/sci/ - Science & Math

[Anonymous No.10674588]

So let me get this straight.

Math is supposed to be this set of rules that always apply equally in every case, and if it fails to apply in a specific case then the rule was just lucky to apply in the other handful of cases and is scrapped aside. For an example, imaginary numbers and their application literally everywhere and no matter how hard you try to find a flaw in them, you can't because they function in every single case without exceptions, therefor, they exist and there's nothing you can do about it.

Fine. So what about division by zero then? How come we pretend this didn't happen and just reflexively go "undefined lmao next" every time it pops up, fully ignoring the fact that it keeps popping up everywhere but no one has any idea what's happening? Is this a problem we're yet to solve and finally find a concrete solution to x/0 that applies everywhere without exceptions or different infinities popping up, or is there no actual solution which means that the entire infrastructure of math is actually flawed and we should start anew?

In both cases this seems like a problem that must be handled NOW, in fact the most important problem of modern math which is probably bottlenecking our next major step in scientific development much like the refusal to use negative numbers and even imaginary numbers were for quite some time. So why the fuck is literally no one looking into it and trying to dig out a solution in the same way there's a legion of PhDs trying to solve the Riemann Hypothesis?

[Anonymous No.10674602]

>>10674588
hsdjdhadh yes yessdssg

[Anonymous No.10674609]

>>10674602

[Anonymous No.10674610]

[math]
1=0\cdot\frac{1}{0}=(1-1)\cdot\frac{1}{0}=\frac{1}{0}- \frac{1}{0}=0
[/math]

[Anonymous No.10674619]

Not how it works. The operation of division a/b is defined as something that satisfies a=bx. Find me x such that 1=0*x. The fact that everything has to work for everything is for theorems. Here we are talking about an arbitrary operation, which was defined by us because it's useful. By its definition, there is no solution to a division by 0 thus it's undefined

[Anonymous No.10674622]

>>10674588
Ok then go solve it fag.

[Anonymous No.10674627]

>>10674588
Your image is a clue as to why this isn't a great unsolved mystery of mathematics. You're just a fucking brainlet.

[Anonymous No.10674632]

Whats the evolutionary advantage of division by zero?

[Anonymous No.10674638]

>>10674588
>"undefined lmao next" every time it pops up
This is wrong. Sometimes we take a/0 to be infinity to simplify things. Some other times, a/0 is replaced by 0. It's not really a problem.

[Anonymous No.10674642]

But you can divide by 0, the value just isn't in the number systems like natural numbers.

[Anonymous No.10674671]

>>10674588
Here's a sketch of how division is obtained.

division isn't a thing. division by x is multiplication by the inverse of x. This is a ring theoretical thing, not a number thing.

Inverses of elements must be constructed. They don't generally exist before construction. For example, if I have the set of integers Z then the inverse of 2 is not in Z. I can then define another kind of inverse (which coincides with the ring theoretical inverse) by eliminating prime factors. For example, 6/3 = 2 because 6 = 3 * 2. It is easy to see that division by zero in this sense is not defined.

It is not that "division by zero is not defined", it is that "we have not defined division by zero". We have done this on purpose. This is not an accident.

[Anonymous No.10674730]

Well let's put it this way. You could interpret i classically as the "square root of -1". But everyone knows that square roots are more useful in the context of quadratics, where every use of them must come in conjugate pairs. And while it makes sense to distinguish positive from negative in the real numbers because they obviously behave differently, such a distinguishment doesnt really make sense for i vs -i, because i isn't really the square root of -1.
It's a placeholder for a root of the equation x^2+1. And the cool thing is that the extension field R[i] ends up being the algebraic closure of R, and is itself algebraically closed, which makes it a useful structure to take into consideration, because it doesn't contradict anything known from the real numbers, it simply extends the ideas. And yet, under that definition -i would be the other root. Which one is i and which one is -i? It doesn't matter because literally every operation ends up staying the same. So we avoid this kind of question completely by leaving it as an arbitrary choice to call one of them i, so that all other roots of any real polynomial can be written in terms of THAT i.
You could also consider the number a, where a is a root of the real valued polynomial x^2+x+1, or in other words a^2+a=-1. It would be equally useful to consider the extension field R[a], because that field is also the algebraic closure of R, and is itself algebraically closed. There would be a similar wrinkle in that theres a conjugate answer as well, but since it can written in terms of a, then we simply choose one of them to be a and the other to be conjugate to that a. Then even i can be written in terms of a, and of course knowing that R[i] is also algebraically closed, we know that a can be written in terms of i.

[Anonymous No.10674737]

>>10674730
(cont.)
So what's the big deal about i? Well first of all x^2+1 has fewer terms than x^2+x+1, so maybe it's better to just write a in terms of i and not vice versa. But what about other polynomials with no real roots? Whats the significance of writing those in terms of i? Well further algebraic inspection quickly shows that every real valued polynomial with know real roots has even degree, an can be factored totally into irreducible quadratics. Therefore roots of real valued polynomials, after some computation has been executed, are really just roots of quadratic which are easily put in terms of our simple primitive element i.

In conclusion, it's not that we just decide to ignore how nonsensical the square root of -1 sounds, because frankly that operation no better than dividing by 0. -1 is just not in the domain. But algebraic factorization of real valued polynomial makes i impossible to ignore, because it will come up no matter what you try to do.
We ignore division by 0 because whatever the result is, it can not be used in algebra. Lots of topology dorks use a point at infinity to compactify the reals, and even they won't divide by 0. Because it doesnt do anything interesting, the result just has to stay there and not be used, if it existed. But none of this shit exists anyway, we only say it does when its useful, so dividing by 0 remains meaningless and hence declaratively invalid.

[Anonymous No.10674850]

>>10674730
>arbitrary choice
AngryGalois.jpg