If you buy from a link in this post, I may earn a commission. This does not affect the price you pay. As an Amazon Associate I earn from qualifying purchases. __Learn more.__

Posted June 30, 2024 By Presh Talwalkar. Read __about me__, or __email me__.

While studying quantum theory, two scientists using string theory have accidentally discovered a new formula for pi, the most famous constant in mathematics. Arnab Saha and Aninda Sinha developed a formula to optimize certain calculations and they were surprised the formula could also calculate the digits of pi. Even more, under a certain limit, the formula equals Madhava’s formula for pi, which was the first recorded infinite series for pi. Here is a link to the study published in Physical Review Letters.. While the work is theoretical at this point, it may find practical applications in the future.

Check out the video for more details on the story.

**Scientists just discovered a new formula for pi accidentally!**

Or keep reading.

.

.

"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.

..

.

.

.

.

M

I

N

D

.

Y

O

U

R

.

D

E

C

I

S

I

O

N

S

.

P

U

Z

Z

L

E

.

.

.

.

**Scientists just discovered a new formula for pi accidentally**

By way of history, pi is a concept that equals the ratio of a circle’s circumference to its diameter. One of the first known estimates comes from Archimedes, who used perimeters of inscribed and circumscribed polygons to bound the circumference of a circle. He came up with 3.14 or 22/7 using a 96 sided polygon. About 500 years later Liu Hui used a slightly different method and tricks using areas of polygons instead of perimeters. He and subsequent mathematicians came up with pi as 3.14159 or 355/113 which was the most accurate approximation of pi for about 100 years. Calculating pi by geometric methods had essentially reached its limit.

Around the late 1300s to early 1400s the mathematician Madhava came up with an entire different approach. He discovered the most wonderful infinite series

π/4 = 1 – 1/3 + 1/5 – 1/7 + …

(The series is unfortunately “named after” Leibniz who was born hundreds of years later than Madhava. The very mathematicians who scrutinize every word of a proof do not have the same zeal for crediting originators, though some are now saying Madhava-Leibniz, so perhaps the convergence on this process is just very slow.)

What an incredible formula that relates pi to the alternating sum of reciprocals of odd numbers! This formula is amazing but is has a flaw: it is very slow to converge. It takes about 5 billion terms to get pi accurate to 10 digits. Madhava actually modified the formula and was able to estimate pi to 10 to 14 digits. In modern times, other infinite series are used and computers do the calculations, so the modern record has over 100 trillion digits of pi computed.

**The new formula**

In the current paper, the Indian scientists developed the following formula:

The meaning is beyond me, but if we take this as a pure mathematical formula, we can try to understand a bit about it. On the right hand side is the familiar factorial *n*! like 4! = 4x3x2x1 = 24. The factorial can be generalized to negative values and fractional values with the gamma Γ function, which appears on the left hand side. For positive integral values the gamma function is a shift of the factorial function with Γ(*n*) = (*n* – 1)!. Also on the right hand side is a closed parenthesis with a subscript *n* – 1, and that indicates the Pochhammer symbol which is a ratio of gamma functions (*a*)_{b} = Γ(*a* + *b*)/Γ(*a*).

The gamma function is an integral

It can be evaluated to particular values like Γ(1/2) = √π and Γ(2) = 1.

So going back to their main formula

If we substitute *s*_{1} = *s*_{2} = -1/2, and simplify, we get the new formula for pi with the parameter λ

Interestingly this formula relates to an ancient pi formula. If you take the limit as λ goes to infinity and divide both sides by 4 you get exactly the Madhava series for pi!

π/4 = 1 – 1/3 + 1/5 – 1/7 + …

While Madhava’s formula is very slow to converge, the new formula can calculate pi to 10 digits using only 30 terms! You can even try it out as the factorial and gamma functions are built-in to spreadsheets. Taking λ = 10 we get:

The formula does rely on the gamma function, and if one could quickly calculate that, one could perhaps calculate Γ(1/2) = √pi and get perhaps a more direct way to get the digits of pi.

There is no claim the formula has practical uses at the present, or that it will break any records. But it is amazing that a string theory formula can relate to a historically important series for pi. It is these accidental discoveries that remind us of the beauty and wonder of mathematics and science.

