9 3 Complex Numbers The Complex Plane Polar

9. 3 Complex Numbers; The Complex Plane; Polar Form of Complex Numbers

In a complex number a is the real part and bi is the imaginary part. When b=0, the complex number is a real number. When a 0, and b 0, as in 5+8 i, the complex number is an imaginary number. When a=0, and b 0, as in 5 i, the complex number is a pure imaginary number.

The Complex Plane Imaginary Axis O Real Axis

Let be a complex number. The magnitude or modulus of z, denoted by As the distance from the origin to the point (x, y). is defined

Imaginary Axis y |z| O x Real Axis

is sometimes abbreviated as

z =-3 + 4 i -3 Imaginary 4 Axis Real Axis

z = -3 + 4 i is in Quadrant II x = -3 and y=4

z =-3 + 4 i 4 Find the reference angle ( ) by solving -3

z =-3 + 4 i 4 -3

Find r:

Imaginary 4 Axis -3 Real Axis

Find the reference angle ( ) by solving

Find the cosine of 330 and substitute the value. Find the sine of 330 and substitute the value. Distribute the r

Write in standard (rectangular) form.

Lesson Overview 9 -7 A

Product Theorem

Lesson Overview 9 -7 B

Quotient Theorem

5 -Minute Check Lesson 9 -8 A

5 -Minute Check Lesson 9 -8 B

Powers and Roots of Complex Numbers

De. Moivre’s Theorem

What if you wanted to perform the operation below?

Lesson Overview 9 -8 A

Lesson Overview 9 -8 B

Theorem Finding Complex Roots roots

Find the complex fifth roots of The five complex roots are: for k = 0, 1, 2, 3, 4.

The roots of a complex number a cyclical in nature. That is, when the points are plotted on a polar plane or a complex plane, the points are evenly spaced around the origin

Complex Plane

Polar plane

Polar plane

To find the principle root, use De. Moivre’s theorem using rational exponents. That is, to find the principle pth root of Raise it to the power

Example Find First express You may assume it is the principle root you are seeking unless specifically stated otherwise. as a complex number in standard form. Then change to polar form

Since we are looking for the cube root, use De. Moivre’s Theorem and raise it to the power

Example: Find the 4 th root of Change to polar form Apply De. Moivre’s Theorem