Download presentation

Presentation is loading. Please wait.

Published byLeo England Modified over 2 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

OK

Trig form of Complex Numbers Objective: Be able to operate with complex numbers, and be able to convert complex numbers into Trig Form and vise versa.

Trig form of Complex Numbers Objective: Be able to operate with complex numbers, and be able to convert complex numbers into Trig Form and vise versa.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on traffic light control using image processing Ppt on six sigma Ppt on eye osmolarity Ppt on chapter 3 atoms and molecules wikipedia Ppt on school management system Ppt on smart tv technology Download ppt on work life and leisure Ppt on conservation of biodiversity Ppt on telephone etiquettes and manners Free ppt on product management