Rectangular and polar forms of complex numbers

The Rectangular and polar forms of complex numbers exercise appears under the Precalculus Math Mission and Mathematics III Math Mission. This exercise continues to understand the connection between the rectangular and polar forms of a complex number.

Types of Problems

There are two types of problems in this exercise:

1. Find the coordinates and plot the point: This problem provides a complex number in polar form. The student is expected to find the coordinate of the point in a rectangular form and plot the appropriate point on the complex plane.

   

   Find the coordinates and plot the point

2. Select the true statements: This problem provides a complex number sketched on a complex plane. The student is expected to select the true statements from a collection of possible statements below the grid.

   

   Select the true statements

Strategies

Knowledge of the polar form of a complex number is encouraged to ensure success on this exercise.

1. This problem requires a student to find the appropriate radius and angle to make a complex number.
2. The radius is the magnitude of the complex number and the angle is the ${\text{arctan}(\frac{y}{x})}$.
3. The rectangular coordinates of a complex number are ${r\text{cos}(\theta)}$ and ${r\text{sin}(\theta)}$ from it's polar form.
4. Answers are to be rounded to the tenths place, one decimal place of accuracy.
5. This problem should always be performed correctly because the answer does not need to be submitted until it is moved to the correct place.
6. At least one statement will be true on the multiple select problem.

Real-life Applications

1. Imaginary numbers are used in chaos theory for predicting chaotic situations, like earthquakes and stock markets.
2. Imaginary numbers can be used to model geometry on the Cartesian plane.
3. The complex plane representation of complex numbers can be used to better understand polar coordinates.
4. Complex numbers have applications as models in non-Euclidean geometries.
5. Complex numbers are used to create fractal images.
6. Complex numbers are useful in studying number theory, which is the study of the positive integers.
7. Complex numbers are a natural extension of the real numbers.
8. The complex plane representation of complex numbers can be used to better understand polar coordinates.