“Mathematics is the music of reason, and Pi is its key” - Roger Nelsen [1]

Few numbers in mathematics hold as much intrigue as Pi. It is a number so fundamental that it transcends disciplines, appearing in pure maths, geometry, physics, engineering, and even probability theory. Despite its simplicity in definition, Pi’s ubiquitous influence stretches far beyond the realm of circles, embedding itself in the very fabric of mathematics and the universe.

Pi is a mathematical constant defined as the ratio of a circle’s circumference to its diameter. It’s first few digits are 3.14159, and its irrational and transcendental number means it cannot be expressed as a rational fraction a/b, where a, b ∈ ℚ. It has an infinite, non-repeating decimal expansion. However, origins begin thousands of years ago. Ancient civilizations, including the Babylonians and Egyptians, sought to approximate the ratio between a circle’s circumference and its diameter. The Babylonians used 3.125, while the Egyptians’ Rhind Mathematical Papyrus suggests a value of 3.1605. Although crude, these approximations were remarkable for their time. [2]

The Greek mathematician Archimedes of Syracuse (c. 287–212 BCE) took a more systematic approach. Ingeniously, he inscribed and circumscribed polygons in and around the circle, so that the circle’s area would be bound by the polygons’ areas. [3] The more edges the polygons have, the more precise the bounds. This derived an approximation for Pi between 3.1408 and 3.142 - a remarkable feat without modern calculus.

The Indian mathematician Madhava of Sangamagrama (c. 1340–1425) developed an early infinite series to approximate Pi,

predating similar work by Leibniz and Gregory in Europe by centuries. Later, the Swiss mathematician Leonhard Euler introduced and popularised the now-famous notation as we know it today, π.

With the advent of computers, calculating Pi to extreme precision has become a challenge of computational power. The current record, as of June 2024, stands at 2.0211229*1014 digits, though only the first few dozen are necessary for most practical applications.[4]

The common, practical applications of pi can be seen everywhere in the world around us. Architects and engineers rely on Pi in designing domes, arches, and circular structures. The famous Pantheon in Rome, with its perfectly circular dome, is a testament to the importance of Pi in construction. Modern bridges and tunnels, including those with circular cross-sections, also depend on Pi-based calculations for structural integrity. Pi emerges naturally in physics, particularly in wave mechanics and electromagnetism. The equation for simple harmonic motion, governing everything from pendulums to alternating current (AC) circuits, incorporates Pi due to the periodic nature of sine and cosine functions.

Additionally, Einstein’s field equations in general relativity, which describe the curvature of spacetime, also involve Pi. The fundamental equation for a black hole’s event horizon—the Schwarzschild radius[5] - is given by:

Rs=2GMc2

which contains Pi indirectly through its derivation from spherical geometry. Pi also has numerous applications in computer algorithms, random number generation, and encryption. The unpredictability of its decimal expansion has led to investigations into whether Pi could serve as a natural source of randomness in cryptographic key generation.

Aside from its practical usefulness, what truly excites mathematicians the most is its unwavering ability to show up everywhere across other various fields of mathematics, like a retired Hollywood star showing up in cameos for the fun of it. Some examples include colliding blocks[6][7] in mechanics and series[8].

In another instance, arguably one of the most famous equations, Euler’s identity[9] links five of the most fundamental mathematical constants: e (Euler’s number), i (the imaginary unit), pi, 1, and 0.

ei+10

Pi also unexpectedly emerges in probability theory. A famous example is Buffon’s Needle Problem, a statistical experiment where a needle dropped on a floor with parallel lines can be used to estimate Pi. The probability of the needle crossing a line is directly proportional to Pi, demonstrating its hidden presence in random events. In statistics, the Gaussian distribution[10] (normal distribution), fundamental to probability theory, includes Pi in its probability density function.

f(x)=122e-(x-)222

This “bell-curve” shape governs countless real-world probabilities, such as standardised exam scores, financial models, quantum mechanics, Bayesian machine learning, and even in medicinal research tracking population distribution. Who would’ve thought Pi turns up here?

Even in computers, Pi plays an unexpected central role in Fourier analysis[11], which decomposes functions into sinusoidal components. This is critical in signal processing, image compression (JPEG), and audio technology. Every time you stream music or watch a video, Fourier transforms—based on Pi—enable efficient data compression and transmission.

Pi is far more than a number- it is a gateway into some of the most profound ideas in mathematics and science. From ancient geometry to modern physics, cryptography, and artificial intelligence, Pi continues to shape our understanding of the universe. While it remains an irrational and infinite number, its applications are anything but abstract; Pi is embedded in the very structures and patterns that govern our world. As computational techniques evolve, Pi will undoubtedly continue to inspire mathematicians and scientists, revealing deeper insights into both mathematics and reality itself.

You’d have to be crazy to think that I could explore all the intricacies about the beauty and simplicity that pi encapsulates. This article serves merely as a springboard, something to spark your interests and invoke your own research, for you to explore and marvel at Pi, a simple irrational number. A universal constant. An invisible thread in this mathematical universe.

References

• [0] - https://unsplash.com/photos/black-steel-letter-b-wall-decor-7h68nZhAZbE

• [1] - https://lavandamichelle.com/2025/03/14/pi-day-10-memorable-quotes-about-pi/

• [2] - https://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus

• [3] - https://www.exploratorium.edu/pi/history-of-pi#:~:text=Archimedes%20knew%20that%20he%20had,brilliant%20Chinese%20mathematician%20and%20astronomer.

• [4] - https://en.wikipedia.org/wiki/Chronology_of_computation_of_%CF%80

• [5] - https://en.wikipedia.org/wiki/Schwarzschild_radius

• [6] - https://www.youtube.com/watch?v=HEfHFsfGXjs

• [7] - https://www.maths.tcd.ie/~lebed/Galperin.%20Playing%20pool%20with%20pi.pdf

• [8] - https://www.youtube.com/watch?v=d-o3eB9sfls

• [9] - https://en.wikipedia.org/wiki/Euler%27s_formula

• [10] - https://en.wikipedia.org/wiki/Normal_distribution

• [11] - https://en.wikipedia.org/wiki/Fourier_analysis