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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 | | ** 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]]. | ||
** Note that the word ''amplitude'' is overloaded and can refer to either the | ** 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>. | ||