Complex Numbers



A complex number is a point in the plane, given by two numbers in an ordered pair: (a,b). We define a way to add two of these "numbers":

(a,b)+(c,d)=(a+c,b+d)

and also to multiply them them

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

We'll also define a length



to be the distance from the point to the point (0,0). And we'll need a function which measures angle up from the positive "x-axis."

Arg((a,b))=acrtan(b/a).

(If you haven't seen "arctan" don't worry, just think of measuring the angle between a line that goes through the point and the right half of the horizonal line that goes through (0,0)).

Addition is straightforward, just add first and second components, respectively. Multiplication may look artificial, but it ends up having the nice properties that:

|(a,b)*(c,d)|=|(a,b)|*|(c,d)|,

and



That is, with multiplication, lengths get multiplied and angels get added, so the strange definition of multiplication is actually quite simple, geometrically.

Question: What is (0,1)*(0,1)?