Complex Numbers: Difference between revisions
Created page with "Complex Numbers ==Basics== A complex number has two components: The real component and the imaginary components, typically denoted by <math>i</math> or <math>j</math>.<br> Here <math>i</math> represents the imaginary number <math>i^2 = -1</math> There are a few common operations on complex numbers: 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 <ma..." |
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Here <math>i</math> represents the imaginary number <math>i^2 = -1</math> | 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 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 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 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>. | * 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== | ==Euler's Formula== | ||
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