Complex numbers

Complex numbers in rectangular form: z=x+yi

Complex numbers in polar form: z=rcos(theta)+rsin(theta)i

i=square root of -1 and cannot be simplified and e^ipi=-1

absolute value of z=square root of x^2+y^2

To multiply complex numbers:

1. Rectangular-FOIL i^2=-1

2. Polar-multiply r and add theta

Example 1:

Express each complex number in polar form. Give angle measures to the nearest degree when necessary.

1+isquare root of 3

r=square root of1^2+isquare root of 3^2

r=square root of 1+3

r=square root of 4

r=+/-2

theta=tan^-1(square root of 3/1)=60degrees

theta=60degrees and 240degrees

z=2cis60degrees

z=2cis240degrees

z=+/-2cos240degrees+2sin240degrees

Example 2:

Express each complex number in rectangular form.

(5cis30degrees)(2cis60degrees)

5(2)=10

60+30=90

10cis90degrees

x=10cos90degrees

y=10sin90degrees

x=0

y=10

0+10i=10i