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]]>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 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.]]>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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