(→Forms of Complex Numbers) |
(→Operations of Complex Numbers) |
||
| Line 10: | Line 10: | ||
== Operations of Complex Numbers == | == Operations of Complex Numbers == | ||
| + | |||
| + | Product: | ||
| + | |||
| + | In rectangular form: <math>z_{1}=(a + bj) , z_{2} =(c + jd)</math> so <math>z_{1}(z_{2}) = (a + jb)(c + jd) = ac - bd + j(bc + ad)</math> | ||
| + | |||
| + | In polar form <math>z_{1}(z_{2}) = \rho_{1}e^{j\Theta_{1}}\rho_{2}e^{j\Theta{2}} = \rho_{1}\rho_{2}e^{j(\Theta_{1}+\Theta{2})}</math> | ||
| + | |||
| + | Division: | ||
| + | |||
| + | In rectangular form: | ||
Revision as of 17:51, 4 September 2008
Forms of Complex Numbers
Rectangular:
The basic form of a complex number in rectangular form is $ z = a + jb $ where $ j = \sqrt{-1} $. $ i $ is sometimes used instead of $ j $
Polar:
The polar representation of a complex number is given by $ \rho cos(\Theta)+j\rho sin(\Theta) = \rho e^{j\Theta} $. This is also known as Euler's Identity. Using the coefficients of the rectangular form $ \rho = \sqrt{a^2 + b^2} $ and $ \Theta = tan^{-1}(b/a) $.
Operations of Complex Numbers
Product:
In rectangular form: $ z_{1}=(a + bj) , z_{2} =(c + jd) $ so $ z_{1}(z_{2}) = (a + jb)(c + jd) = ac - bd + j(bc + ad) $
In polar form $ z_{1}(z_{2}) = \rho_{1}e^{j\Theta_{1}}\rho_{2}e^{j\Theta{2}} = \rho_{1}\rho_{2}e^{j(\Theta_{1}+\Theta{2})} $
Division:
In rectangular form:
