Theorem 1.2.12: Multiplying complex numbers geometrically

 If z = r1 cis(s) and w = r2 cis(t) in polar coordinates then z*w = r1 r2 cis(s + t) and zn = rn cis(ns) Context

The formulas means:

• two complex numbers are multiplied by multiplying their length and adding their angles
• a complex number is raised to a power by raising the length to the power and multiplying the angle
 Multiplying vectors z and w

The proof is an exercise in trig identities. Recall that:

(1)       cos(x + y) = cos(x)cos(y)-sin(x)sin(y)
(2)       sin(x + y) = sin(x)cos(y)+cos(x)sin(y)

Now

z*w = r1 cis(s) r2 cis(t) =
= r1(cos(s)+ i sin(s)) r2(cos(t) + i sin(t)) =
= r1 r2 (cos(s)+ i sin(s))(cos(t) + i sin(t)) =
= r1 r2 (cos(s) cos(t) - sin(s)sin(t)) + i(cos(s)sin(t) + sin(s)cos(t)) =
= r1 r2 (cos(s+t) + i sin(s+t)) =
= r1 r2 cis(s+t)

which proves the first assertion. As for the second, first note that

z2 = z*z = r cis(t) r cis(t) = r2 cis(2t)

by our previous result. Now you should be able to finish the proof by carefully writing down an induction proof yourself.

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Interactive Complex Analysis, ver. 1.0.0
(c) 2006-2007, Bert G. Wachsmuth