Complex Numbers

Complex Numbers

Basics

A complex number has two components: The real component and the imaginary components, typically denoted by \(\displaystyle i\) or \(\displaystyle j\).
Here \(\displaystyle i\) represents the imaginary number \(\displaystyle i^2 = -1\)

There are a few common operations on complex numbers:
Consider a complex number \(\displaystyle c = a + bi\),

• The conjugate of is \(\displaystyle c^* = a - bi\), also sometimes denoted as \(\displaystyle \bar{c}\).
• The squared norm is \(\displaystyle |c|^2 = c * c^* = a^2 + b^2\). Or \(\displaystyle |c| = \sqrt{a^2 + c^2}\).
• The angle is \(\displaystyle \angle c = \arctan(b, a)\).
• The exponential representation is \(\displaystyle c = |c| e^{i \theta} = |c| (\cos(\theta) + i \sin(\theta))\) where \(\displaystyle \theta = \angle c\).
  • In this representation, \(\displaystyle \theta\) is known as the argument and \(\displaystyle |c|\) is the modulus or absolute value.
  • This analagous to polar coordinates representation for 2d vectors (angle and magnitude).
  • Note that the word amplitude is overloaded and can refer to either the complex number, the absolute value, or the angle depending on context.
• Addition and multiplication are as usual, but with \(\displaystyle i^2 = -1\).

Euler's formula

Main article: Wikipedia: Euler's formula

Euler's formula states: \[ e^{ix} = \cos(x) + i \sin(x) \]

Properties

• The conjugate is \(\displaystyle e^{-ix}\) since cosine is symmetric and sine is odd (i.e. \(\displaystyle sin(-x) = -sin(x)\))

Euler's identity

Main article: Wikipedia: Euler's identity

Euler's identity states: \[ e^{i \pi} + 1 = 0 \]

Resources

• https://web.stanford.edu/~boyd/ee102/complex-primer.pdf Brief review of complex numbers