Why can't division by zero numbers be another type of number that expands the world of algebra just like imaginary...

why can't division by zero numbers be another type of number that expands the world of algebra just like imaginary, transcendental, negative numbers have?

Other urls found in this thread:

en.wikipedia.org/wiki/Riemann_sphere
wolframalpha.com/input/?i=2/10/5/2
en.wikipedia.org/wiki/Wheel_theory
www2.math.su.se/reports/2001/11/2001-11.pdf
twitter.com/NSFWRedditImage

Because it just can't.

Because it'd completely break things and you could prove 1 = 2, for example.

Because you could divide any type of number by zero and it would undefined

That's only if you assume field axioms still hold moron.

>L'Hôpital

You think you're a genius for thinking about that basic shit, don't you?

I'm browsing Veeky Forums since it was created back in 2010 and there's at least FIVE such threads per month of people who think that they're the next Euler and post exactly the same shit. Not only that, but you're also too late as well - en.wikipedia.org/wiki/Riemann_sphere

Then what would be point? Give up all the properties of 0?

Also, it's brainlet, newfaggot.

Wanna get proved incorrect? alright:
[eqn]({{x}\over{0}})\cdot 0=x\\
({{x}\over{2\cdot 0}})\cdot 0=({{x}\over{0}})\cdot 0=x\\
({{x}\over{2\cdot 0}})\cdot 0={x\over 2}[/eqn]

This is a contradiction, thus dividing by an unspecified zero is invalid.

then how is this possible?

...

its not possible because going from line 1 to line 2 is complete bullshit

that's like saying how is this possible:

x*0 = 0
x = 0/0

I'm pretty sure that you can devise a group where the multiplicative identical element can be "divided" from its elements. It wouldn't be as interesting as C tough.

It can be, but it would cause your field to be trivial.
Suppose 0 has a multiplicative inverse in your field, then a = a*1 = a*0*0^(-1). But 0 = 0 + 0, so a*0*0^(-1) = a*(0+0)*0^(-1) = a*0*0^(-1) + a*0*^(-1), and as we have shown, a = a*0*^(-1), so, a*0*^(-1) + a*0*0^(-1) = a + a. By transitivity, a = a + a, so, if we add (-a) to both sides, we get 0 = a for all elements a in your field.

I'm sure you mean field or maybe ring. If you have a group you just have one operation and by definition, every element can be divided in a group. Even the identity element.

wolframalpha.com/input/?i=2/10/5/2

sqrt(144), cos(0), sin(pi/2), ln(e^2), e^0

It's Like I'm Working With Children-Fucking-All-Day-DingGone.

太陽禮物的視線全部P> 1光!

where does it say one must only use one insult? fuck off

It already is? It's simply a part of the concept of infinity.

All of the numbers OP lists satisfy the field axioms. Division by zero does not. It's a different thing to build from another set of axioms, then. So, you're wrong, retard. So, you fuck off. Remember to neck yourself, you waste of human life.

It can and is in meme algebra like wheel theory, but to have an inverse for zero means that some numbers, when multiplied by zero, aren't zero, and as such we lose the very interesting fact of a multiplicative annihilator. Also, to define division by zero requires the introduction of "numbers" that don't behave in any of the ways normal numbers do, so many properties are true "for all except this division by zero crap", which is much less useful than a proper [math]\forall[/math]. See for example the list of exceptions to normal algebra in en.wikipedia.org/wiki/Wheel_theory
tl;dr to do so would require abandoning too many rules of algebra to be useful.
Further reading: www2.math.su.se/reports/2001/11/2001-11.pdf

It is, it just isn't useful at all. Try to find three real world applications for wheel theory, I'll wait.
Basically though the issue is that the scope of things you can technically do is much larger than the scope of things that actually matter. Forcing division by zero to work is an intellectual dead end.

A group can't have an annihilator element (like zero is for multiplication) if it has more than one element, which is to say for any group [math]G: |G| \not = 1[/math]
there does not exist a [math]0\in G : \forall a\in G: a*0 = 0[/math].
If there was, then the inverse of 0 must satisfy that [math]0^{-1}*0 = e[/math] (where e is the identity).
But by the definition of 0, [math]0^{-1}*0=0[/math], so [math]e=0[/math]. Consequently, for any [math]a\in G[/math], we find that [math]a*0=a*e\implies 0=a[/math].
So if there exists an element [math]a\not =0[/math], which is to say [math]a\not = e[/math], there can't exist an annihilator, precisely because of its relationship with the inverse.

That's a retarded question, since such an extension of the real numbers already exists.

But IIRC it is not a field anymore (unlike the complex numbers) and can't be easily made into one, which makes it not that useful.

>zero numbers
wat

Thanks user

>All of the numbers OP lists satisfy the field axioms.
so?
>Division by zero does not
my point exactly
>It's a different thing to build from another set of axioms, then. So, you're wrong
by this line of reasoning you may as well say quaternions are "wrong" because their construction requires giving up commutativity

You've completely missed the point. How's undergrad, faggot?

