Learning Objectives:

Math Tutoring on Complex Numbers

If we assume then . This lead to a new set of numbers called **Imaginary Numbers** and then the set of **Complex Numbers**.

Imaginary Numbers are those in the form , where ‘a’ is a negative real number.

**Complex Numbers** are those numbers written in the form , where **‘a’** and **‘b’** are real numbers and **.**

The expression ** is** called **Standard Form of a Complex Number**. Therefore

(C)** Complex: ** and (I) **Imaginary: **

*Addition and Subtraction of Complex Numbers in Math Tutoring:*

** & **

*Multiplying Complex Numbers:*

To divide Complex Numbers following steps should be taken:

1. Write the problem in fractional form.

2. Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

As a **Complex Number** ** **form has two components, we can graph a **Complex Number** similar to how we graph vectors, except we use the **Complex Plane** instead of the **Coordinate Plane**.

The Complex Number** **** **and angle **θ** can be given in trigonometric form:

, where **‘r’** represents the **Magnitude** of **‘z’** called **Modulus** and **‘θ’** is often called the **Argument.**

Every nonzero **Complex Number** has exactly **‘n’** distinct complex roots. For any positive integer **‘n’**, the **Complex Number ** is a **n ^{th} root** of the

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**Hook Questions:**

*1. **What are Complex Numbers?*

*2. **What is the trigonometric form of Complex Numbers?*

*3. **State and prove De Moivre’s theorem?*

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