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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:

Zeta function

The argument s can be complex, but there is a finite answer for any s where the real part is greater than one. But what we want to know is what happens if s equals negative one.

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 gamma (s) times zeta (s) which is valid down to real part of s greater than negative two. Then you can find not only

the sum of all numbers from one to infinity is negative one twelfth

but also

sum of one infinite number of times is negative one half.

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.

Random FAQ Comments (3)

3 Responses to “The answer is Problem 12.4”

  1. Anita says:

    Why can’t you watch porn like a *normal* person? :)

  2. GP says:

    Who says I don’t?

  3. Anita says:

    Greg, I love you like a brother. And some things are just not meant to be shared between siblings.

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