Complex Numbers: Difference between revisions
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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 }} | ||
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* 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 | ==Euler's identity== | ||
Euler's | {{main | Wikipedia: Euler's identity}} | ||
Euler's identity states: | |||
<math display="block"> | <math display="block"> | ||
e^{i \pi} + 1 = 0 | e^{i \pi} + 1 = 0 | ||