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Zero/zero = ? 1 or zero?

Extending Zeta function (sum of series) to negative numbers or complex numbers allow larger application in other fields (engineering, crypto, analysis…). It does not mean that the sum in original definition will converge to negative numbers.

What is impressive is that one can use sloppy wrong math on series and they still find the correct answer from the analytic extension !

A similar case is extending n! = 1*2*….*n to complex numbers:

(1/2)! = (√π)/2

These are undefined, so one has to be clear about the definitions

Last Edited by Ibra at 17 Jun 10:40

Peter wrote:

But surely this is playing with words i.e. BS. If you live in the physical world, there is no way that adding up positive numbers can produce a negative sum!

That’s what I said, wasn’t it? In reality there is obviously no way, but in cases where there is positive numbers do not have any sum “in reality”, it can make sense to *define* a sum.

Mathematics does not have to have a direct connection to reality to be useful.

So no, it is not BS.

Ibra wrote:

What is impressive is that one can use sloppy wrong math on series and they still find the correct answer from the analytic extension !

There is nothing “sloppy” or “wrong” about that extension of the Riemann zeta function.

If there is any sloppiness, then it is in applying common sense to infinite sums. Since you can’t have an infinite sum in realty, all infinite sums are just definitions and common sense does not apply.

gallois wrote:

Zero/zero = ? 1 or zero?

Neither one. Anything divided by zero equals to infinity!

0 / 0 = oo

0/0 is undefined. If you try to evaluate it in a computer, you will get “nan” (not a number) as the result. (The universally used IEEE floating point format has some special non-numeric values – iirc inf, minf, nan).

However lim(n→0) n/n = 1, which is as good an answer as any.

Peter wrote:

Complex numbers are just a tool in engineering. Nobody pretends they are real.

Quantum mechanics, the most accurate theory invented wrt real world measurements, is all based on complex numbers. Are they real or not? Who knows. Nevertheless, the common conception is that complex numbers are not real. According to Elon Musk we live in a simulation, so nothing is real in any case Reality is fake.

is all based on complex numbers

Well, not really. It happens that complex numbers provide a useful way to describe quantum mechanics, just as they do to describe what is going on in electrical engineering. So yes in a way, because if you want to understand quantum stuff without re-inventing it all yourself, you will be confronted with complex numbers. But there’s nothing about the subject itself that requires them.

UdoR wrote:

Neither one. Anything divided by zero equals to infinity!

It’s not that simple.

Anything *non-zero* divided by zero gives infinity. This definition makes sense as the limit of x / y is infinity when x is (positive) non-zero and y approaches zero (from the positive side)..

However the limit of x / y when neither x nor y are constant and *both* approach zero varies depending on x and y. E.g. the limit of 2n/n as n approaches zero is 2. Thus you can’t just define 0 / 0 to be infinity (or zero or one or anything else). It depends.

Last Edited by Airborne_Again at 17 Jun 11:03

Airborne_Again wrote:

If there is any sloppiness, then it is in applying common sense to infinite sums

Yes that’s what I had in mind: the sum method in YT video is sloppy math, the analytical extension is proper math