
Complex numbers
I'm in a number systems class, which has us prove things we already know from scratch.
I need to prove that Multiplication on complex numbers are associative.
We have defined complex numbers to be ordered pairs and definition of multiplication and addition of complex numbers is as follows:
(x1, y1)(x2, y2) = (x1x2  y1y2, y1x2 + x1y2) and
(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)
I must show [(x,y)(z,w)] (a,b) = (x,y) [(z,w)(a,b)]
Any advice?

$\displaystyle [(x,y)(z,w)](a,b)=(xzyw,xw+yz)(a,b)=$
$\displaystyle =(axzaywbxwbyz,bxzbyw+axw+ayz)$ (1)
$\displaystyle (x,y)[(z,w)(a,b)]=(x,y)(azbw,bz+aw)=$
$\displaystyle =(axzbxwbyzayw,bxz+axw+ayzbyw)$ (2)
From (1) and (2) we have equality.