article Spring 2023

The Road to Unraveling the Secrets of the Universe starts with … -1/12

One of the most important values in all of physics is -1/12, and the number comes about only by staring infinity in the face and accepting it as nothing more than a convenient mathematical idea.

If you were to add all the natural numbers – every positive integer from one to infinity – what would your sum equal? A billion? A quintillion? Perhaps the illustrious Googolplex? But each of these requires the mathematician to stop adding at a finite number, rather than looking infinity in the face. When you summon the courage to break away from the elementary way of thinking about numbers, the dust clears, and we find the sum of all the natural numbers to be a mere -1/12.

In 1859, German mathematician Bernard Riemann worked off the questions asked and solved a century prior by Swiss mathematician Leonard Euler. Euler solved this very question using a method that would eventually be called the Riemann Zeta Function, which sums all the natural numbers from 1 to raised to the same integer power.

The Riemann Zeta Function evaluated for s = -1 may seem simple enough, but these sums each play important roles in the development and understanding of cutting-edge physics, from string theory to thermodynamics. The roots of these revolutionary developments can all be traced back to the same place: -1/12.

So, why is this? While the Riemann Zeta Function can be used to complete the proof, Antonio Padilla, a professor of physics at the University of Nottingham in Nottingham, United Kingdom, insists that we “needn’t get our knickers in a twist” to solve this problem. In Brady Haran’s podcast Numberphile, Padilla and his fellow physics professor, Ed Copeland, prove this phenomenon.

While this proof has stirred up some controversy among physicists and mathematicians for its use of shortcuts, Copeland and Padilla manipulate just a few sums and yet again arrive at that one magical number. 

This result may seem confusing and the opposite of intuitive. Haran ponders at the end of the podcast, “If I sat here and punched into a calculator until the end of everything and then hit enter [on the sum of all of the natural numbers], would I [reach -1/12?]”  But the answer brings us back to the very concept of infinity.

Infinity is exactly that – a concept. It is not a real thing, and therefore, it is not possible for any calculator to spit out that result. Physics has yet to measure an example of infinity in nature, so infinity is nothing but a tool to simplify and drive discovery. Calculations to find the critical number of dimensions in String Theory rely on the result of infinity being -112. The Stefan-Boltzmann Law, which tells us about the energy flux across a black-body, also depends on our abilities as physicists to look infinity in the face and acknowledge it as nothing more than a tool toward advancing the knowledge of humanity.

S1 = 1-1+1-1+… = 1/2
S2 = 1-2+3-4+…
2Ss = 1+-2+1+3-2+-4+3+…=1-1+1-1+… = 1/2
S2 = 1/4
S = 1+2+3+4+…
S-S2 = 0+4+0+8+0+12+… = 4S
-3S = 1/4
S = -1/12

Above you can see the (controversial) proof that Copeland and Padilla use to prove that all whole numbers sum to -1/12.

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