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 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. | 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. | ||
+ | |||
+ | So <math>z=re^{i\theta}</math> looks like: | ||
+ | <asy> | ||
+ | import markers; | ||
+ | pair A,B,C,D,E; | ||
+ | A=(0,1); | ||
+ | B=(0,0); | ||
+ | C=(1,0); | ||
+ | D=(1/2,sqrt(3)/2); | ||
+ | E=(1/2,0); | ||
+ | draw(A--B--C); | ||
+ | label("Im",2*A/3,W); | ||
+ | label("Re",2*C/3,S); | ||
+ | dot("$z$",D,NE); | ||
+ | draw(B--D,blue); | ||
+ | draw(E--D,red); | ||
+ | draw(B--E,green); | ||
+ | label("$r$",(B+D)/2,p=blue,NW); | ||
+ | label("$\sin\theta$",(E+D)/2,p=red,E); | ||
+ | label("$\cos\theta$",(B+E)/2,p=green,S); | ||
+ | markangle("$\theta$",radius=13,C,B,D,ArcArrow,orange); | ||
+ | </asy> | ||
==See also== | ==See also== | ||
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{{stub}} | {{stub}} | ||
+ | |||
+ | [[Category:Complex numbers]] |
Latest revision as of 22:51, 20 February 2022
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.
So looks like:
See also
This article is a stub. Help us out by expanding it.