Adding a Third Dimension to the Complex Plane so We can Divide by Zero

[Asterisk Propernoun]

This was just a little expirement that went on inside my head while I was on one of my walks. It seemed interesting at the time, so I'm going to share it with you.

 

We have an x axis that represents all real numbers, and a y axis that represets all imaginary numbers, i^2=-1. The idea of a whole-nother number line sounds fascinating; and to some, literally unbelievable. However, dispite some people's disbelief, the imaginary number line and the complex plane is perfectly sound from a mathematical standpoint. If we managed to solve something impossible like -1^0.5, then maybe we can do the same with another impossible function like a/0.

 

A person (I can't remember who) once said that smart people learn through trial and error, while smarter people learn through other people's trial and error. So, I started out by showing the marks on the z axis as -1(1/0), 0(1/0), 1(1/0), and so on. However, I quickly realized that this would simplify to -1/0, 0/0, 1/0, and so on. Well, I made a third dimension. Now the questions were what good was it and what are it's properties?

 

Well, let's start by comparing 1/0 to -1^0.5.

 

A few people who don't accept the complex plain may argue that -1^0.5 just equals 1. However, when you find the square root of a number, you should be able to reverse the process by squaring.

 

25^0.5=5

5^2=25

 

You cannot do this with a real number only argument to -1^0.5.

 

-1^0.5=1

1^2=1

 

If the person you were arguing debating with was desprate to hold on to their claims, they may say that 1^2= +/- 1.

 

Please, tell me what's wrong with this picture:

 

1*1=-1

 

So, I concluded in my head that, if you wanted -1^0.5 to work, you needed to use the imaginary unit i. I am now going to compare the problems with using real numbers for -1^0.5 with 1/0.

 

Again, you should be able to reverse what you've done.

 

10/5=2

2*5=10

 

And again, this won't work if 1/0=, say, 0. This was something I didn't take in to consideration on my first attempt to divide by zero.

 

10/0=0

0*0=0

 

The question now is, can the third dimension I added fix this? I believe so. However, it may be difficult to wrap your head around the solution.

 

10/0= the tenth place on my axis. For now, let's just call that place 10g.

10g*0=10

 

So when you multiply g by 0, you get 1? Well, when you think about it; yea, why not? Wouldn't you agree that *0 cancels out /0 the same way *5 cancels out /5? Getting a nonzero number through multiplication by zero is no less believable than saying you multiplied two of the same signs together and got a negative number.

 

Another problem a mathematician on this forum told me was that, if we were to make it where any number divided by zero equaled the same number, then that opens up the possibility for any number to equal any other number:

 

2/0=0

3/0=0

Therefore, 2=3

 

This doesn't happen with my idea.

 

2/0=2g

3/0=3g

Therefore, 2=/=3.

 

So, is this perfectly sound like the complex plain, or does it have problems?

Edited May 29, 2014 by Asterisk Propernoun

[Endy0816]

If I represented zero as a fraction, what would be its denominator?

[Greg H.]

Demonstrate how to divide a cake into 0 equal parts, while retaining the ability to reassemble them into a whole.

 

That's dividing by zero.

[studiot]

There is already a rich mathematics extending complex numbers to as many dimensions as you like, called quaternions.

 

http://en.wikipedia.org/wiki/Quaternion

[John Cuthber]

There is already a rich mathematics extending complex numbers to as many dimensions as you like, called quaternions.

 

http://en.wikipedia.org/wiki/Quaternion

A rich beautiful mathematics that I totally don't understand.

However, I suspect you still can't divide by zero.

[Asterisk Propernoun]

If I represented zero as a fraction, what would be its denominator?

 

It's denominator could be any real number.

 

0/1

0/2

0/3.1415...

[ajb]

Division by zero is not necessarily "outlawed" in mathematics, just when you can define it you cannot also keep the axioms of a field or ring. So, adding a new "number" to the complex numbers may in part solve division by zero, but I am confident that the structure you get is not a field. Thus what you get is not really what we would like to call a "number".

 

So I suggest first testing your extended complex numbers to see what kind of algebraic structure you really have.

[Endy0816]

 

It's denominator could be any real number.

 

0/1

0/2

0/3.1415...

 

so what happens when you bring that denominator up on top, to put it another way, what is the numerator of your 'g'?

 

How would that in turn impact anything multiplied by it?

Edited May 30, 2014 by Endy0816

[`hýsøŕ]

Demonstrate how to divide a cake into 0 equal parts, while retaining the ability to reassemble them into a whole.

