The number Pi (π) has a long and fascinating history, extending from Ancient Greece to modern computer chips. Although the constant does not depend on the Earth or our species, it naturally arises when dividing the circumference of a circle by its diameter, making it universally applicable to any civilization that understands geometry.
From the earliest times, peoples in various regions noticed the constant proportion in circles of different sizes, which led to the first approximations of π, even without a formal explanation. The oldest records date back to Mesopotamia, where Babylonian clay tablets from around 1900 BC indicate the use of an approximation equivalent to 3.125, useful for engineering and construction.
Subsequently, the Egyptians refined this knowledge. The Rhind Papyrus, dated to about 1650 BC, details a method for calculating the area of a circle, resulting in an approximation close to 3.1605. It is important to note that, for much of the last two millennia, π was not seen as a sequence of digits; the Hindu-Arabic numeral system and decimal places only spread throughout the West starting in the 12th century.
Marcelo Viana, a mathematician and director-general of the National Institute of Pure and Applied Mathematics (Impa), explains that the initial search was more focused on fractions that approximated π.
The Contribution of Archimedes and Other Cultures
One of the most notable approximations came from the Greek mathematician and philosopher Archimedes. Around 250 BC, he developed an extremely precise method that served as a reference for almost two thousand years. Archimedes constructed concentric polygons inside and outside a circle, progressively increasing the number of sides. By reaching polygons with 96 sides, he used the perimeter of the outer figure to estimate a value greater than the actual one, and the perimeter of the inner figure to obtain a lower value. Thus, he determined that π was contained between the fractions 223/71 and 22/7.
In the 3rd century, in China, Liu Hui expanded Archimedes' method, reaching an approximation of 3.1416 with a polygon of 3,072 sides. Later, in the 5th century, the Chinese scholar Zu Chongzhi presented the fraction 355/113, an approximation that would only be surpassed a thousand years later.
In India, around the 14th century, Madhava of Sangamagrama introduced a completely new approach. He demonstrated that π could be derived from an infinite series of alternating sums, starting with 1, subtracting 1/3, adding 1/5, subtracting 1/7, and so on. This idea was revolutionary because it took π out of the purely geometric field and placed it into mathematical operations, forming the basis of modern calculus.
Pi as an Irrational and Transcendental Number
In the 18th century, Johann Heinrich Lambert questioned whether there was any exact fraction to represent π. In 1761, he proved that such a fraction does not exist, classifying π as an irrational number. Rational numbers can be written as simple fractions and, when converted to decimals, either terminate or form repeating decimals. Pi, in its decimal representation, continues infinitely without ever repeating a pattern.
In 1882, Ferdinand von Lindemann elevated the classification of π, proving that it is a transcendental number. This means that there is no algebraic equation, no matter how complex, capable of generating its exact value. This discovery ended the ancient challenge of squaring the circle, which required constructing a square with the same area as a circle using only a ruler and compass, something impossible due to the nature of π.
The Computational Era and the Mysteries of Pi
With technological advancement, the focus shifted to the ability to calculate more digits of π. Before computing, progress was slow. However, in 1949, the ENIAC calculated 2,037 digits in less than three days, and a little over a decade later, the number exceeded 100 thousand digits. In the 1970s, the mark of one million decimal places was reached.
In November 2025, researchers from StorageReview and Micron Technology set a record by calculating 314 trillion decimal places of π, using a high-performance computer for 110 days. This process now serves as a stress test for hardware and software. An example of this occurred in 1994, when a defect in Intel's Pentium processor was detected by errors in calculations involving π.
Although mathematicians know trillions of digits, adding more billions rarely reveals significant new information, as a few dozen decimal places are sufficient for almost all practical applications. However, scientific curiosity persists: many suspect that π is a normal number, which would imply that any finite sequence of numbers, such as a CPF or a ZIP code, would appear somewhere in its decimal expansion.
The proof that π is a normal number remains a challenge, just like demonstrating that certain mathematical propositions are true but unprovable within a logical system, as shown by Kurt Gödel in 1931. Beyond its theoretical nature, π is omnipresent, helping to describe phenomena such as sound waves, light, planetary motion, and even the distribution of population heights. Mathematician Marcelo Viana describes this presence as one of the most fascinating characteristics of the number, citing the Basel Problem, where Leonhard Euler demonstrated in 1735 that the infinite sum of the reciprocals of the squares of integers results exactly in π²/6, a fact that challenges the apparent connection to circular geometry.