COMPLEX NUMBERS (8.3)

Complex Numbers and Their Representations

Definition of Complex Numbers
- A complex number is expressed in the form:
  $z = x + i\beta$
  where:
- $x$ is the real part,
- $\beta$ is the imaginary part,
- $i$ is the imaginary unit (defined as $i = ext{sqrt}(-1)$ ).
Geometric Interpretation of Complex Numbers
- The complex number $z$ can be represented as a point in the Cartesian coordinate system where the x-axis is the real part and the y-axis is the imaginary part.
- It can also be described in polar coordinates using radius and angle:
- Modulus (Magnitude)
  - The distance of the point from the origin, calculated using the equation:
    $r = ext{sqrt}(x^2 + y^2)$
  - This modulus is denoted as:
    $|z| = r$
- Argument (Angle)
  - The angle made with the positive x-axis, denoted as:
    $heta = ext{atan2}(y, x)$
Rectangular and Polar Forms
- Rectangular Form:
- Standard representation: $z = x + i\beta$
- Polar Form:
- Can be expressed as:
  $z = r ext{cis}( heta)$
  where:
  $ext{cis}( heta) = ext{cos}( heta) + i ext{sin}( heta)$ .
Key Characteristics
- Modulus and argument are critical in converting to polar form.
- Example of rewriting in polar form:
- Given $z = r ext{cis}( heta)$
- You can represent it as:
  $z = r( ext{cos}( heta) + i ext{sin}( heta))$
Advantages of Polar Form
- Simplifies multiplication and division of complex numbers:
- For multiplication:
 $z1 imes z2 = r1 r2 ext{cis}( heta1 + heta2)$
- For division:
 $rac{z1}{z2} = rac{r1}{r2} ext{cis}( heta1 - heta2)$
Trigonometric Relationships
- Recognize that the cosine function is even and sine is odd.
- Therefore, in polar forms:
- If an angle is negative, alter the signs properly while using cosine and sine functions.
Example Calculations
- Finding Modulus:
- If $z = -1 + 3i$ :
  Calculate:
  $|z| = ext{sqrt}((-1)^2 + (3)^2) = ext{sqrt}(1 + 9) = ext{sqrt}(10)$
- Finding Argument:
- For the same $z$ , determine:
  $heta = ext{atan2}(3, -1)$
- Converting to Polar Form:
- Then express:
  $z = ext{sqrt}(10) ext{cis} heta$

Operations with Complex Numbers

Addition and Subtraction
- For complex numbers much like regular algebra:
- $z1 + z2 = (x1 + x2) + i(\beta1 + \beta2)$
Multiplication
- Use as follows:
 $z1 imes z2 = (x1x2 - \beta1\beta2) + i(x1\beta2 + \beta1x2)$
Division
- When dividing, utilize the complex conjugate:
- $rac{z1}{z2} = rac{z1 imes ext{conjugate}(z2)}{|z_2|^2}$ .

Complex Roots of Unity

Definition
- The complex roots of unity are the solutions to the equation: $z^n = 1$ , where n is an integer.
- Generally, n roots of unity can be expressed as:
  $ext{cos} rac{2k heta}{n} + i ext{sin} rac{2k heta}{n}$ for $k = 0, 1, …, n-1$ .
Characteristic Values
- The first three cube roots of unity:
- $z_0 = 1$
- $z_1 = ext{cos} rac{2 ext{pi}}{3} + i ext{sin} rac{2 ext{pi}}{3}$
- $z_2 = ext{cos} rac{4 ext{pi}}{3} + i ext{sin} rac{4 ext{pi}}{3}$
Practical Utilization
- Roots of unity facilitate computations for polynomial roots and simplify understanding of periodic functions in trigonometry and calculus.

Summary of Key Concepts

The modulus and argument of a complex number provide insight into its geometric interpretation.
Polar form offers a method for simplifying operations with complex numbers.
Understanding complex roots of unity assists in advanced mathematical contexts.

General Exam and Homework Preparation Tips

Practice Problems
- Engage in both direct algebraic manipulation and polar representations for complex numbers.
- Ensure fluency in converting between rectangular and polar forms.
Calculators
- When testing, use calculators in radian mode when calculating trigonometric functions related to complex numbers.
Review Scope
- Ensure focus on understanding definitions and theorems associated with complex numbers, particularly in solving polynomial equations and transformations in complex analysis.