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Difference between revisions of "Exponential form"

 
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Every complex number z is the sum of a real and an imaginary component, z=a+bi.  If you consider complex numbers to be coordinates in the complex plane with the x-axis consisting of real numbers and the y-axis pure imaginary numbers, then every point z=a+bi can be graphed as (x,y)=(a,b).  We can convert z into polar form and re-write it as <math>z=r(\cos\theta+i\sin\theta)=r cis\theta</math>, where r=|z|.  By Euler's formula, which states that <math>e^{i\theta}=\cos\theta+i\sin\theta</math>, we can conveniently (yes, again!) rewrite z as <math>z=re^{i\theta}</math>, which is the general exponential form of a complex number.
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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.
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 +
So <math>z=re^{i\theta}</math> looks like:
 +
<asy>
 +
import markers;
 +
pair A,B,C,D,E;
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A=(0,1);
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B=(0,0);
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C=(1,0);
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D=(1/2,sqrt(3)/2);
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E=(1/2,0);
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draw(A--B--C);
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label("Im",2*A/3,W);
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label("Re",2*C/3,S);
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dot("$z$",D,NE);
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draw(B--D,blue);
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draw(E--D,red);
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draw(B--E,green);
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label("$r$",(B+D)/2,p=blue,NW);
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label("$\sin\theta$",(E+D)/2,p=red,E);
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label("$\cos\theta$",(B+E)/2,p=green,S);
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markangle("$\theta$",radius=13,C,B,D,ArcArrow,orange);
 +
</asy>
 +
 
 +
==See also==
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* [[Trigonometry]]
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* [[Trigonometric identities]]
 +
 
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{{stub}}
 +
 
 +
[[Category:Complex numbers]]

Latest revision as of 21:51, 20 February 2022

Every complex number $z$ is the sum of a real and an imaginary component, $z=a+bi$. If you consider complex numbers to be coordinates in the complex plane with the $x$-axis consisting of real numbers and the $y$-axis pure imaginary numbers, then any point $z=a+bi$ can be plotted at the point as $(a,b)$. We can convert $z$ into polar form and re-write it as $z=r(\cos\theta+i\sin\theta)=r cis\theta$, where $r=|z| = \sqrt{a^2 + b^2}$. By Euler's formula, which states that $e^{i\theta}=\cos\theta+i\sin\theta$, we can conveniently (yes, again!) rewrite $z$ as $z=re^{i\theta}$, which is the general exponential form of a complex number.

So $z=re^{i\theta}$ 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

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