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 | + | Every [[complex number]] z is the sum of a [[real]] and an [[imaginary]] component, z=a+bi. If you consider complex numbers to be [[coordinate]]s 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. |
| + | |||
| + | ==See also== | ||
| + | * [[Trigonometry]] | ||
| + | * [[Trigonometric identities]] | ||
Revision as of 17:57, 23 June 2006
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 
, where r=|z|. By Euler's formula, which states that 
, we can conveniently (yes, again!) rewrite z as 
, which is the general exponential form of a complex number.
