more points to halve the error. While inefficient for calculating digits, it beautifully demonstrates how randomness can solve deterministic problems and is widely used in physics simulations.
Nilakantha Series
ฯ = 3 + 4/(2ร3ร4) - 4/(4ร5ร6) + 4/(6ร7ร8) - ...
Discovered by the Indian mathematician Nilakantha Somayaji around 1500 AD, this series converges much faster than Leibniz - approximately O(1/nยณ). Each term divides by three consecutive integers, causing rapid decay. About 10 terms give you 6 correct digits!
Machin's Formula
ฯ/4 = 4รarctan(1/5) - arctan(1/239)
Discovered by John Machin in 1706, this formula is remarkably efficient. Using the Taylor series for arctan, each term for arctan(1/5) decreases by a factor of 25, and for arctan(1/239) by a factor of ~57,000! Machin used this to calculate 100 digits of Pi by hand. It was the basis for Pi calculations for over 200 years.
Why Such Different Convergence Rates?
The dramatic differences in convergence rates come from how quickly each term in the series shrinks. Leibniz terms shrink as 1/n (slow), Nilakantha as 1/nยณ (fast), and Machin's arctangent terms shrink exponentially (extremely fast). This is why modern record calculations use Machin-like formulas or even more sophisticated algorithms like the Chudnovsky algorithm, which adds 14 digits per term!
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