The Fascinating History of Pi

Introduction

Pi (π) is perhaps the most famous number in all of mathematics. Defined as the ratio of a circle's circumference to its diameter, pi appears everywhere—from geometry and trigonometry to physics, engineering, and statistics.

Pi is an irrational number: it cannot be expressed as a fraction of two integers, and its decimal representation goes on forever without repeating. Today, we know trillions of digits of pi, but for most practical purposes, 3.14159 is more than sufficient.

Ancient Approximations

Ancient Egypt (c. 1650 BCE)

The Rhind Mathematical Papyrus shows Egyptians used an approximation of (16/9)² ≈ 3.16 for pi. This was remarkably close to the true value, given the tools available.

Ancient Babylon (c. 1900 BCE)

Babylonian mathematicians used 25/8 = 3.125 as their approximation. Clay tablets show they understood the relationship between a circle's circumference and diameter quite well.

Ancient India (c. 600 BCE)

Indian texts from the Shulba Sutras approximate pi as 3.125 (the same as Babylon), suggesting possible cultural exchange along ancient trade routes.

Ancient China (c. 250 CE)

Chinese mathematician Liu Hui calculated pi to 3.14159 using a polygon-inside-circle method. Later, Zu Chongzhi approximated pi to 7 decimal places (between 3.1415926 and 3.1415927)—a record not broken in the West for over 800 years.

The Greek Contribution

Archimedes (c. 250 BCE)

Archimedes of Syracuse made one of the most significant contributions. He used a 96-sided polygon inscribed in a circle and calculated that:

Archimedes' method—calculating the perimeters of inscribed and circumscribed polygons—remained the primary approach for computing pi for centuries.

The Symbol π

The Greek letter π wasn't used to represent this ratio until 1706, when Welsh mathematician William Jones introduced it. The symbol was popularized by Swiss mathematician Leonhard Euler in the 1730s, and it stuck.

The choice of the Greek letter π was natural because it's the first letter of "periphery" (περιφέρεια in Greek), and it's also the first letter of "perimeter" in Greek.

The Race for More Digits

17th-19th Centuries

• 1573: German mathematician Valentine Otho calculated pi to 6 decimal places
• 1701: John Machin computed 100 digits using a new formula
• 1844: Zacharias Dase calculated 200 digits
• 1873: William Shanks spent 20 years calculating 707 digits (later found to have an error after digit 527)

The Computer Age

The invention of computers revolutionized the calculation of pi:

• 1949: ENIAC computer calculated 2,037 digits in 70 hours
• 1958: Over 10,000 digits calculated
• 1983: One million digits reached
• 2019: Google calculated 31.4 trillion digits
• 2022: Researcher calculated 100 trillion digits

Formulas for Pi

Leibniz's Formula (1673)

Nilakantha's Formula (15th century)

Modern Formulas

Modern algorithms use formulas like:

Pi in the Modern World

Memorizing Pi

Memorizing digits of pi has become a popular challenge. The record for most digits memorized is held by Rajveer Meena, who recited 70,000 digits in 2019 (taking about 10 hours!).

Mnemonic devices help, like this famous poem where the number of letters in each word gives digits:

"How I need a drink, alcoholic of course, after the heavy lectures involving quantum mechanics."

(3.14159265358979...)

Pi Day

Pi Day is celebrated on March 14th (3/14) because the date resembles 3.14. Schools and math enthusiasts around the world celebrate with pi-themed activities, contests, and of course, pie!

Uses of Pi Today

• Engineering: Designing wheels, gears, pipes, and any circular objects
• Physics: Calculations involving waves, circles, and periodic phenomena
• Statistics: The normal distribution curve involves pi
• Navigation: GPS calculations use pi
• Cryptography: Used in certain encryption algorithms

Is Pi Normal?

One of mathematics' great unsolved questions is whether pi is "normal"—that is, whether every possible string of digits appears with equal frequency in its infinite decimal expansion. We believe pi is normal, but no one has ever proven it.

If pi is normal, then within its digits, you could find your birthdate, your phone number (encoded), and even complete copies of Shakespeare (if properly encoded).

Key Takeaways

• Pi = circumference/diameter ≈ 3.14159...
• Ancient civilizations approximated pi with surprising accuracy
• Archimedes established bounds: 223/71 < π < 22/7
• The symbol π was introduced in 1706 by William Jones
• Modern computers have calculated trillions of digits
• Pi appears in countless real-world applications