Pi, represented by the Greek letter π, is one of the most important mathematical constants. It is defined as the ratio of a circle's circumference to its diameter. No matter how large or small the circle is, this ratio is always the same constant value, approximately 3.14159.
Throughout history, many ancient civilizations attempted to calculate the value of pi. The Babylonians, around 2000 BCE, approximated pi as 3.125. The ancient Egyptians used a value of approximately 3.16. These early approximations were remarkably close to the true value, considering the limited mathematical tools available at the time.
The ancient Greek mathematician Archimedes developed a method to calculate pi using polygons. He inscribed and circumscribed regular polygons around a circle. The inscribed polygon has a smaller perimeter than the circle, while the circumscribed polygon has a larger perimeter. As we increase the number of sides, both polygons approach the circle's circumference, giving us better approximations of pi.
Modern mathematics has developed sophisticated methods to calculate pi with extraordinary precision. The Leibniz formula uses an infinite series, while Machin's formula provides faster convergence. Today, computers have calculated pi to trillions of decimal places. Pi appears everywhere in mathematics and science: in wave equations, Fourier transforms, probability distributions, and signal processing. Its ubiquity makes it one of the most important constants in all of mathematics.