How Indian Mathematicians Calculated The Value Of Pi

Mathematics in India has a rich history. Indian mathematicians Madhava and Aryabhata made very significant contributions in finding the exact value of π (pi).

Professor K Ramasubramanian, who teaches at the Department of Humanities and Social Sciences, IIT Bombay, has wonderfully explained the history of calculation of pi in India in the above video.

Of the various infinite series representation of pi, one of the oldest is today ascribed to is Gregory and Leibniz.

“But almost three hundred years before Gregory and Leibniz's formula came to the fore, there was a series which was codified in the form of verse by an Indian mathematician Madhava,” he says.

“If the series were to be given a name which honours the founder, then it should be called Madhava series instead of Gregory-Leibniz series,” he said.

So what was the context in which people wanted to know the value of Pi in the Indian tradition?

One of the reasons why Indians would have wanted to know the value of pi, would have been for the construction of altars for havana. According to Professor Ramasubramanian, each household was supposed to have three of them—one square, one circular, and third semi-circular one. The only condition was that the area of all the three altars be the same. This would have made it necessary for the society to be aware of the value of pi.

The other Indic mathematical genius to calculate the value of pi, upto four digits after decimal, was Aryabhata. In the the video embedded, Prof Ramasubramanian quotes and explains the exact verses of Aryabhata where he deals with the value of pi.

Cosmic Triangles Open a Window to the Origin of Time

Cecile G. Tamura

A close look at fundamental symmetries has exposed hidden patterns in the universe. Physicists think that those same symmetries may also reveal time’s original secret.

The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities

(https://www.quantamagazine.org/the-origin-of-time-bootstrapped-from-fundamental-symmetries-20191029/?fbclid=IwAR3Ty4_RN5CpbSXTVHdRaGRvs2HEltqypC3FKPNEu6qjR6jDu49w6WWSUAw)

“We look at patterns in space today, and we infer a cosmological history in order to explain them.

The approach has the potential to help explain why time began, and why it might end.

As Arkani-Hamed put it, “The thing that we’re bootstrapping is time itself.”

"One curious pattern cosmologists have known about for decades is that space is filled with correlated pairs of objects: pairs of hot spots seen in telescopes’ maps of the early universe; pairs of galaxies or of galaxy clusters or superclusters in the universe today; pairs found at all distances apart.

You can see these “two-point correlations” by moving a ruler all over a map of the sky. When there’s an object at one end, cosmologists find that this ups the chance that an object also lies at the other end."

"The simplest explanation for the correlations traces them to pairs of quantum particles that fluctuated into existence as space exponentially expanded at the start of the Big Bang. Pairs of particles that arose early on subsequently moved the farthest apart, yielding pairs of objects far away from each other in the sky today. Particle pairs that arose later separated less and now form closer-together pairs of objects. Like fossils, the pairwise correlations seen throughout the sky encode the passage of time — in this case, the very beginning of time."

A Map of the Start of Time

In 1980, the cosmologist Alan Guth, pondering a number of cosmological features, posited that the Big Bang began with a sudden burst of exponential expansion, known as “cosmic inflation.” Two years later, many of the world’s leading cosmologists gathered in Cambridge, England, to iron out the details of the new theory. Over the course of the three-week Nuffield workshop, a group that included Guth, Stephen Hawking, and Martin Rees, the future Astronomer Royal, pieced together the effects of a brief inflationary period at the start of time. By the end of the workshop, several attendees had separately calculated that quantum jitter during cosmic inflation could indeed have happened at the right rate and evolved in the right way to yield the universe’s observed density variations.

To understand how, picture the hypothetical energy field that drove cosmic inflation, known as the “inflaton field.” As this field of energy powered the exponential expansion of space, pairs of particles would have spontaneously arisen in the field. (These quantum particles can also be thought of as ripples in the quantum field.) Such pairs pop up in quantum fields all the time, momentarily borrowing energy from the field as allowed by Heisenberg’s uncertainty principle. Normally, the ripples quickly annihilate and disappear, returning the energy. But this couldn’t happen during inflation. As space inflated, the ripples stretched like taffy and were yanked apart, and so they became “frozen” into the field as twin peaks in its density. As the process continued, the peaks formed a nested pattern on all scales.

After inflation ended (a split second after it began), the spatial density variations remained. Studies of the ancient light called the cosmic microwave background have found that the infant universe was dappled with density differences of about one part in 10,000 — not much, but enough. Over the nearly 13.8 billion years since then, gravity has heightened the contrast by pulling matter toward the dense spots: Now, galaxies like the Milky Way and Andromeda are 1 million times denser than the cosmic average. As Guth wrote in his memoir (referring to a giant swath of galaxies rather than the wall in China), “The same Heisenberg uncertainty principle that governs the behavior of electrons and quarks may also be responsible for Andromeda and The Great Wall!”

Sai Baba Family rare photoஓம் ஶ்ரீசாயியை பணிவோம்