## Theorem 1.2.12: Multiplying complex numbers geometrically |

If |

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 vectorszandw

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 = r_{1}cis(s) r_{2}cis(t) =

= r_{1}(cos(s)+ i sin(s)) r_{2}(cos(t) + i sin(t)) =

= r_{1}r_{2}(cos(s)+ i sin(s))(cos(t) + i sin(t)) =

= r_{1}r_{2}(cos(s) cos(t) - sin(s)sin(t)) + i(cos(s)sin(t) + sin(s)cos(t)) =

= r_{1}r_{2}(cos(s+t) + i sin(s+t)) =

= r_{1}r_{2}cis(s+t)

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

z^{2}= z*z = r cis(t) r cis(t) = r^{2}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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