Chapter Contents

• Complex Numbers
• 1. Basic Definitions
• 2. Basic Operations
• 3. Graphical Representation
• 4. Polar Form
• Convert polar to rectangular using calculator
• 5. Exponential Form
• 6. Products and Quotients
• 7. Powers and Roots
• 8. AC Circuit Definitions
• 9. Impedance and Phase Angle
• 10. Reactance and Angular Velocity
• 11. AC Circuit Exercises
• 12. Parallel AC Circuits
• Vocabulary
• Fractals
• Complex Numbers Problem Solver

• Comments, Questions?

Feelings about Math

Recommendation

Easy to understand algebra lessons on DVD. Try before you commit. More info:
MathTutorDVD.com

Online Algebra Solver

Solve your algebra problem step by step!

Online Algebra Solver »

Share

Share this page.

From the math blog...

4. Polar Form of a Complex Number

by M. Bourne

We can think of complex numbers as vectors, as in our earlier example. [See more on Vectors in 2-Dimensions].

We have met a similar concept to "polar form" before, in Polar Coordinates, part of the analytical geometry section.

Our aim in this section is to write complex numbers in terms of a distance from the origin and a direction (or angle) from the positive horizontal axis.

We find the real (horizontal) and imaginary (vertical) components in terms of r (the length of the vector) and θ (the angle made with the real axis):

From Pythagoras, we have: r² = x² + y² and basic trigonometry gives us:

x = r cos θy = r sin θ

Multiplying the last expression throughout by j gives us:

yj = jr sin θ

So we can write the polar form of a complex number as:

x + yj = r(cos θ + j sin θ)

r is the absolute value (or modulus) of the complex number

θ is the argument of the complex number.

There are two other ways of writing the polar form of a complex number:

r cis θ [means r (cos θ + j sin θ)]

r ∠ θ [means once again, r (cos θ + j sin θ)]

NOTE: When writing a complex number in polar form, the angle θ can be in DEGREES or RADIANS.

Example 1

Find the polar form and represent graphically the complex number 7 - 5j.

Answer

Example 2

Express 3(cos 232°+ j sin 232°) in rectangular form.

Answer

Exercises

1. Represent 1 + j√3 graphically and write it in polar form.

Answer

2. Represent √2 - j√2 graphically and write it in polar form.

Answer

3. Represent graphically and give the rectangular form of
6(cos180°+ jsin180°).

Answer

4. Represent graphically and give the rectangular form of 7.32 ∠ -270°

Answer