Complex Numbers and Their Forms

Ashley, Pearl, Anjali, Farzana

Focus : Converting Between Forms!

Main Ideas...

​

1. Different Forms of Complex Numbers
1. Cartesian Form
2. Polar Form ( aka Trigonometric Form / Modulus-Argument Form)
3. Euler’s Form (Exponential form)

​

• Vocabulary Words
• Arithmetic
• Modulus : the magnitude, distance from the origin
• Argument : (direction) angle measure

​

Cartesian Form...

z = x + yi

• This form helps us to easily visualize the corresponding vector on the complex plane.

​

​

​

Polar Form..

Z = r (cos + sin i )

AKA z = r cis

​

• Uses a complex number’s modulus and argument!

​

• r = Modulus (magnitude)

​

• = argument(direction/angle from positive x-axis)

Finding the Modulus and Argument

Argument

Modulus

�Same as with vectors! ^

Euler’s Formula

​

Euler discovered…

→ eⁱ = cis

​

So…

Polar Form : z = r cis

• Substitute

Euler’s form : z = r e ⁱ

Converting Cartesian to Polar

The hardest conversion, but you can do it!

​

3 Key Steps (It helps to know the unit circle!)

1. Find the Modulus �
2. What quadrant is the complex number in?

​

• Find the Argument �

​

1. Find the Modulus.�

​

​

​

​

2. What quadrant are we in? → First quadrant!

3. Find argument…

​

​

Unit circle!

Now that we know the modulus and argument

Polar form! Yay!

​

EXAMPLE PROBLEM #2

GIVEN THAT |z| = 2√5, FIND THE COMPLEX NUMBER ‘z’ THAT SATISFIES THE EQUATION:

STEPS TO SOLVE:

​

Determine the modulus of the complex number.

​

​

Substitute ‘a + bi for z and ‘a - bi ‘ for z* into the equation.

​

Solve.

​