 

That's dividing by zero.

this is gunna sound ridiculous but just for crazed speculation's sake, what if you divided the cake into arbitrarily small pieces such that they were subject to the laws of quantum mechanics, and if not observed directly, the pieces would be described by a wavefunction and so would have no distinct position, as the wavefunctions would all superpose on one another. could that be considered dividing something into 0 parts? xD

 

(im not serious here hehe, im not even convinced that would be the right QM-ical description, but idk maybe if that's right then you could form like a particular branch of maths which deals with what'd happen if that was a valid way to divide something by 0 o.o)

[Greg H.]

this is gunna sound ridiculous but just for crazed speculation's sake, what if you divided the cake into arbitrarily small pieces such that they were subject to the laws of quantum mechanics, and if not observed directly, the pieces would be described by a wavefunction and so would have no distinct position, as the wavefunctions would all superpose on one another. could that be considered dividing something into 0 parts? xD

 

(im not serious here hehe, im not even convinced that would be the right QM-ical description, but idk maybe if that's right then you could form like a particular branch of maths which deals with what'd happen if that was a valid way to divide something by 0 o.o)

I would have never thought of it that way. +1 for an original idea. Like you, I have no idea if that's even a possibility, but it's definitely a new way of looking at the problem.

[`hýsøŕ]

thanks, hehe

 

i'll try now to see if i can get any kind of coherent answer (will edit this post if i find anything cool)

 

edit: btw i think that since quantum mechanics is an empirical thing, involving it in mathematics would mean that you get like a mathematical structure that only makes sense 'in' our universe but.. well if it exists might as well try finding it xD

Edited June 1, 2014 by `hýsøŕ

[ajb]

Really we have a mathematical question and so no need to think about any physics. Any structure that allows division by zero, and there are some studied structures such as the projective real line, will not lead to structures that satisfy the nice rules of standard numbers.

There is the concept of a wheel (new to me I must say), where division by zero is okay.

Edited June 1, 2014 by ajb

[John Cuthber]

As far as I can tell, for anything like the traditional definitions of zero ( the identity element under addition) and division (the inverse of repeated addition) division by zero will always be undefined,

Essentially, it's like asking how many times I did nothing before it made a difference.

 

Of course, if you choose different definitions for division and zero you can divide by zero- but is it still a valid use of the words?

Edited June 1, 2014 by John Cuthber

[ajb]

Of course, if you choose different definitions for division and zero you can divide by zero- but is it still a valid use of the words?

I think in a modern setting it is valid. Of course in making such definitions you have to loose something and that is the basic structure of "numbers".

[John Cuthber]

So, you can divide by zero- as long as you don't want to use numbers.

 

Haven't you thrown the baby out with the bathwater?

[ajb]

Haven't you thrown the baby out with the bathwater?

It depends on what you want to do...but generically I would agree with you. Setting things up so that division by zero is okay throws away a lot of the nice structure you would like to keep.

 

You lose things like 0x =0 for all elements and the group structure of addition. Maybe you loose other things also depending on how you modify your set and operations.

[`hýsøŕ]

what about if division is really a bunch of separate operators that give the same result when used on real or complex numbers in general, apart from 0.

 

what i mean by that is like... one way of looking at division is the idea with the cake where you try to distribute pieces of a cake between people and situations like that. there the idea of dividing 1 cake into 0 parts is undefined.

 

but suppose there's some other way of looking at division which isn't really division at all, like the limit of 1/x as x goes to 0 from either the positive or the negative direction. what if doing this is like doing a different type of division, which instead of just two numbers, would have arguments of two numbers and a sign (the direction which you approach 0 from). the answer to this type of division would be infinity or -infinity depending on the direction you approach 0 from.

because if this was another division it'd look like the normal division when you use it on a non-divide-by-zero situation. like if you did the limit of 1/x as x goes to 2, you'd get 1/2, which is the same as the answer you'd get using cake type of division. this type of division would also work for 0/0, 0^0, 1/infinity.

 

i think it'd be good if that was a new operator just because then it sorta solves the mystery of 1/0=? --> 1=0 * ?, because that's a problem with the cake approach but you wouldn't be able to just multiply by by the x in the limit approach: ?=lim[x->0] 1/x ---///---> ? * x=lim[x->0] 1

 

so like if that's true and there are 2 division operators, what if there are more?

 

or would this new type of division just be a generalization of the cake approach :S

Edited June 1, 2014 by `hýsøŕ