The History and Wonders of Pi (π): Tracing the Circle Across Centuries

May 26, 2025

The History and Wonders of Pi (π): Tracing the Circle Across Centuries

Have you ever wondered how ancient mathematicians, with no calculators or computers, managed to estimate one of the most famous mathematical constants—π (pi)? This seemingly simple number, defined as the ratio of a circle’s circumference to its diameter, has fascinated minds for thousands of years. Today, let's journey through time and discover how different civilizations and brilliant minds uncovered the value of π in remarkable ways.

What is Pi (π)?

Pi (π) is the constant ratio of a circle's circumference (C) to its diameter (d):

\pi = \frac{C}{d}

Its decimal expansion is:

\pi \approx 3.141592653589793...

π is an irrational number (its decimal never ends or repeats) and transcendental (not a solution of any non-zero polynomial equation with rational coefficients).

Ancient Civilizations and Early Approximations

1. Ancient Egypt (~1650 BCE) – Rhind Papyrus

Egyptian scribe Ahmes gave:

\pi \approx \left(\frac{16}{9}\right)^2 = 3.1605

2. Babylon (~1900–1600 BCE)

Babylonians approximated:

\pi \approx 3.125

The Greek Legacy: Geometry and Logic

3. Archimedes (287–212 BCE)

Using inscribed and circumscribed polygons (up to 96 sides), he found:

3.1408 < \pi < 3.1429

4. Ptolemy (c. 150 CE)

Using trigonometry:

\pi \approx 3.1416

India and the Islamic Golden Age

5. Aryabhata (476–550 CE)

Gave:

\pi \approx 3.1416

6. Madhava of Sangamagrama (14th century)

Madhava of Sangamagrama, a brilliant 14th-century Indian mathematician and astronomer, is credited with discovering an early form of what we now call the Leibniz series for π—centuries before it appeared in Europe.

He derived the infinite series:

$\pi = 4\left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots\right)$

This is known today as the Gregory–Leibniz series, but Madhava discovered it long before James Gregory (1671) or Gottfried Wilhelm Leibniz (1676).Even before calculus was invented. Why it is not that famous i think because it is pretty slow give us an answer.

The Madhava–Leibniz series for π is extremely slow to converge. Here's how many terms you need to reach different levels of accuracy:

Decimal Places of π	Required Terms (n) in the Series
1 (≈ 3.1)	~10 terms
2 (≈ 3.14)	~100 terms
3 (≈ 3.142)	~1,000 terms
4 (≈ 3.1416)	~10,000 terms
5 (≈ 3.14159)	~100,000 terms
6 (≈ 3.141593)	~1,000,000 terms

for further information please check the next blog named as Powell’s Pi Paradox and the Ancient Indian Wisdom That Predicted It.

7. Al-Khwarizmi and Al-Kashi

Al-Khwarizmi: $\pi \approx 3.1416$
Al-Kashi: Accurate to 16 decimal places using trigonometric methods.

Europe and the Rise of Calculus

8. Ludolph van Ceulen

Computed 35 digits of π using polygons; his result is engraved on his tombstone.

9. Isaac Newton

Calculated 15 digits using arctangent series. How he did i will be sharing this the blog named Understanding Pi: From Ancient Geometry to Newton's Revolutionary Leap.

10. Srinivasa Ramanujan: The Master of Series

Ramanujan developed incredibly fast-converging formulas:

\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^{\infty} \frac{(4k)!(1103 + 26390k)}{(k!)^4 396^{4k}}

Still used in modern π computation algorithms!

I will be further explaining how we got this in a separate series of blogs based on Ramanujans discoveries in maths.

11. Computers and the Digital Age

1949: ENIAC computed π to 2,037 digits.
2022: Google Cloud calculated π to 100 trillion digits!

Wonders of Pi in Mathematics

Euler’s Identity

One of the most beautiful equations:

e^{i\pi} + 1 = 0

It connects five fundamental constants:

$e$ (Euler’s number)
$i$ (imaginary unit)
$\pi$
1 (multiplicative identity)
0 (additive identity)

Wonders of Pi in Probability: Buffon’s Needle Problem

In the 18th century, Buffon proposed:

Drop a needle on a floor with equally spaced horizontal lines. What's the chance it intersects a line?

Surprisingly, the probability is:

P = \frac{2}{\pi}

Yes—you can estimate π just by randomly dropping needles! This was one of the earliest examples of using random sampling to calculate π.

🌟 Wonders of Pi in Computation: The Arithmetic-Geometric Mean (AGM) and Fast Algorithms

📘 What Is the AGM?

The Arithmetic-Geometric Mean (AGM) is a powerful iterative method for calculating Pi (π) with high accuracy. It starts with two numbers:

a₀ = 1 (arithmetic mean)

b₀ = 1 / √2 ≈ 0.7071 (geometric mean)

At each step, we compute:

aₙ₊₁ = (aₙ + bₙ) / 2

bₙ₊₁ = √(aₙ × bₙ)

As these values converge, we use them to estimate π with remarkable precision.

🔬 How the AGM Approximates Pi: A Simple Example
📌 Initial Setup

Refined by Jonathan and Peter Borwein in the 1980s, this method is not only fast, but deeply connected to important mathematical fields like elliptic integrals and analytic number theory.

Start with these values:

a₀ = 1

b₀ = 1 / √2 ≈ 0.7071

t₀ = 1 / 4

p₀ = 1

🔁 After One Iteration

We calculate the next set of values:

a₁ = (1 + 0.7071) / 2 = 0.8536

b₁ = √(1 × 0.7071) ≈ 0.8409

t₁ = 0.25 − 1 × (1 − 0.8536)² = 0.2285

p₁ = 2 × 1 = 2

🧮 Estimating Pi

Using the formula:

π ≈ ((a₁ + b₁)²) / (4 × t₁)

That gives us:

π ≈ (1.6945)² / (4 × 0.2285) ≈ 2.870 / 0.914 ≈ 3.1405

Even after just one iteration, we get a highly accurate value of Pi!

⚡ Why the AGM Method Is Powerful

Each iteration doubles the number of accurate digits of π.

Compared to traditional series (like the Leibniz formula), this method converges extremely quickly.

It’s used in modern algorithms to calculate millions (and even trillions) of digits of Pi.

🧠 Theoretical Depth: A Link to Elliptic Integrals

The Borwein brothers didn’t just invent a fast method — they uncovered a deep mathematical relationship between:

Pi (π)

Elliptic integrals

Modular functions

Analytic number theory

This beautifully bridges computational techniques with profound theoretical mathematics.

🌍 Wonders of Pi in Nature and Science

Pi appears in countless places in the real world and in science:

Geometry, circles, and trigonometry

Signal processing and wave mechanics

Einstein’s field equations and quantum physics

DNA structure and resonance in music

Spiral patterns in flowers, hurricanes, and galaxies

🎉 Fun Fact: Pi Day!

Celebrated on March 14 (3/14)
Coincides with Einstein’s birthday
Celebrated worldwide with math events, pie eating, and more!

Conclusion: The Endless Circle

From the sands of ancient Egypt to the circuits of supercomputers, the story of π is one of humanity’s most enduring mathematical journeys. Whether you’re admiring Euler’s elegant identity or simulating needles on a lined floor, π never ceases to amaze. It connects ancient geometry with modern computation, randomness with precision, and curiosity with discovery.