Argand diagram transformations

Let&nbsp;\(z=x+iy\) and \(w=u+iv\), where&nbsp;\(x,y,u\) and \(v\) are real numbers. Then
\[zw = (x+iy)(u+iv) = (xu-yv)+i(xv+yu).\]
This definition looks very contorted and complicated because it seems far removed from the usual multiplication of real numbers, or is it?

Let \(z=x+iy\) and \(w=u+iv\), where \(x,y,u\) and \(v\) are real numbers. Then

\[zw = (x+iy)(u+iv) = (xu-yv)+i(xv+yu).\]

This definition looks very contorted and complicated because it seems far removed from the usual multiplication of real numbers, or is it?

Modulus, Argument, and Operations on Complex Numbers

Complex Numbers

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Multiplication of complex numbers

Multiplication of complex numbers