Download presentation

Presentation is loading. Please wait.

Published byLeo England Modified over 3 years ago

1
8.3 THE COMPLEX PLANE & DEMOIVRE’S THEOREM Precalculus

2
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

3
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

4
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
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

6
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

7
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
8 Find Complex Roots

Similar presentations

Presentation is loading. Please wait....

OK

8.2 Trigonometric (Polar) Form of Complex Numbers.

8.2 Trigonometric (Polar) Form of Complex Numbers.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google