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 the size of the circle, this ratio always equals pi. The value of pi is approximately 3.14159, but it's an irrational number that continues infinitely without repeating. Throughout history, mathematicians have developed various methods to calculate pi with increasing precision.
One of the earliest methods for calculating pi was developed by Archimedes around 250 BCE. His ingenious approach used inscribed and circumscribed polygons to approximate a circle. By calculating the perimeters of these polygons, he established upper and lower bounds for pi. The inscribed polygon, shown in blue, always gives a value less than pi, while the circumscribed polygon in red gives a value greater than pi. As the number of sides increases, both polygons approach the circle, and their perimeters converge toward the true value of pi. Using 96-sided polygons, Archimedes determined that pi lies between three and ten seventy-firsts and three and one seventh, which is approximately 3.1408 to 3.1429. This was an extraordinary achievement for his time.
By the 17th century, mathematicians developed more efficient methods using infinite series to calculate pi. One of the earliest was the Leibniz formula from 1674, which states that pi over 4 equals 1 minus one-third plus one-fifth minus one-seventh, and so on. While elegant in its simplicity, this series converges extremely slowly. As you can see in the graph, even after 10 terms, we're still far from the true value of pi over 4, which is approximately 0.785. In fact, you would need about 10,000 terms to get just 4 correct decimal places! In 1706, John Machin discovered a much faster formula using arctangent functions. His formula, pi over 4 equals 4 times arctangent of one-fifth minus arctangent of one over 239, converges much more rapidly. Machin used this to calculate pi to 100 decimal places. Today, mathematicians use even more sophisticated algorithms based on similar principles to compute pi to trillions of digits.
The Monte Carlo method offers a completely different approach to calculating pi, using probability and random sampling. Imagine a circle with radius r inscribed within a square with side length 2r. The ratio of the circle's area to the square's area is pi divided by 4. To estimate pi, we randomly generate points within the square and count how many fall inside the circle. As we generate more points, the ratio of points inside the circle to the total number of points approaches pi over 4. Therefore, pi approximately equals 4 times the number of points in the circle divided by the total number of points. In our simulation, blue dots represent points inside the circle, while red dots are outside. This method is conceptually simple and easy to implement, but it converges slowly. You need millions of random points to get just a few decimal places of accuracy. However, it's a great illustration of how probability can be used to solve deterministic mathematical problems.
To summarize what we've learned about calculating pi: Pi is defined as the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. Throughout history, mathematicians have developed increasingly efficient methods to calculate this fundamental constant. We've seen Archimedes' method using inscribed and circumscribed polygons, which gave us the first accurate approximation. Later, infinite series formulas like the Leibniz series and Machin's formula enabled much more precise calculations. The Monte Carlo method showed us how probability can be used to estimate pi. Today, mathematicians use advanced algorithms based on these principles to compute pi to trillions of digits. Despite all this progress, pi remains an irrational number that continues infinitely without repeating. Its importance extends far beyond geometry into virtually every field of mathematics, physics, engineering, and science.