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- Thread starter JungleJesus
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Landau

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Mark44

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You can interleave the x and y coefficients to get a single number.

For example, if z = 0.1357 + 0.2468i, maps to the real number 0.12345678.

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Hurkyl

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In other words, you want the line to be homeomorphic to the plane....If I can have a bijection, I would like it to be continuous.

There's an easy impossibility proof: the ordered field axioms imply that xI need to be able to use it to define an ordering of the complex numbers that satisfies the properties of order on the real number line (the order axioms).

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There's an easy impossibility proof: the ordered field axioms imply that x^{2}>0 for all nonzero x....

Sorry about that. I know exponentiation will not preserve order, but the lexicographic ordering that I am familiar with only allows order to be preserved if a constant is added to both sides. If a > b and c > 0, then I would like ac > bc. The lexicographic ordering fails here due to the properties of 0.

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Hurkyl

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The ordered field axioms imply that x*x > 0 for any nonzero x.Sorry about that. I know exponentiation will not preserve order, but the lexicographic ordering that I am familiar with only allows order to be preserved if a constant is added to both sides. If a > b and c > 0, then I would like ac > bc. The lexicographic ordering fails here due to the properties of 0.

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The ordered field axioms imply that x*x > 0 for any nonzero x.

Yikes. This is not the way to go. Drop the order axioms. I just want a way to say that any two complex numbers satisfy trichotomy (<, >, =). I was hoping to do this by mapping C onto R via a one-to-one function and then comparing the real numbers produced by that function.

Strictly speaking, if I have z and q (both complex) and G() is my one-to-one (preferably continuous) function of C onto R, then I want to be able to say that:

1.) G(z)=G(q) or

2.) G(z)>G(q) or

3.) G(z)<G(q)

In this fashion, I could define z>q, z<q, and z=q in a similar fashion to the real numbers.

Ideally (this may be far-fetched), the properties of this complex ordering would be generalizations of order in the real numbers. From the complex orders, the real orders could be derived by simply assuming that Im(z) = Im(q) = 0.

As far as that goes, I don't know what's possible and what has already been done.

I do know that ordering the complex numbers is considered to be a null point. I also think, but cannot prove, that any such ordering as the one I described will not be unique.

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I'm thinking more on the abstract side of things. I would like to arrive at a complex ordering that yields the real ordering as a special case.

I'm also beginning to look at complex analysis as a major area of study for me. The function G() which I described earlier would surely have some powerful properties. Because G() is one-to-one, it could also offer a unique way to build the complex numbers from the reals. This is big for me. Either system could, in principle, construct the other, given only the definition of G(). The way I see it, that has to mean something...

Also, G() may be able to help derive some complex order axioms which are more general than, say, ac > bc. That's the idea here; there has to be

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