A circular number with a difference: Physicists have discovered a new mathematical approach to the circular number pi – by pure chance. They were actually looking for a formula that could be used to calculate the amplitude of vibrating strings. But they also found a new mathematical series formula to calculate π. This formula allows a relatively quick estimate of the value of the circuit number.
The circle number π is a universal natural constant. Not only does it describe the relationship between the circumference of a circle and its diameter, but it also plays a role in many physical phenomena – from vibrations to Heisenberg’s uncertainty principle. Mathematically speaking, Pi is almost everywhere, but at the same time it is difficult to understand. Because a transcendent irrational number has an infinite number of seemingly arbitrary decimal places.
Approaching Bay
The decimal places of the circuit number are usually calculated using various approximations, for example in the form of mathematical molecular formulas (strings). One of the earliest string representations of Pi was developed over 600 years ago by the Indian mathematician Sangamagrama Madhava. Its molecular formula approximates the circle number by alternately adding and subtracting all the fractions with odd denominators – but it requires millions of fractions to get close to the value of pi.
But now two Indian physicists have discovered a much shorter and simpler series formula that can be used to determine pi by chance. “Our goal was never to get a new view of Pi. Instead, we used quantum theory to develop a model that has fewer and more precise parameters for how particles interact,” explains Aninda Sinha of the Indian Center for High Energy Physics in Bangalore.
Oscillating strings and Euler’s beta function
Specifically, Sinhan and his colleague Arnab Piya Saha were looking for a way to calculate the amplitudes of vibration of strings — the hypothetical “strings” that, according to string theory, form the basis of all elementary particles in the universe. How these strings vibrate determines the properties of molecules such as their energy and mass. “String field theory suggests the existence of a field-like representation of string amplitudes, which operates not only at lower energies but also at higher masses,” the physicists explain.
In searching for this equation, Sinha and Saha turned to two physics-mathematical tools. The first is Euler’s beta function, a function of two complex numbers that can also be used to calculate the amplitudes of simpler strings, the researchers explain. They combined this with the Feynman diagram, which describes the exchange of energy between two interacting particles. In the course of their complex mathematical calculations, physicists were finally able to find the solution they were looking for to describe the string.
New series for me
But that was not all: Saha and Sinha also discovered a new formula for the circular number Pi. This formula or molecular chain is related to the Madhava chain, but is much more efficient. “While the Madhava series needs five billion terms to reach ten decimal places, lambda values between 10 and 100 do so in 30 terms,” the physicists wrote. Combining the parameters in this molecular formula makes it possible to quickly determine the value of Pi and include it in equations.
“Physicists (and mathematicians) have ignored this until now because they didn’t have the right computational tools,” Sinha explains. He and his team have been developing these technologies in just the past three years. “Researchers actually looked at this approach in the early 1970s, but quickly abandoned it because it was too complicated.” However, it is another way to approach this irrational number. (Physical Review Letters, 2024; doi: 10.1103/PhysRevLett.132.221601)
Source: Indian Institute of Science (IISc)
June 24, 2024 – Nadja Podbrigar
“Prone to fits of apathy. Zombie ninja. Entrepreneur. Organizer. Evil travel aficionado. Coffee practitioner. Beer lover.”
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