Well, the sum of all positive numbers is equal to negative one twelfth afterall. Or at least I found the part in the textbook that explains it. The downside is that it’s not explained so much as there’s a problem telling the reader to figure it out for themselves. Problem 12.4. It was assigned as a homework problem a few weeks ago and everything, so in theory I should understand it.
It has something to do with the zeta function, defined as:
The argument can be complex, but there is a finite answer for any where the real part is greater than one. But what we want to know is what happens if .
Apparently you do some fancy calculus trickery (analytic continuation), which I won’t try to explain because frankly it’s mostly lost on me, to get an equation for which is valid down to . Then you can find not only
For the record, that has nothing to do with string theory. It’s just mathematics. Crazy complex variable calculus, to be sure, but still just mathematics.
In english, not only is the sum of all positive numbers (1+2+3+…) equal to negative one twelfth, but if you add one to itself an infinite number of times (1+1+1+1…) you get negative one half.
Bet you didn’t see that coming.