3

This is probably very basic, but I need to learn more about this:

What is this operation? 

(a, b) operation (c, d) = (a * c - b * d, a * d + b * c)

And where can I learn more about the topic?

Thanks.

Jason
  • 3,563

1 Answers1

11

This comes from the definition of the complex numbers as ordered pairs $(x,y)$ of reals, with the natural addition and the somewhat less natural multiplication of the post.

Note that $$(a+bi)(c+di)=ac-bd +i(ad+bc).$$

Among the many advantages of this definition of complex number is the fact that we do not need to talk about mysterious square roots of $-1$. Our objects are ordered pairs of "ordinary" reals.

André Nicolas
  • 507,029
  • Thanks. So the operation would be the multiplication of the complex number pairs? – Jason Jan 11 '14 at 06:40
  • Under the identification of the complex number $x+iy$ with the ordered pair $(x,y)$, the operation is ordinary multiplication of two complex numbers. – André Nicolas Jan 11 '14 at 06:42
  • 1
    Additional advantages: real numbers are just numbers with the second component = $0$. Also, (0,1) opreation (0,1) = (-1,0), and there you have the square root of $-1$. – user44197 Jan 11 '14 at 06:46