Why the most famous number in mathematics seems to hide the secrets of the universe

Every year, on March 14, mathematicians, scientists, and science enthusiasts celebrate a curious and symbolic date: Pi Day. The choice is far from random. In the date format used in several countries—especially in the United States—March 14 is written as 3/14, which corresponds exactly to the first digits of the most famous number in mathematics.

That number is π, a fundamental mathematical constant representing the ratio between the circumference of a circle and its diameter.

\pi \approx 3.14159265358979323846\ldots

The symbol π became widely used in the eighteenth century, but the constant itself has accompanied humanity for thousands of years. Ancient civilizations had already noticed that there was a fixed proportion between the diameter of any circle and its circumference, regardless of the circle’s size.

What no one imagined at that time was that this seemingly simple number would become one of the keys to understanding phenomena ranging from basic geometry to the fundamental laws governing the universe.

In this special article for Cangando Grilo, we explore the fascinating history of π, understand why it appears in unexpected areas of mathematics, and uncover a few curiosities so surprising that they almost seem to challenge the logic of reality itself.

What is π and why does it exist?

In simple terms, π is the constant that defines the relationship between two measurements of any circle:

• the circumference

• the diameter

If you measure the circumference of a circle and divide it by its diameter, the result will always be π.

This relationship can be expressed mathematically as:

Circumference = π × diameter

Or, equivalently:

Circumference = 2πr

This value never changes. A small circle, a plate, a car wheel, or even the orbit of a planet all obey the exact same ratio.

This means π is not merely an interesting number—it is a fundamental property of geometry itself.

An infinite and unpredictable number

One of the most fascinating characteristics of π is that it is an irrational number.

This means its decimal representation:

• has infinitely many digits

• never repeats in a predictable pattern

Unlike numbers such as 1/3 (0.3333…) or 1/7 (0.142857…), π never falls into a repeating cycle.

Even with the most powerful computers in the world, mathematicians can only compute increasingly long approximations.

Today, more than one hundred trillion digits of π have been calculated.

Interestingly, these massive calculations serve more as tests of computational power and algorithms than practical needs. For almost any real-world calculation, only a few dozen decimal places are necessary.

The millennia-long history of π

Long before modern mathematics, ancient civilizations already had approximate values for π.

The Babylonians, around four thousand years ago, used an approximation of 3.125. The Egyptians, around 1650 BCE, used a value close to 3.16.

One of the greatest advances came from Archimedes, the famous Greek mathematician of the third century BCE.

Archimedes used polygons inscribed inside and outside circles to estimate π with remarkable precision for his time. With this geometric method, he concluded that π was between:

3.1408 and 3.1429.

For the technology of that era, this achievement was extraordinary.

For centuries afterward, mathematicians continued refining approximation methods, until the development of calculus and modern computing enabled dramatic progress.

π appears where no one expects it

Perhaps the most intriguing feature of π is the fact that it appears in areas of mathematics that seem completely unrelated to circles.

A classic example was discovered by the mathematician Leonhard Euler in the eighteenth century.

He solved a problem known as the Basel problem, which involves the infinite sum of fractions:

\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}

The result of summing infinitely many simple numbers revealed something surprising: π appears once again.

This type of phenomenon occurs repeatedly throughout mathematics, suggesting that π is deeply connected to the fundamental structure of numbers themselves.

The most beautiful equation in mathematics

Another striking appearance of π occurs in the famous Euler’s identity, often regarded as the most elegant equation in mathematics.

It connects five fundamental mathematical constants:

• e

• π

• i

• 1

• 0

The relationship is expressed as:

e^{i\pi} + 1 = 0

This equation unites seemingly distant concepts such as exponential growth, complex numbers, and trigonometry into a single concise expression.

For many mathematicians, it represents a kind of mathematical masterpiece.

The universe can be measured with only a few digits of π

Although π has infinitely many decimal places, in practice only a handful are needed for extremely precise calculations.

With only 15 decimal digits, engineers can compute circumferences with enough precision for virtually any practical application.

With about 39 decimal digits, it would be possible to calculate the circumference of a circle the size of the observable universe with an error smaller than the diameter of an atom.

This means that the trillions of digits calculated today are essentially demonstrations of computational power rather than practical necessities.

Calculating π without using circles

One of the most surprising aspects of mathematics is that π can be calculated through formulas that do not involve circles at all.

For example, a series discovered by Gottfried Wilhelm Leibniz shows that π can emerge from an infinite sum of fractions involving only odd numbers:

\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots

Another extraordinary example was discovered by the Indian mathematician Srinivasa Ramanujan, who developed extremely efficient series for calculating π.

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

Each additional term in this series yields several new correct digits of π, making it remarkably powerful for high-precision calculations.

π may contain all possible information

There exists a fascinating mathematical hypothesis known as the normal number.

If π truly is a normal number, its digits would contain every possible sequence of numbers.

This would mean that somewhere within the decimal expansion of π one could theoretically find sequences representing:

• every book ever written

• every possible phone number

• every possible musical composition

• every possible digital image

In theory, every conceivable piece of information could appear somewhere in the digits of π.

Of course, locating such sequences would be practically impossible because they would occur unimaginably far within the expansion.

Nevertheless, this idea reinforces the almost mysterious nature of this number.

Pi Day

The celebration of Pi Day began officially in 1988 at the Exploratorium, a well-known science museum in San Francisco.

The date, March 14 (3/14), was chosen as a natural tribute to the first digits of π.

Since then, the day has become a worldwide celebration of mathematics and science.

In schools, universities, and research centers, Pi Day is often celebrated with:

• lectures

• mathematical challenges

• digit memorization contests

• and, naturally, pies—a playful pun on the English pronunciation of π.

Interestingly, March 14 is also the birthday of Albert Einstein, one of the greatest physicists in history.

A simple number that reveals the complexity of the universe

In the end, π is much more than a mathematical constant.

It appears in:

• geometry

• physics

• wave theory

• statistics

• quantum mechanics

• cosmology

Whenever nature involves cycles, oscillations, rotations, or symmetry, π tends to emerge.

This suggests that the number is not merely a mathematical curiosity but perhaps a reflection of the very structure of the universe itself.

Perhaps that is why π has fascinated mathematicians and scientists for millennia.

And it is exactly why, every year on March 14, the world celebrates this extraordinary number that connects mathematics, science, and mystery.