How do I 'compare' fields like the set of all real numbers and the set of all rational numbers?

I have to compare three fields, Q, R, and the set of complex numbers, C. I understand that Q is a subset of R, and R is a subset of C, and I also understand that there are many properties and theorems that arise from R that don't in Q. For example, that a rational number multiplied by an irrational number is irrational, and so on, but I'm really not sure if there's something larger that I'm missing, like some bigger, encompassing thing to talk about in terms of the differences and similarities in the three fields. Help?

How do I 'compare' fields like the set of all real numbers and the set of all rational numbers?
the normal way of comparing mathematical structures is by setting up a mapping between them that preserves the structure. The canonical mappings I:Q-%26gt;R by I(x) = x, and F:R-%26gt;C by F(x) = (x, 0) are both 1-1 but not onto, and preserve the field operations. When you have an isomorphism (a field-operation preserving map that is 1-1 and onto, then you are saying that the different structures are just different ways of looking at the same thing. For examply G:C-%26gt;R^2 by G(x+iy) = (x,y) is an isomorphism between the complex numbers are the real plane.





another way to compare sets is by cardinality. the rationals are countable, while the reals and complexes are not.


A proof that the reals in [0,1) are not countable:


Suppose they were. Let S%26lt;n%26gt; be an enumeration of the reals (as decimals between 0 and 1). Now construct the real number S whose nth digit is (5 if the nth digit of S%26lt;n%26gt; is 0 or 9; the %26lt;n%26gt;th digit of S%26lt;n%26gt;+1 otherwise). Then S is not in our list, since for all n, it has a different nth digit from S%26lt;n%26gt;.


R and C have the same cardinality, although one isa line, the other a plane. The mapping here H:C-%26gt;R takes (x+iy) into the real whose odd digits are those of x and whose even digitsare those of y.