 # Unravelling the Pi Paradox: The Pioneering Role of Ancient Indian Mathematicians

In the annals of mathematical history, few figures shine as brightly as Madhava of Sangamagrama, the Indian mathematician and astronomer who revolutionized our understanding of mathematical concepts. Born in the 14th century in what is now Kerala, India, Madhava’s groundbreaking work laid the foundation for the development of calculus, centuries before Western mathematicians like Leibniz and Newton entered the scene. His pioneering work on infinite series and trigonometric functions marked a significant leap forward in the mathematical sciences, setting the stage for future advancements.

Madhava’s most notable contribution, however, lies in his exploration of the mathematical constant Pi (π). His innovative approach to approximating Pi, known as the Madhava-Leibniz series, was able to compute Pi up to 11 decimal places. This was a remarkable achievement for the time and a testament to the advanced state of Indian mathematics during his era. But Madhava’s work on Pi did not stop at mere approximation. He delved deeper, uncovering hidden patterns and correction terms that allowed for even more precise calculations. This article, “Unraveling the Pi Paradox: The Pioneering Role of Ancient Indian Mathematician,” seeks to shed light on Madhava’s profound contributions to the understanding of Pi and the paradoxes he uncovered along the way.

The Powell’s Pi Paradox, a formula for Pi built from odd numbers, has been a cornerstone in the computation of Pi. By summing an infinite number of terms, it can compute Pi to many digits. This formula reveals a surprising pattern of consecutive digits that repeat multiple times before deviating. This paradoxical observation, discovered in 1983, has been a source of intrigue for mathematicians worldwide.

However, the paradoxical property of Pi is not a modern discovery. Ancient Indian mathematical discoveries, including the Leibniz formula, provide an explanation. The Leibniz formula, credited to Western mathematicians Leibniz and Gregory, was actually discovered by the Indian mathematician Madhava of Sangamagrama 200 years prior. This fact, though overlooked for centuries, has recently been acknowledged by the global mathematical community.

Madhava, along with other ancient Indian mathematicians, made significant strides in advanced calculus concepts. They discovered the power series expansions for inverse tangent, sine, and cosine, centuries before Leibniz and Newton. Madhava, in particular, found hidden patterns in a series that allowed him to speed up convergence and approximate Pi as approximately equal to 4.

By the 14th century, Indian mathematicians had developed even better approximations for Pi, such as 355/113, which coincided with Pi up to 7 digits. By analyzing the differences between the partial sums and 355/113, they found patterns that improved the approximation of Pi. This discovery suggested a smarter way to choose the last number to add in Madhava’s formula for Pi.

The ancient Indian mathematicians also discovered a method to improve the approximation of Pi by adding or subtracting fractions and using correction terms. They discovered a refinement of the correction term that improved the approximation of Pi from 3 to 15 digits. However, they were unable to discern any further patterns.

The Pi paradox in 14th century India revealed that strings of 0s and 9s coincided with the regions of coincidence between Pi and the millionth partial sum. This led to the discovery of correction terms that explain these coincidences. Madhava’s third correction term failed to explain the remaining coincidences of the millionth sum and Pi, but there is a fourth correction term and an infinite sequence of these correction term refinements that can account for these coincidences.

Medieval Indian mathematicians made significant discoveries in calculus, including correction terms for Pi, which were later used to create new formulas for Pi. Many of their original records have been lost, but Madhava’s mathematical discoveries, as documented by his disciples, included complete proofs and correction terms for Pi. These correction terms in the sequence can be transformed into new infinite series formulas for Pi through algebraic manipulation.

The first two terms of the series in the old Indian manuscripts were written in simple and beautiful ways, expressed in verse rather than concise mathematical language. This poetic approach to mathematics is a testament to the rich and vibrant history of mathematical discovery in ancient India.

In conclusion, the ancient Indian mathematicians’ contributions to the understanding of Pi and calculus are profound and significant. Their work, though often overlooked, laid the foundation for many modern mathematical principles and continues to inspire mathematicians today.

## Madhava-Leibniz series One of the most significant contributions of Indian mathematicians to the approximation of Pi is the Madhava-Leibniz series. This series is given by:

Pi/4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11 + ….

If you calculate the first few terms of this series and multiply by 4, you get the following approximations for Pi:

4 * (1) = 4

4 * (1 – 1/3) = 2.666…

4 * (1 – 1/3 + 1/5) = 3.466…

4 * (1 – 1/3 + 1/5 – 1/7) = 2.895…

As you can see, the more terms you add, the closer you get to the value of Pi, which is approximately 3.14159.

However, Madhava discovered that you can get a much better approximation of Pi by adding a correction term to the series. The correction term is of the form 1/N, where N is the number of terms. For example, if you add the first four terms of the series and then add the correction term 1/4, you get:

4 * (1 – 1/3 + 1/5 – 1/7 + 1/4) = 3.339…

This is a much closer approximation to Pi than without the correction term.

Madhava and other Indian mathematicians discovered further correction terms that improved the approximation of Pi even more. For example, they found that by adding a term of the form 1/N^2, where N is the number of terms, they could improve the approximation of Pi from 3 to 15 digits.

These discoveries of patterns and correction terms in the series for Pi were significant advancements in calculus. They laid the groundwork for the development of the concept of a limit, which is a fundamental concept in calculus. The work of Madhava and other Indian mathematicians on the approximation of Pi is a testament to the advanced state of mathematics in ancient India. 