# Rectangular and Polar Forms of Complex Numbers Worksheets

Complex numbers are very theoretical aspect of mathematics. They are combination of real and imaginary numbers. This type of math is used to understand the limitations of all different types of electronic devices for both safety and productivity purposes. Until now we have worked with these expressions written in what called rectangular form. We can also use a different form of notation that can give us a little more information about the expression called polar notation. These worksheets and lessons will explore different geometrical methods for representing complex numbers and operations between them.

### Aligned Standard: HSN-CN.B.4

- 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.

- Answer Keys - These are for all the unlocked materials above.

### 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.

- Quiz 1 - We can do this by using these formulas: x = r cos Θ and y = r sin Θ
- Quiz 2 - Use the formula: r = √x
^{2}+y^{2}(To find r.) - Quiz 3 - The last step is always to plug in your numbers into this equation.

### 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. Imaginary numbers complicated just about all forms of math because they are values that express in the terms of the square root of a negative number. One technique of working with complex numbers is by using vectors, and to apply this method, you must have a complete understanding of complex number notations. Complex numbers behave much like a two dimensional vector. This allows us to use vector operations to help us make sense of these values.

There are two other forms of notation of these equations, and these are rectangular and polar forms.

**Rectangular Form ** - We are commonly accustomed to seeing complex numbers written in what is called rectangular coordinate form. When a complex number is written in its horizontal and vertical components, we call this the rectangular form of complex numbers. Working off of the rectangular form given by: z = x + yi. Here; x is the horizontal component, and y is the vertical component.

**Polar Form** - The polar form of complex numbers includes its length or magnitude and angle of its vector. It gives you a little more to work with. This is especially useful when you are working with powers and roots of a complex number. The polar form is given by: z = r cos Θ + i r sinΘ = r(cos Θ + i sin Θ) Here, r is the magnitude of the complex number, and Θ is the angle of its vector. To calculate r: r^{2} = a^{2} + b^{2}.

There is a conversion formula that will allow you to convert rectangular notation into polar coordinates:

r = √(x^{2} + y^{2}) and Θ = tan^{-1} (y/x).