8.3 THE COMPLEX PLANE & DEMOIVRE’S THEOREM Precalculus

C OMPLEX P LANE X-axis: real axis (x) Y-axis: imaginary axis (yi) Complex plane: plane determined by these 2 axes To graph the complex number (x+yi), plot the ordered pair (x,y) 2

P OLAR F ORM OF C OMPLEX N UMBER cos Ө = x/r sin Ө = y/r x = rcos Ө y= rsin Ө tan Ө = b/a Ө = arctan (b/a) Complex number form: z = x+yi Polar form: z = (rcos Ө )+(rsin Ө )i = r(cos Ө +i sin Ө ) r is the modulus (magnitude) of z and Ө is the argument of z. 3 x + yi θ y x r

P LOTTING A P OINT IN THE C OMPLEX P LANE AND W RITING C OMPLEX NUMBERS IN P OLAR F ORM Example Graph each complex number and write in Polar form. i (a) 1 + i (b) 3 - 4i 4 x

5 Plotting a point in the Complex Plane and Converting from Polar form to Rectangular form Example Plot the point in complex plane and write in rectangular form. z = 3(cos 60° + i sin 60°) i

P RODUCT AND Q UOTIENT OF C OMPLEX N UMBERS IN P OLAR F ORM The theorem states: To multiply 2 complex numbers, multiply the moduli and add the arguments To divide 2 complex numbers, divide the moduli and subtract the arguments If the 2 complex numbers z 1 and z 2 are in the polar form: then … z 1 z 2 =r 1 r 2 [cos( Ө 1 + Ө 2 )+i sin( Ө 1 + Ө 2 )] (Multiplication) and… (z 1 /z 2 ) = (r 1 /r 2 ) [cos( Ө 1 - Ө 2 )+ i sin( Ө 1 - Ө 2 )] (Division) (z 2 ≠0) 6

D E M OIVRE ’ S T HEOREM Repeated use of the multiplication formula gives a useful formula for raising a complex number to a power n for any positive integer n. Let z be a complex number written in polar form. z=r(cos Ө +i sin Ө ) Then … 7

8 Find Complex Roots