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#1 2009-05-17 03:40:41

random_fruit
Member
Registered: 2008-12-25
Posts: 39

Three dimension maths

One could say the real number line is one dimensional.  Then, one could say that imaginary numbers give us a second dimension, and complex numbers allow us to 'explore' the plane.  So, what piece of maths allows us to 'move' into a three-dimensional space?

I know we could just give 3 real numbers (x,y,z) which describe a point in 3D.  I could for myself work out 3 separate lines/planes/solids using 3 sets of one-dimensional co-ordinates.  But, that NOT what I want.  What I wish is an idea as 'clever' or as 'complicated' or even as 'simple' as imaginary numbers which takes us from two to three dimensions.

Has this ever been done?  I so, I have yet to hear of it.  Mind you, that's not to say either way... 'cos my maths education pretty much stopped at age 18 before university.  But I am curious that as we live in a 3D world I have not yet come across a 3D maths.

Can an answer be put in 'simple' terms, such as I can recall from 1975 when I finished 'A' level maths in the UK?  Thanks,

random_fruit

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#2 2009-05-17 06:04:21

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Three dimension maths

There's no similar way to pull a third dimension from the complex numbers, because the complex numbers are algebraically closed.

In the reals, x² + 1 is made of real coefficients but has no real roots. The imaginary number i can then be defined as a root of this polynomial, and we then get complex numbers by making linear combinations of 1 and i.

However, given a polynomial

, with complex coefficients, this will always have n complex roots. There are no "gaps" in this number system and so no third dimension for it to go to.


Why did the vector cross the road?
It wanted to be normal.

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#3 2009-05-17 06:26:59

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Three dimension maths

There's no similar way to pull a third dimension from the complex numbers, because the complex numbers are algebraically closed.

It is important to keep in mind that this is one way to look at the complex numbers, but not the only way.  The complex numbers form a 2 dimensional vector space over the real numbers, with multiplication defined in a funny way.  They actually have a lot more properties than that, but this is just one view.

The question is can we do the same thing with a third component, extending the complex numbers?  The answer is a rather surprising no.  We would have to lose a fundamental algebraic property to do so (I think it was the associative property, but I need to look this up).  However, as discovered by Hamilton, we can do such a thing with four components, and these are known as the Quaternions.  And so we have 1, 2, 4, and the next number is the Octonions.

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


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#4 2009-05-17 06:54:11

random_fruit
Member
Registered: 2008-12-25
Posts: 39

Re: Three dimension maths

Thank you both.  smile The words "algebraically closed" tell me what I needed to know, but I hadn't made the leap that rules out 3D.  when I have some time to spare I think I had better look Quaternions up in Wiki and hope I can understand some of that. dizzy

random fruit

Last edited by random_fruit (2009-05-17 07:05:48)

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