**References**

Press release

https://iisc.ac.in/events/iisc-physicists-find-a-new-way-to-represent-pi/

Paper

https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.132.221601

Sabine Hossenfelder video

https://www.youtube.com/watch?v=rGd7Db52w1Q

References

https://www.iflscience.com/physicists-accidentally-discover-a-whole-new-way-to-write-pi-74768

https://phys.org/news/2024-06-physicists.html

https://www.indiatoday.in/science/story/indian-iisc-physicists-untangle-new-pi-series-that-could-change-maths-forever-2555611-2024-06-20

https://www.msn.com/en-in/entertainment/tv/indian-physicists-untangle-new-pi-series-that-could-change-maths-forever/ar-BB1oyGde

https://www.ynetnews.com/health_science/article/bk7d5ibir

https://mathoverflow.net/questions/473931/possible-new-series-for-pi

https://math.stackexchange.com/questions/4937730/is-the-new-series-for-a-big-or-even-medium-deal

https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80

## Published by

### PRESH TALWALKAR

I run the MindYourDecisions channel on YouTube, which has over 1 million subscribers and 200 million views. I am also the author of The Joy of Game Theory: An Introduction to Strategic Thinking, and several other books which are available on Amazon.

(As you might expect, the links for my books go to their listings on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.)

By way of history, I started the Mind Your Decisions blog back in 2007 to share a bit of math, personal finance, personal thoughts, and game theory. It's been quite a journey! I thank everyone that has shared my work, and I am very grateful for coverage in the press, including the Shorty Awards, The Telegraph, Freakonomics, and many other popular outlets.

I studied Economics and Mathematics at Stanford University.

People often ask how I make the videos. Like many YouTubers I use popular software to prepare my videos. You can search for animation software tutorials on YouTube to learn how to make videos. Be prepared--animation is time consuming and software can be expensive!

Feel free to send me an email [emailprotected]. I get so many emails that I may not reply, but I save all suggestions for puzzles/video topics.

### MY BOOKS

If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.

Book ratings are from January 2023.

(US and worldwide links)

https://mindyourdecisions.com/blog/my-books

**Mind Your Decisions** is a compilation of 5 books:

(1) The Joy of Game Theory: An Introduction to Strategic Thinking

(2) 40 Paradoxes in Logic, Probability, and Game Theory

(3) The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias

(4) The Best Mental Math Tricks

(5) Multiply Numbers By Drawing Lines

**The Joy of Game Theory** shows how you can use math to out-think your competition. (rated 4.3/5 stars on 290 reviews)

**40 Paradoxes in Logic, Probability, and Game Theory** contains thought-provoking and counter-intuitive results. (rated 4.2/5 stars on 54 reviews)

**The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias** is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 4.1/5 stars on 33 reviews)

**The Best Mental Math Tricks** teaches how you can look like a math genius by solving problems in your head (rated 4.3/5 stars on 116 reviews)

**Multiply Numbers By Drawing Lines** This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 4.4/5 stars on 37 reviews)

**Mind Your Puzzles** is a collection of the three "Math Puzzles" books, volumes 1, 2, and 3. The puzzles topics include the mathematical subjects including geometry, probability, logic, and game theory.

**Math Puzzles Volume 1** features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 112 reviews.

**Math Puzzles Volume 2** is a sequel book with more great problems. (rated 4.2/5 stars on 33 reviews)

**Math Puzzles Volume 3** is the third in the series. (rated 4.2/5 stars on 29 reviews)

### KINDLE UNLIMITED

Teachers and students around the world often email me about the books. Since education can have such a huge impact, I try to make the ebooks available as widely as possible at as low a price as possible.

Currently you can read most of my ebooks through Amazon's "Kindle Unlimited" program. Included in the subscription you will get access to millions of ebooks. You don't need a Kindle device: you can install the Kindle app on any smartphone/tablet/computer/etc. I have compiled links to programs in some countries below. Please check your local Amazon website for availability and program terms.

US, list of my books (US)

UK, list of my books (UK)

Canada, book results (CA)

Germany, list of my books (DE)

France, list of my books (FR)

India, list of my books (IN)

Australia, book results (AU)

Italy, list of my books (IT)

Spain, list of my books (ES)

Japan, list of my books (JP)

Brazil, book results (BR)

Mexico, book results (MX)

### MERCHANDISE

Grab a mug, tshirt, and more at the official site for merchandise: **Mind Your Decisions at Teespring**.