The imaginary unit and complex numbers

The  exercise appears under the Algebra II Math Mission. This exercise practices operations and understanding of complex numbers.

Types of Problems
There are two types of problems in this exercise:


 * 1) Determine which numbers are real and which are not: This problem has a list of numbers involving the i, the imaginary unit. The student is asked to determine which of the numbers are real, and which are complex non-real.Tiuacn1.png
 * 2) Simplify the complex number: This problem has an expression that utilizes the imaginary unit. The student is asked to simplify the expression to it's simplest form.Tiuacn2.png

Strategies
Knowledge of complex number operations are encouraged to ensure success on this exercise.
 * 1) Adding and subtracting complex numbers is performed by collecting like terms.
 * 2) Multiplication of complex numbers if done with the distributive property.
 * 3) Division is done by rationalizing the denominator.
 * 4) The powers of the complex units are i, -1, -i or 1 depending on if the remainder when the exponent is divided by four is 1, 2, 3 or 0.
 * 5) Some calculators can be used to perform operations on complex numbers.

Real-life Applications

 * 1) Complex numbers have applications as models in non-Euclidean geometries.
 * 2) Complex numbers are used to create fractal images.
 * 3) Complex numbers are a natural extension of the real numbers.