Complex Numbers: Difference between revisions

Revision as of 00:15, 15 February 2023

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 \(\displaystyle \theta = \angle c\).
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. \(\displaystyle sin(-x) = -sin(x)\))

Resources

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

@@ Line 7: / Line 7: @@
 Here <math>i</math> represents the imaginary number <math>i^2 = -1</math>
-There are a few common operations on complex numbers:
+There are a few common operations on complex numbers:<br>
-Consider a complex number <math>c = a + bi</math>
+Consider a complex number <math>c = a + bi</math>,
 * The conjugate of is <math>c^* = a - bi</math>, also sometimes denoted as <math>\bar{c}</math>.
 * The squared norm is <math>|c|^2 = c * c^* = a^2 + b^2</math>. Or <math>|c| = \sqrt{a^2 + c^2}</math>.
 * The angle is <math>\angle c = \arctan(b, a)</math>.
 * The exponential representation is <math>c = |c| e^{i \theta}</math> where <math>\theta = \angle c</math>.
+* Addition and multiplication are as usual, but with <math>i^2 = -1</math>.
 ==Euler's Formula==