Mathematicians have discovered the value of pi through various methods, most notably through geometric approximation techniques and, later, infinite series and computational algorithms.
Here's a breakdown of how it was achieved:
Early Geometric Methods: Archimedes' Approach
One of the earliest and most influential methods for approximating pi was developed by Archimedes. His method involved inscribing and circumscribing regular polygons around a circle.
- Inscribed Polygons: He drew a regular hexagon inside a circle. The perimeter of this hexagon is less than the circumference of the circle.
- Circumscribed Polygons: He drew a regular hexagon outside the circle. The perimeter of this hexagon is greater than the circumference of the circle.
- Increasing Sides: Archimedes successively doubled the number of sides of the polygons (to 12, 24, 48, and finally 96 sides).
- Calculating Perimeters: By calculating the perimeters of these polygons, he obtained increasingly accurate lower and upper bounds for the circumference of the circle, and thus for pi.
- Result: Archimedes proved that 223/71 < π < 22/7 (approximately 3.1408 < π < 3.1429).
In essence, he "squeezed" the value of pi between two known values, progressively narrowing the range.
Infinite Series
Later mathematicians discovered infinite series that converge to pi. These series provide a means to calculate pi to a high degree of accuracy. Some examples include:
- Leibniz Formula: π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ... (This converges very slowly)
- More efficient series: Many other series converge much faster, enabling more precise calculations.
Computational Algorithms
With the advent of computers, mathematicians developed sophisticated algorithms to calculate pi to trillions of digits. These algorithms are based on infinite series and other mathematical formulas and leverage the computational power of modern computers.
Summary
The determination of pi's value has evolved from geometric approximations to sophisticated mathematical formulas and computational methods, each building upon previous knowledge and technological advancements.
