Is commutativity of real and imaginary units an axiom in complex analysis?

[McLaren Rulez]

Hi,

In complex analysis, is it an axiom that iy=yi where y is real? Or can this result be proved somehow? Thank you.

 

[micromass]

Well, it depends on how you define the product. The most elementary way to define complex numbers is by defining the to be couples (a,b) with addition

(a,b)+(c,d)=(a+c,b+d)
(a,b)(c,d)=(ac-bd,bc+ad)

Here a real number is of the form (a,0) and i is (0,1). It can now easily be checked tht

(a,0)(0,1)=(0,a)=(0,1)(a,0).

 

[McLaren Rulez]

Thank you micromass. That is a nice way of looking at it