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

[popsicle_3000]

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

[reuben_tate]

Wait what?

[popsicle_3000]

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

[reuben_tate]

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]

what reuben said

Originally Posted by popsicle_3000

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

[All_That_Chaz]

A few years ago, I was offered a free masters in math. As in, this prof liked me and wanted to teach me and eventually research with me. However, I was just back in school to get the necessary math credits so I could teach high school math. I was broke as it was and needed to get working, so I turned it down and started teaching.

This thread makes me regret that decision. I want to take all that non-euclidean nonsense and real analysis and modern algebra and topology and all that garble. I feel like I really missed something.

[stargroup100]

Originally Posted by All_That_Chaz

A few years ago, I was offered a free masters in math. As in, this prof liked me and wanted to teach me and eventually research with me. However, I was just back in school to get the necessary math credits so I could teach high school math. I was broke as it was and needed to get working, so I turned it down and started teaching.

This thread makes me regret that decision. I want to take all that non-euclidean nonsense and real analysis and modern algebra and topology and all that garble. I feel like I really missed something.

It comes down to your thirst for knowledge and the ability to appreciate some of the amazing properties of purely logical constructs.

If you have both then yeah you probably missed something. You've probably "wasted" far more time than I have, and even I feel depressed about not spending more time working on this kind of stuff when I was younger.