[justin_]

He probably means the algebraic structure of a field. "A field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do."[0]

You might be tempted to think of complex numbers as "just" being 2-dimensional real vectors (x, y). Looks pretty similar to how you can plot a complex number a + ib at point (a, b) on a 2D plane. But importantly, division is defined on a field, which is not necessarily true for vectors. For any complex number (except 0), you can find another complex number that multiplies with it to give 1, the multiplicative identity.

You _can_ think of complex numbers as being "made of" real numbers though. a and b above are just real numbers. Complex numbers are the two-dimensional normed division algebra over the reals[1].

[0] https://en.wikipedia.org/wiki/Field_(mathematics) [1] https://ncatlab.org/nlab/show/normed+division+algebra

[nathan_compton]

Exactly. But I want to point out that the field character of the complex numbers is not some incidental quality which happens to distinguish them from 2-vectors. Its absolutely essential to their mathematical character and usage and it also distinguishes them from other complexes we might want to form that behave in a real number like fashion. For instance, there is no way to form a field over the three vectors. In general, one has to give up more and more structure as the dimensions go up.

I think that the obsession with quantum mechanics containing complex number is a little overblown. Quantum Mechanics is fundamentally about a defining a formalism which preserves the ability to simultaneously keep track of the physical symmetries in a system and the probabilities of particular outcomes of measurement. In many situations complex numbers provide a useful way to do this because of the symmetries involved (eg spin 1/2) but in other situations other symmetry groups are required. The appearance of complex numbers is no more (or less, I suppose) mysterious than the appearance of SU(3) in nuclear physics or SU(2)xU(1) in electroweak physics. Its just a matter of what symmetries you have and how many outcomes a measurement can have (roughly).