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== | ||
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:
![[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]](http://latex.artofproblemsolving.com/c/b/0/cb026a6e4281ccaa5a6caae131415b551f51c1e5.png)
See also
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