1+2+3+4... = -1/12

[jaz_pup]

Re: 1+2+3+4... = -1/12

I'm (a bit) surprised so many people seem to have taken complex analysis.


I find it interesting that so many people get so heated over what they believe is the right answer regarding math. It's like that one problem floating around involving the order of operations...I'm pretty sure there have been like legit fist fights over that one...probably.

[21992]

Re: 1+2+3+4... = -1/12

Funny we were just discussing this in my calculus class. In fact the infinite series of positive numbers is divergent (infinity or undefined). This instance is kind of like how my calc teacher told us that taking the integral of natural log yielded 1=0 . But then there came the +C that corrected the equation. I think something like that will come up but for now it can't be explained. I find it funny that the tool we created to better understand our universe, math, we can't even fully figure out ourselves.

[21992]

Re: 1+2+3+4... = -1/12

A quick google search https://www.youtube.com/watch?v=w-I6XTVZXww

[benguino]

Re: 1+2+3+4... = -1/12

e.g.

sum(n: n=1 to infinity) = x (here x is assumed to be -1/12 but that doesn't matter)
=> sum(n: n=0 to infinity) = x
=> sum(n-1: n=1 to infinity) = x
=> sum(n: n=0 to infinity) - sum(n-1: n=1 to infinity) = 0
=> sum(1: n=1 to infinity) = 0
=> sum(1: n=0 to infinity) = 0
=> sum(1: n=0 to infinity) - sum(1: n=1 to infinity) = 0
=> 1=0

There's a lot of assumptions being made here and a lot of things done without justification. The terms in an infinite series are like the flesh of a cacti...they don't want to be ****ed with.

However, there is a lesson to be learned here. The lesson being that this way of defining how to given values to divergent series isn't "nice" and doesn't have the "nice" properties like the ones we see with convergent series (see this and this). We know with convergent series we can add term-by-term...but your thing shows that that is clearly not the case anymore (or else otherwise we get the absurdity that 0=1).

To be honest...this way of defining how to assign values to divergent series doesn't sit too well me either. It works well for complex analysts since analytic functions are "nice". However, this definition doesn't have a lot of the "nice" properties one is looking so if you find a "nicer" one let me know

[stargroup100]

Re: 1+2+3+4... = -1/12

I find it interesting that so many people get so heated over what they believe is the right answer regarding math.

I don't really care about what people believe is the right answer. Whether or not they believe doesn't make the answer right or wrong.

What I do care about however, is how people approach and understand math. There are way too many misconceptions about the nature of mathematics and how it's applied.

Example of what I mean:

I think something like that will come up but for now it can't be explained. I find it funny that the tool we created to better understand our universe, math, we can't even fully figure out ourselves.

"it can't be explained"
Actually this topic has been explored pretty well. It's nothing new for mathematicians, and while there are questions that are still unanswered, we generally have a pretty good idea of what goes on here.

Math can be thought of as a tool we created to better understand our universe, this is fine. There should be absolutely nothing surprising about not being able to fully figure out mathematics as a whole. Just because you created something doesn't necessarily mean you understand it completely, and there are countless examples of this in the real world. Why is math special in that it needs to be completely understood?

Mathematics in many ways has infinite complexity in many dimensions. Not only are there probably an infinite number of systems and objects, but different mathematical objects behave differently under different circumstances/domains/etc. There are no "concrete rules" in mathematics. You absolutely must follow the rules you set up within your own system, but none of these rules are universal to all systems. In fact, complete axiomization of all mathematics has already been proved to be impossible (See Gödel's incompleteness theorems).

What is important is not whether or not we fully understand math, it's how much we understand and how we use it. Over time we build upon our existing knowledge and expand it, which then results in more applications to real life. There are always going to be things we don't know; it's literally impossible to know everything. Our objective is not to have perfect knowledge, but to increase our knowledge as much as possible.

[jaz_pup]

Re: 1+2+3+4... = -1/12

I think this problem is too advanced and we should find something fascinating in math that's in the realm of understanding for everyone. Any ideas?

[stargroup100]

Re: 1+2+3+4... = -1/12

I think this problem is too advanced and we should find something fascinating in math that's in the realm of understanding for everyone. Any ideas?

The details surrounding this result is quite advanced, yes, but that doesn't mean you shouldn't find it fascinating. You really only need to know one thing to appreciate this fact: In some contexts this result is justified. You can go one step further: You can use analytic continuation to expand the domain of a function. One step further: This result is related to the behavior of the riemann-zeta function. You can go as far as you like, but the first statement alone allows us to appreciate the surprising result.

When you find something fascinating, it doesn't have to be something you can understand. There are lots of facts about the world, not just mathematics, that are amazing, and many of them even top academics don't understand yet.

It's more important to first understand mathematics as a whole, the things I've mentioned in my last post. Once people understand these basic ideas, there will be far fewer misunderstandings about math, far more people that will appreciate it, and it will contribute greatly to people being better at math.

