Complex Numbers: Difference between revisions
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Complex Numbers | Complex Numbers | ||
==Basics== | ==Basics== | ||
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* 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} = |c| (\cos(\theta) + i \sin(\theta))</math> where <math>\theta = \angle c</math>. | ||
** In this representation, <math>\theta</math> is known as the [[Wikipedia: Argument | argument]] and <math>|c|</math> is the modulus or [[Wikipedia: Absolute_value#Complex_numbers | 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 <math>i^2 = -1</math>. | * Addition and multiplication are as usual, but with <math>i^2 = -1</math>. | ||
==Euler's | ==Euler's formula== | ||
{{ main | Wikipedia: Euler's formula }} | {{ main | Wikipedia: Euler's formula }} | ||
Euler's formula states: | Euler's formula states: | ||
<math display="block> | <math display="block"> | ||
e^{ix} = \cos(x) + i \sin(x) | e^{ix} = \cos(x) + i \sin(x) | ||
</math> | </math> | ||
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===Properties=== | ===Properties=== | ||
* The conjugate is <math>e^{-ix}</math> since cosine is symmetric and sine is odd (i.e. <math>sin(-x) = -sin(x)</math>) | * The conjugate is <math>e^{-ix}</math> since cosine is symmetric and sine is odd (i.e. <math>sin(-x) = -sin(x)</math>) | ||
==Euler's identity== | |||
{{main | Wikipedia: Euler's identity}} | |||
Euler's identity states: | |||
<math display="block"> | |||
e^{i \pi} + 1 = 0 | |||
</math> | |||
==Resources== | ==Resources== | ||
* [https://web.stanford.edu/~boyd/ee102/complex-primer.pdf https://web.stanford.edu/~boyd/ee102/complex-primer.pdf] Brief review of complex numbers | * [https://web.stanford.edu/~boyd/ee102/complex-primer.pdf https://web.stanford.edu/~boyd/ee102/complex-primer.pdf] Brief review of complex numbers | ||
Latest revision as of 00:23, 2 February 2024
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
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