Here is a fundamental difference between real numbers and complex numbers. Real numbers are ordered while complex numbers are not. More specifically, there is an order relationship among real numbers, which is a total order. i.e., for any two real numbers a and b, one of the following holds: a < b or a > b or a == b.
The standard definition of complex numbers doesn’t involve any such order relationship. For two complex numbers a and b, we can just say, either a == b or a != b.
Is it possible to establish a total order on complex numbers? It turns out that it is indeed possible. One of the possibilities is to use the so-called dictionary order or lexicographical order. Exactly how? Let’s say the two complex numbers are a = x + i y and b = u + i w respectively. We compare the real part first. If x < u, we say a < b. If x > u, we say a > b. But, if x == u, we look at the imaginary part. Now, if y < w, we say a < b; y > w, we say, a > b. Finally, if y == w, we say a == b. Recall that x is already equal to u.
We can easily verify that this relationship imposes a total order on the set of complex numbers. So, why don’t we see this used often? You would notice that this order relation gives undue priority to the real parts of the complex numbers being compared. There is no natural justification behind giving this importance to the real part. Thus, this order can be used only in those contexts, where the application provides some justification behind giving importance to the real part.
The natural question at this point is: Is this the only way to provide a total ordering on complex numbers? Of course not. As a straight-forward alternative, you could compare the imaginary parts first, and then compare the real parts. You can even use the polar form of complex numbers for lexicographical comparison. Compare the magnitudes first. If the magnitudes are equal, then compare the angles.
The three different ways discussed above suggest that we must pick up some representation of complex numbers and compare the components one by one to achieve a total ordering. This is reminiscent of the way strings are compared in programming languages like C or the way words are listed in the dictionaries. In every representation, we are comparing exactly two components one by one.
So, is it possible to achieve a total ordering on complex numbers by using just one component in some representation of complex numbers? While you are wondering about it, a different way to pose the question would be: is it possible to establish a one-to-one relationship between real numbers and complex numbers? Or, if you think of real numbers and complex numbers as groups, then are they isomorphic to each other?
Using the axiom of choice, one can show that real numbers and complex numbers are isomorphic as additive groups. They are both vector spaces over rational and AC gives bases of these two vector spaces of cardinalities 𝑐 and 𝑐×𝑐=𝑐, so they are isomorphic as vector spaces over rational.
REFERENCE: An Order on Complex Numbers
QUESTION I: How would an explicit isomorphism between Real Numbers and Complex Numbers look like?
QUESTION II: Is it possible to have an isomorphism between a set and its power set?
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