Welcome to Numberholic! I’m your host, Sejari, and you’re watching the fifth episode of Proven Yet Hard to Believe. In this series, we explore groundbreaking discoveries and proofs that shaped the world of mathematics, yet remain incredible to grasp. Today’s topic is one of precision and brilliance: the Madhava-Leibniz series for pi. A formula so accurate, it was proven back in 1673. But here’s the twist—this series was discovered in India over 200 years earlier by Madhava! Are you ready to delve into the journey of this remarkable formula? Let’s get started!
There exists a precise formula for pi—the Leibniz series, proven in 1673. But, remarkably, this discovery predates Leibniz by 200–300 years. An Indian mathematician named Madhava first uncovered this formula, which is why it is often called the Madhava-Leibniz series.
Let’s derive this formula using concepts accessible at a high school level.
First, we begin with the formula for the sum of a geometric series.
After rearranging the equation,
we integrate both sides.
The right-hand side involves no significant difficulties.
We can bypass a direct calculation by introducing inequalities to simplify the process.
However, the left-hand side presents a challenge.
At the high school level, it requires the substitution method of integration, specifically trigonometric substitution, to solve.
Even historically, this step was challenging. Let’s see how Leibniz approached it.
He used the following diagram to tackle the problem.
Before diving deeper, let’s first clarify two key ideas.
The first is the concept of breaking down a curve into a sum of infinitesimally short straight lines—or segments. This idea inspired not just Leibniz, but also researchers like Fermat, his contemporary Newton, and many others of the time.
The second is the term “moment,” which we’ll simplify in this video as the “distance-length product.” This concept played a significant role in applying the principles of levers to mathematics.
In Leibniz’s proof, the distance-length product appears as x times ds,
where x times ds represents a tiny area—a mathematical infinitesimal.
For now, think of it as just a tool for deriving the equations.
Now, let’s begin. Consider a circle with a radius of 1.
On the arc AT, select a point B and consider an infinitesimally short segment of length ds in the tangential direction. Draw the tangent at B and extend it to meet the tangent at A at a point P. From B, drop a perpendicular to the diameter OA, meeting it at D. Let the length of OD be x, and the length of AP be y. Extend the segment on the line containing BD by the length y.
As point B moves along the arc AT, repeating this construction creates a graph.
Next, let’s establish the relationship between x and y.
Using the Pythagorean theorem in triangle OBD, we can calculate the length of BD.
Label the angles, and observe that triangle OBD is similar to triangle CBD. Using the similarity ratio, we obtain an equation relating x and y.
Now, consider an infinitesimally small triangle at B, with the hypotenuse as the segment of length ds. This triangle is similar to triangle OBD, allowing us to derive a relationship involving the distance-length product.
This infinitesimal triangle is also similar to another triangle in the diagram, giving us another equation for the distance-length product.
Interestingly, the red rectangle’s area in the diagram equals the distance-length product.
Now we sum these distance-length products for every point B on the arc AT.
Using the first equation, we quickly reach the result.
Using the second equation, summing these infinitesimal quantities corresponds to integration. The result represents the yellow-shaded area in the diagram.
But this area can also be expressed as the area of a square with side length 1, minus the green-shaded region.
Using the original relationship we derived, we rewrite the integral in a new form.
Finally, we arrive at the Madhava-Leibniz series formula.
This formula is a stunning representation of pi. Yet, even with the power of modern computers, it converges very slowly, which is its primary drawback.
And there you have it—the fascinating story behind the Madhava-Leibniz series for pi! A formula that connects the brilliance of ancient Indian mathematics with the ingenuity of European thinkers like Leibniz. Thank you for joining me on this journey. Don’t forget to like, subscribe, and share your thoughts in the comments below. Until next time, this is Sejari, signing off from Numberholic. See you soon!