You might think that if you simply started adding the natural numbers, 1 plus 2 plus 3 and so on all the way to infinity, you would get a pretty big number.
So it came as a shock to many people when, in a recent video, a pair of physicists purported to prove that this infinite series actually adds up to … minus 1/12.
More than 1.6 million people have viewed this calculation, which plays a key role in modern physics and quantum theory; the answer, as absurd as it sounds, has been verified to many decimal places in lab experiments.
Even the makers of the video — Brady Haran, a journalist, and Ed Copeland and Antonio Padilla, physicists at the University of Nottingham in England — admit there is a certain amount of “hocus-pocus,” or what some mathematicians have called dirty tricks, in their presentation. But there is broad agreement that a more rigorous approach to the problem gives the same result, as shown by a formula in Joseph Polchinski’s two-volume textbook “String Theory.”
So what’s going on with infinity? “This calculation is one of the best-kept secrets in math,” said Edward Frenkel, a mathematics professor at the University of California, Berkeley, and author of “Love and Math: The Heart of Hidden Reality.” “No one on the outside knows about it.”
The first one down this road was the great 18th-century mathematician Leonhard Euler, who was born in Switzerland but did most of his work in Berlin and St. Petersburg, Russia. In 1749, he used a bag of mathematical tricks to solve the problem of adding the natural numbers from 1 to infinity. Clearly, if you stop adding anywhere along the way — at a quintillion (1 with 18 zeros after it), say, or a googolplex (10 to the 100th power zeros) — the sum will be enormous. The problem with infinity is that you can’t stop. You never get there. It’s more of a journey than a destination.
In modern terms, Frenkel explained, the gist of the calculations can be interpreted as saying that the infinite sum has three parts: One blows up when you go to infinity, one goes to zero, and minus 1/12. The infinite term, he said, is just thrown away.
New york Times