[Zapmeister]

Re: 1+2+3+4... = -1/12

i'm not seeing it. why should/would i appreciate the result "1+2+3+4+stuff = -1/12", or find it "fascinating", if by analytic continuation any series that has a rule for the terms can be "summed" by taking a power series or (in your case here) a dirichlet series and extending it outside its radius of convergence. if it's about the applications of the actual value -1/12 then why bother with writing it out like that instead of writing it as zeta(-1) and using properties of the zeta function.

have at me.

[jaz_pup]

Re: 1+2+3+4... = -1/12

The details surrounding this result is quite advanced, yes, but that doesn't mean you shouldn't find it fascinating. You really only need to know one thing to appreciate this fact: In some contexts this result is justified. You can go one step further: You can use analytic continuation to expand the domain of a function. One step further: This result is related to the behavior of the riemann-zeta function. You can go as far as you like, but the first statement alone allows us to appreciate the surprising result.

When you find something fascinating, it doesn't have to be something you can understand. There are lots of facts about the world, not just mathematics, that are amazing, and many of them even top academics don't understand yet.

It's more important to first understand mathematics as a whole, the things I've mentioned in my last post. Once people understand these basic ideas, there will be far fewer misunderstandings about math, far more people that will appreciate it, and it will contribute greatly to people being better at math.

I never said it wasn't fascinating, I just said we should discuss something more easily understandable that is fascinating. And I think I understand enough math, I gave the suggestion so that we could all collectively have a more useful discussion since it might be something more accessible as not many people have studied (as far as I know) the Riemann-Zeta function.

[stargroup100]

Re: 1+2+3+4... = -1/12

-.-

by analytic continuation any series that has a rule for the terms can be "summed" by taking a power series or (in your case here) a dirichlet series and extending it outside its radius of convergence.

if you learn this fact first then maybe it's not so fascinating. someone that doesn't know this could be fascinated by the fact that that sum can be justified.

when I first learned about calculus, the idea that you can find the area under any polynomial curve in trivial time was mindblowing to me. now I take it for granted because I'm so familiar with basic integrals, but that didn't make it any less surprising when I first learned about it

if it's about the applications of the actual value -1/12 then why bother with writing it out like that instead of writing it as zeta(-1) and using properties of the zeta function.

because a layman wouldn't understand anything about the zeta function unless you used layman terms.

stuff

ah I misunderstood you. my bad.

[popsicle_3000]

Re: 1+2+3+4... = -1/12

All i've seen from you is "some guy who is awesome said"

[benguino]

Re: 1+2+3+4... = -1/12

Wait what?

[popsicle_3000]

Re: 1+2+3+4... = -1/12

I haven't seen him actually attempt to prove his point/theory. stargroup that is

[benguino]

Re: 1+2+3+4... = -1/12

The reason the exact proof is not provided is because the proof requires knowledge in a field that most people on this site would not be familiar with (primarily complex analysis). If you read the thread however, it has been said numerous times that it can be justified using something called analytic contiunation which I and some other people in the thread have given everyone a flavor of.

[stargroup100]

Re: 1+2+3+4... = -1/12

what reuben said

I haven't seen him actually attempt to prove his point/theory. stargroup that is

What point/theory would you like me to prove? If you're talking about analytic continuation I'm not an expert. Google it yourself to find all the information you could want about it.

If you're talking about the notion that different objects in different domains can have different properties and therefore many different statements can be justified/proven under certain contexts/conditions/systems, here's an example:

Polynomials of degree n with real (or even complex) coefficients have exactly n compelx roots, but not necessarily n real roots. Certain applications of polynomials, such as most geometry problems, disregard the complex solutions because many kinds of complex lengths/measurements cannot be physically realized (you can't draw a line i units long). However, the result is fundamentally important to understanding polynomials and is used extensively in other fields of math. Here's a directly related result:

Exponentiation is an example of a function that has different behaviors in different domains. The exponent of e^n tells us to multiply e by itself n times. This makes sense for integers, and if we consider the inverse of this function, also makes sense for rational numbers. However, we can easily plot this relation and make a smooth line to extend it to real numbers. Once we find a real definition for the exponential function, we can observe its properties and extend it to complex numbers (by Taylor expansion for instance) and get results such as e^x = cos x + i sin x.
However, this is where it gets a bit interesting. If x is a rational number, e^x has d solutions, where d is the denominator of x in simplest form. This result is directly related to the extension of polynomials to complex numbers we mentioned above. If x is irrational, then it "probably" (I'll explain in a second) only has one solution in the complex plane. Once we extended our definition, the behavior of rational numbers in this function has changed! Even more bizarre is when you take the relation e^(a*ln(x)) = x^a. Here, we have a case where even though the exponent could be irrational, and yet we can change the base to make the exponent rational again!

And this is only scratching the surface of complex analysis, and complex numbers are just one type of mathematical object. Other mathematical objects include matrices, vectors, tensors, sets, groups, operators, rings, partitions, spaces, manifolds, functions, geometric figures, etc. Even today, we're still investigating all of the properties of these objects and new ones that we've conceptualized.


I know I'm not directly replying to anyone's specific ideas, so I wrote this more for myself as practice. I figured I might as well post it if anyone is interested in reading it.

EDIT: I just realized that I pretty much just did a layman's reiteration of one of reuben's earlier posts. I was wondering why something was bugging me when I posted this LOL