In-Depth Notes on Complex Numbers and Arguments

Complex Numbers and Their Arguments

• Complex Number Definition: A complex number is represented as $z = x + yi$ where $x$ is the real part and $y$ is the imaginary part.

• Argument of a Complex Number: The argument of a complex number, denoted as ${Arg}(z)$, refers to the angle formed between the positive direction of the x-axis and the line representing the complex number in the complex plane.

  • Example: The argument for $z = x + yi$ is determined as the angle $theta$ where $an( heta) = \frac{y}{x}$.

• Standard Value of Argument: There is an essential consideration regarding multiple values of the argument: for a complex number, one cannot consistently select a singular value due to the periodic nature of tangent functions.

  • To address this, we typically measure angles counter-clockwise from the positive x-axis, restricting the argument to the interval $(-\pi, \pi]$.

Polar Coordinates of Complex Numbers

• Conversion to Polar Form: Every complex number can also be represented in polar form as:
  $z = re^{i\theta}$
  where $r$ is the magnitude (or modulus) defined as $r = \sqrt{x^2 + y^2}$ and $\theta$ is the argument of the complex number.

• Important Properties:

  1. The product of two complex numbers in polar form: $(re^{i\theta})(se^{i\phi}) = (rs)e^{i(\theta + \phi)}$

  2. Raising to a power: $(re^{i\theta})^n = r^n e^{in\theta}$

De Moivre's Theorem

• Theorem Statement: For any integer n, De Moivre's theorem states that:
  $(cos(\theta) + i\sin(\theta))^n = cos(n\theta) + i\sin(n\theta)$
  This allows for easier calculation of powers of complex numbers expressed in polar form.

• Example Application:

  • Given a complex number $z = re^{i\theta}$, then:

  • Calculating the nth power gives:
    $z^n = r^n e^{in\theta}$

Roots of Complex Numbers

• Finding nth Roots: If $z$ is a non-zero complex number represented as $z = re^{i\theta}$, the nth roots are found by:
  $w_k = r^{1/n} e^{i(\frac{\theta + 2\pi k}{n})}$, where $k = 0, 1, 2, …, n-1$.

  • This results in n distinct roots that are evenly spaced around a circle in the complex plane.

• Example of 5th Roots: For n = 5, the 5th roots of a complex number will form a regular pentagon in the complex plane, each separated by an angle of $\frac{2\pi}{5}$ radians.