Difference between revisions of "Exponential form"
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| − | Every [[complex number]] <math>z</math> is the sum of a [[real]] and an [[imaginary]] component, <math>z=a+bi</math>. If you consider complex numbers to be [[coordinate]]s in the [[complex plane]] with the <math>x</math>-axis consisting of real numbers and the <math>y</math>-axis [[pure imaginary number]]s, then any point <math>z=a+bi</math> can be plotted at the point as <math>(a,b)</math>. We can convert <math>z</math> into [[polar form]] and re-write it as <math>z=r(\cos\theta+i\sin\theta)=r cis\theta</math>, where <math>r=|z| = \sqrt{a^2 + b^2}</math>. By [[Euler's formula]], which states that <math>e^{i\theta}=\cos\theta+i\sin\theta</math>, we can conveniently (yes, again!) rewrite <math>z</math> as <math>z=re^{i\theta}</math>, which is the general exponential form of a complex number. | + | Every [[complex number]] <math>z</math> is the sum of a [[real]] and an [[imaginary]] component, <math>z=a+bi</math>. If you consider complex numbers to be [[coordinate]]s in the [[complex plane]] with the <math>x</math>-axis consisting of real numbers and the <math>y</math>-axis [[pure imaginary number]]s, then any point <math>z=a+bi</math> can be plotted at the point as <math>(a,b)</math>. We can convert <math>z</math> into [[polar form]] and re-write it as <math>z=r(\cos\theta+i\sin\theta)=r cis\theta</math>, where <math>r=|z| = \sqrt{a^2 + b^2}</math>. By [[Euler's identity|Euler's formula]], which states that <math>e^{i\theta}=\cos\theta+i\sin\theta</math>, we can conveniently (yes, again!) rewrite <math>z</math> as <math>z=re^{i\theta}</math>, which is the general exponential form of a complex number. |
==See also== | ==See also== | ||
* [[Trigonometry]] | * [[Trigonometry]] | ||
* [[Trigonometric identities]] | * [[Trigonometric identities]] | ||
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{{stub}} | {{stub}} | ||
Revision as of 10:30, 3 March 2007
Every complex number 
is the sum of a real and an imaginary component, 
. If you consider complex numbers to be coordinates in the complex plane with the 
-axis consisting of real numbers and the 
-axis pure imaginary numbers, then any point can be plotted at the point as 
. We can convert into polar form and re-write it as 
, where 
. By Euler's formula, which states that 
, we can conveniently (yes, again!) rewrite as 
, which is the general exponential form of a complex number.
See also
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