> Everything can be done without complex numbers by using matrices. A simple way to convince yourself of this is to realize a complex number is just a pair of r

> Everything can be done without complex numbers by using matrices. A simple way to convince yourself of this is to realize a complex number is just a pair of real numbers at the end of the day. Slightly tangential, but whilst this is a perfectly valid way of thinking about things, to me it seems to be mixing up concepts (e.g. real numbers, complex numbers, matrices, etc.) with encodings-of-concepts-using-other-concepts (in software terms, it's conflating an interface with a particular implementation). In particular, I wouldn't say that a complex number is a pair of real numbers. I'd rather define complex numbers by their algebraic laws (e.g. i × i = -1), and note that some other concepts (like pairs of reals, certain matrices, clifford algebras, etc.) can satisfy these same laws. Similar to how fractions can be encoded as a pair of integers (where "(a, b)" encodes "a ÷ b"), integers can be encoded using a pair of naturals (where "(a, b)" encodes "a - b"), etc. yet we rarely claim that, say, -3 is the pair (0, 3). Likewise a vector/matrix/tensor isn't a 1/2/N dimensional arrangement of numbers. This is a total nit-pick, but notice that you're larger point is basically arguing the same thing w.r.t. quantum mechanics (that it doesn't rely on complex numbers, since other representations exist) ;)