## Rectangular and Polar Forms of Complex Numbers

#### Aligned To Common Core Standard:

**High School** - HSN-CN.B.4

What are Rectangular and Polar Forms of Complex Numbers?
The number system in mathematics is vast, and there are a variety of different forms of numbers that students come across and deal with. The most common ones that you will have to learn about include real numbers, imaginary numbers, rational and irrational numbers, whole numbers, integers, and complex numbers.
Students find the concept of complex numbers a bit tricky. These numbers are a combination of real and imaginary numbers. One technique of working with complex numbers is by vectors, and to apply this method, you must have a complete understanding of complex number notations.
There are two forms of notation of complex numbers, and these are rectangular and polar forms.
**Rectangular Form of Complex Numbers** - When a complex number is written in its horizontal and vertical components, we call this the rectangular form of complex numbers. The rectangular form of complex numbers ids given by;
z = x + yi
Here; x is the horizontal component, and y is the vertical component.
**Polar Form of Complex Numbers** - The polar form of complex numbers includes its length or magnitude and angle of its vector. The polar form of complex numbers is given by;
z=r cos β+(r sin β)i
Here, r is the magnitude of the complex number, and β is the angle of its vector.
To calculate r;
r^{2} = a^{2} + b^{2}
a is the horizontal component, and b is the vertical component.
To find the angle;
tan β = b/a
Other relations that we use to represent a complex number in the polar form includes;
cos β =a/r, sin β =b/r. These worksheets and lessons will explore different geometrical methods for representing complex numbers and operations between them.

### Printable Worksheets And Lessons

- Polar to Rectangular Form Step-by-step Lesson- We are all use to rectangular forms, but polar coordinates make a lot of sense and it's easy to see why engineers favor their use.

- Guided Lesson - Switching between polar and rectangular and vice versa. A good way to work it.

- Guided Lesson Explanation - I find rectangular to polar much easier. A lot less square roots to worry about with this one.

- Practice Worksheet - Looks simple, but this one will take you a while to get right.

- Matching Worksheet - Match the converted form of the coordinates.

#### Homework Sheets

We have you jump back and forth between the two formats.

- Homework 1 - Rectangular coordinates are Cartesian coordinates, i.e., (x, y) coordinates.
- Homework 2 - There are known formulas for converting x and y to polar forms. Those are displayed on this sheet.
- Homework 3 - Those values (x and y) give a vertical distance, y, and a horizontal distance, x.

#### Practice Worksheets

You will find a detailed explanation with each of the files.

- Practice 1 - Convert the following rectangular coordinates to polar form. (3, 11)
- Practice 2 - Polar Coordinates are in the form (r, Θ). r = the distance from the origin. Θ = angle relative to the x axis.
- Practice 3 - Convert the following polar coordinates to rectangular form.

#### Math Skill Quizzes

There is a straight rectangular and polar quiz. Then we mix them together.