Beyond the complex number plane (math nerds/geeks, I mean geniuses, this is for you)

38 posts

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First, please do not be offended by the fact that I said the word geek/nerd, because I am a math geek/nerd MYSELF. So if by some reason you take this as an insult, then I’m also insulting myself.

Second, in math, I’m only about Grade 11 level (but I do have some ideas that are whackier than probably Einstein – again, do not be offended). So do not say anything like “omfg u retard” if I get something wrong in University-level calculus.

What I’m considering here is adding another number line, or axis, to the existing complex number plane. To do that, I need another unit, or type of number, for the third axis, in order to form the “hypercomplex number space”. There have been many attempts to find hypercomplex numbers, but most them do not form a field, and the best one, the quaternion, is not commutative, and I believe that if the complex plane is commutative, then its further extension should also be commutative.

So what I am asking you here is this: What should this “hypercomplex unit” be? The unit for imaginary numbers is i, the square root of -1, and there are certain things that always occur when doing operations on both units (e.g. 1 divided by i is -i). So if a third hypercomplex unit is established, what will happen if you do operations on the three units?

 
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So do not say anything like “omfg u retard”

first time visiting the SD forum i see. You will see we are much more level headed than that.

 
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Yes, you are correct. I am glad to hear that.

The part I’m most confused with is this: (r = Real, i = Imaginary)

r * r = r
r * i = i
i * i = r

The above law applies mutually to each other (r<=>i). So what’s the according law of the third hypercomplex unit? Does the rule go around in a ring (r<=>i<=>h<=>r) or what?

 
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Just guessing as i have no idea what you are talking about

but

h*h=r
h*r=h
h*i=i

Could you maybe so some sort of graphic representation of what your talking about.

 
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Bart the genius – http://video.aol.com/video-detail/the-simpsons-the-simpsons-hardy-har-har/3127817278

 
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Never mind. I’m just trying to figure out the laws for multiplication. Forget exponentiation for now as I have no idea what is 2^i.

It could also be true that a 3D number space will not work, and any numbering system that works must be a power of 2 (like 1D, 2D, 4D, 8D etc, hence the quaternion, octonion, etc)

Jitters, remove your wall of text in this thread.
http://www.kongregate.com/forums/2/topics/37848?page=4

 
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Originally posted by Blood_Shadow:

Never mind. I’m just trying to figure out the laws for multiplication. Forget exponentiation for now as I have no idea what is 2^i.

Jitters, remove your wall of text in this thread.

http://www.kongregate.com/forums/2/topics/37848…

I don’t negotiate with terrorists.
Now stop cross contaminating threads before you get yourself banned.

 
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I wholeheartedly agree with my good friend jitters.

 
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Oh my god, geez, didn’t you READ why I wanted you to remove it? It takes a lot of space, and it is an unnecessary quote!
That’s it. If you still refuse to remove it, I’m calling an admin to remove it. Seriously, how hard is it to edit a post?

 
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You’ll notice that there are operations on R that don’t map to R for the entire set of real numbers – the square root of a negative number, for instance, maps to C, and it is this fact that we use to define the basic “unit” length of a complex number.

Is there any such operation that will map R or C to H?

 
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OH CRAP. I forgot about that… That makes it even more complicated…

Are there any illegal operations on the numbers 1, -1, i, -i? That’s probably where I should look…

On another thought… Maybe I need to invent some sort of new operation…

 
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Why do we need this?

 
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To satisfy our (my) curiosity.

 
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Originally posted by Blood_Shadow:

To satisfy our (my) curiosity.

Wouldn’t it be a dash?

Basically you are saying it is either;

To satisfy our curiosity.

Or;

To satisfy our my curiosity.

I think it’s correctly phrased as;

To satisfy our/my curiosity.

 
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Does it really matter? But what I was saying is, I’m saying our curiosity, but what I really meant is my curiosity.

 
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Originally posted by Blood_Shadow:

Does it really matter? But what I was saying is, I’m saying our curiosity, but what I really meant is my curiosity.

And you couldn’t just say that why?

 
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It doesn’t matter, because it is not important.

The operation beyond exponentiation is called tetration… Any illegal operations on that?

Tetration is iterated exponentiation. All regular tetrations must use positive integers, otherwise the result is undefined…
All right. I will call the two numbers involved in tetration the base and the tetrate. The tetrate must be a positive integer. So the hypercomplex unit could possibly be made using negative numbers as the tetrate…

 
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I think you need to read moar Wikipedia – like this, as well as everything it links to – and then take a few college courses in field theory, and then ask on an actual math forum. I don’t think you’re going to get any useful discussion on this topic here.

 
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Yes, I’ve read the Wikipedia article. But, as it says, none of them form a field.

I believe that to go beyond complex numbers, you need to go beyond exponentiation. But no one have linked hypercomplex numbers with tetration yet.

 
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Originally posted by Pink_Fuzzy_Bunny:

Why do we need this?

2nd.

 
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Define it as you like, and see what happens. As long as you can’t derive any contradictions, it will be sound. Given the closed nature of the complex plane, as already noted, you probably won’t be able to find something to solve a problem inherent in C.

 
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I believe that to go beyond complex numbers, you need to go beyond exponentiation. But no one have linked hypercomplex numbers with tetration yet.

And I believe that even the most cursory reading of Wikipedia will show you that this is impossible because of the fundamental theorem of algebra – that is, the field of complex numbers is algebraically closed. You cannot extend it and form another field.

 
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I believe that C is closed in all algebraic operations. Therefore, there is not a way to find a great new unit out of it. However, the one ugly asymmetry in mathematics is that you can multiply by 0 yet not divide by it. Just a hunch, but maybe start there? Find some consistent system in which to have a new unit h which represents 1/0 and form a consistent algebra.

EDIT: Sorry, didn’t read Einar’s post, forgive my redundancy.

 
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A field being algebraically close does not imply that it’s impossible to extend it. The quaternions extend the complex set with 4 dimensions; the octonions further extend that with 8 dimensions; and the sedenions have 16.

Further, all of the following are “complex extensions”:

I haven’t investigated why they are said to do so though. If you can be bothered to find out, please post here on it.

 
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I didn’t say it was impossible to extend it; I said that it was impossible to extend it and form another field. This is important, because fields are nice to work with and there are some useful proofs that only hold on fields.

Indeed, the original poster specifically mentioned quaternions, and seemed to disregard them because they are not commutative. The point of the fundamental theorem of algebra is that you cannot extend the complex number space in a way that forms another field; your extension will always lack some attribute of a field. (you can find the list of those attributes here on Wikipedia.)

I’m not saying that these things aren’t interesting or sometimes useful, I’m just saying that what the OP wants is impossible.