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 \(\displaystyle c = |c| e^{i \theta}\) where \(\displaystyle \theta = \angle c\).
- In this representation, \(\displaystyle \theta\) is known as the Argument and
|c|is the modulus or absolute value. - Note that the word amplitude is overloaded and can refer to either the entire complex number, the absolute value, or the angle depending on context.
- In this representation, \(\displaystyle \theta\) is known as the Argument and
- Addition and multiplication are as usual, but with \(\displaystyle i^2 = -1\).
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
Euler's identity states: \[ e^{i \pi} + 1 = 0 \]
Resources
- https://web.stanford.edu/~boyd/ee102/complex-primer.pdf Brief review of complex numbers