Where's the point?

They can and are. That kind of number is infinite, and infinities don't work the same way as finite numbers, but mathematicians have studied them. Though as others have pointed out, that branch of mathematics doesn't seem to be useful yet.

And FWIW 0/0 = ±∞

>why can't division by zero numbers be another type of number that expands the world of algebra just like imaginary, transcendental, negative numbers have?
this post makes me seriously mad. its so fucking full of uneducated pseudointellect.

when will idiots like you understand that the reason you CANNOT divive by zero (pleb term for "there is no multiplicative inverse to zero") is because zero in the mathematical construct called "field" is the neutral element of addition. From this follows that zero cannot have a multiplicative inverse 0^-1.

This is completely different to introducing an "imaginary" number which is just a convenient notation for R^2 with certain properties.


Read a fucking calculus book for once. its LITERALLY within the first 40 pages.

How is that possible? I am serious.

>How is that possible? I am serious.
its not. multiplication by zero is not bijective therefore you cannt invert it.

Jesus Christ this thread. This happens every single time when someone asks something not directly answered in undergraduate textbooks, on here and other sites (like Stackexchange).
>Why can't we make some new number to make this possible?
>Because it's literally impossible!
>Because then these arbitrary rules of normal numbers don't apply to it!
>Because of we use normal number rules we get this contradiction!
And then finally, someone comes along and says you can, it's called wheel theory.

This happens every fucking time I Google something 'nonstandard', a bunch of retards calling the guy asking the question dumb and saying it's impossible, until eventually someone actually answered the question and shows it is possible. And even then you will still get people going 'actually, it's impossible, I learned about it in my first year of community college' after someone showed it is possible because they don't read the previous replies. This same thing happened to a tread about extracting energy from the earth's rotation as well. If you don't know the answer, then shut the fuck up.

(I'm not talking about the posts showing its not a field and such, since they motivate why we don't use wheel theory , I'm talking about the other retards.)

this
/thread

>why can't division by zero numbers be another type of number that expands the world of algebra just like imaginary, transcendental, negative numbers have?
Because as soon as you extend a number system such that something akin to division by zero is possible, it necessarily (this can be proven) behaves to differently from normal number systems that it isn't really "division" anymore. Formally, you would have to sacrifice at least one of the Field axioms, which define what the concept of "division" means. So as soon as you can do something-like-division to zero, it isn't really division anymore as that word is usually taken to mean.

Case in point is Wheel theory. It defines something-like-division that allows for division by zero; but as a consequence, does not have the property that x / x = 1, or that x * 0 = 0. Which means it's not really what we generally mean by "division", formally represented by the fact that Wheels do not satisfy the Field axioms (which require that x * 0 = 0, for example).

Per above, I reject your claim that wheels allow for division by zero. They allow SOMETHING by zero alright, but I think calling that something "division" would be wrong.

i was asking the same few years ago, and the math fags freakedout on it, i don't see a problem with this if properly executed.

You get a field as the sub-algebra when you ignore infinity and perp though, and /x then just becomes the inverse. Saying it's not division is like saying i^2 isn't to the power because it's negative. This happens with every generilization of an existing algebra, it's the whole point.

no it is not, X * 0 = 0 x

0x=0 is not an axiom of a field you retard.

Nor did I claim it is. I said the field axioms *require* that 0x=0, which is true (it is a theorem under the field axioms).

you dont have enough information.

x/0 is undefined, not enough info to determine what it really is.

>Saying it's not division is like saying i^2 isn't to the power because it's negative.
Not so. The crucial difference is that division is defined axiomatically, whereas the property of exponentiation you refer to follow from a particular construction.

There is no axiomatization of exponentiation that needs to be weakened to allow complex numbers, so claiming that i^2 is not a power is incorrect. What IS true is saying that i - 1 < i is meaningless because the complex numbers is not an ordered field -- because the order axioms are exactly those that you DO need to sacrifice in allowing the complex numbers.

>This happens with every generilization of an existing algebra, it's the whole point.
Generalization of an algebra generally sacrifices some concepts that exist in the more specific algebra, and retains others. That means that the sacrificed concepts have become meaningless, or different in meaning (and therefore in name), in the generalized algebra. If you generalize away the defining properties of division, what remains is something that is NOT division even if it shares some structure.

So you just arbitrary decide what counts as devotion? Is the binary operator in a non commuting group multiplication? It's can't be because we all know multiplication commutes, since obviously it's that way with real numbers. Yet why do people call it multiplication? What about in magma's where it doest even associate? It's still called multiplication. Some algebras have different left and right inverses, are those not inverses since it's not double sided? In wheel theory we have an operator identical to division except for the new numbers and 0, yet in this special case it's not a generilization of the operator, but a new one?