The succession of enlargements from ℤ (the system of the integers) to ℚ (that of the rational numbers) and thence to ℝ (the real numbers) and then ℂ (the complex numbers) is generally seen as one of increasing richness of structure. But gains are often accompanied by losses: for instance the step from ℝ to ℂ leads to the glories of complex analysis but away from ordinal properties once thought essential to any concept of number.

As far as group structure is concerned there are gains and losses at each stage of the progression. The following exposition is intended to demonstrate this without assuming more than a minimal knowledge of groups and of the theoretical foundations of analysis. Beyond some elementary arithmetical algebra, and an acquaintance with standard notation and vocabulary for maps in general, essentials taken for granted include the idea of a homomorphism (of groups), the existence and fundamental properties of the exponential function (of a real variable), and the Fundamental Theorem of Algebra (which is actually a theorem in complex analysis).