This is my answer to the same question posed on the Mathematics Stack Exchange. It is therefore licenced under CC-BY-SA 3.0.

Question

As I know when you move to “bigger” number systems (such as from complex to quaternions) you lose some properties (e.g. moving from complex to quaternions requires loss of commutativity), but does it hold when you move for example from naturals to integers or from reals to complex and what properties do you lose?

Answer

The most important ones as I see it:

• Naturals to integers: lose well-orderedness, gain “abelian group” (and, indeed, “ring”).
• Integers to rationals: lose discreteness, gain “field”.
• Rationals to reals: lose countability, gain “Cauchy-complete”.
• Reals to complexes: lose a compatible total order, gain the Fundamental Theorem of Algebra.