Complex numbers arose in order to solve polynomial equations such as
or
which are not satisfied by real numbers. It’s assumed that a complex number has the form a + bi where a, b are real numbers and i, called imaginary unit, has the property that i2 = -1. Complex numbers are defined as follows.
- Addition.
- Subtraction.
- Multiplication.
- Division.
The ordinary rules of algebra has been used except that replaces i2 by -1 wherever it occurs. The commutative, associative and distributive laws also apply to complex numbers. It’s called a and b of a + bi the real and imaginary parts, respectively. Two complex numbers are equal if and only if their real and imaginary parts are respectively equal.
A complex number z = x + iy can be considered as a point P with coordinates (x, y) on a rectangular xy plane called in this case the complex plane or Argand diagram. If the line would be constructed from origin O to P and let ρ be the distance OP and φ the angle made by OP with the positive x axis, you could have from Figure
and could write the complex number in so-called polar form as
It’s often called that ρ the modulus or absolute value of of z and denote it by |z|. The angle φ is called the amplitude or argument of z abbreviated arg z. It could be also written
where
is called the conjugate of
.
If you write two complex numbers in polar form as
then
![\displaystyle z_1z_2 = \rho_1\rho_2[\cos(\phi_1 + \phi_2) + i\sin(\phi_1 + \phi_2)]\\\vspace{0.2 in} \frac{z_1}{z_2} = \frac{\rho_1}{\rho_2}[\cos(\phi_1 - \phi_2) + i\sin(\phi_1 - \phi_2)]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+z_1z_2+%3D+%5Crho_1%5Crho_2%5B%5Ccos%28%5Cphi_1+%2B+%5Cphi_2%29+%2B+i%5Csin%28%5Cphi_1+%2B+%5Cphi_2%29%5D%5C%5C%5Cvspace%7B0.2+in%7D++%5Cfrac%7Bz_1%7D%7Bz_2%7D+%3D+%5Cfrac%7B%5Crho_1%7D%7B%5Crho_2%7D%5B%5Ccos%28%5Cphi_1+-+%5Cphi_2%29+%2B+i%5Csin%28%5Cphi_1+-+%5Cphi_2%29%5D&bg=T&fg=000000&s=0 "\displaystyle z_1z_2 = \rho_1\rho_2[\cos(\phi_1 + \phi_2) + i\sin(\phi_1 + \phi_2)]\\\vspace{0.2 in} \frac{z_1}{z_2} = \frac{\rho_1}{\rho_2}[\cos(\phi_1 - \phi_2) + i\sin(\phi_1 - \phi_2)]")
Also if n is any real number, you have
![\displaystyle z^n = [\rho(\cos\phi + i\sin\phi)]^n = \rho^n(\cos n\phi + i\sin n\phi)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+z%5En+%3D+%5B%5Crho%28%5Ccos%5Cphi+%2B+i%5Csin%5Cphi%29%5D%5En+%3D+%5Crho%5En%28%5Ccos+n%5Cphi+%2B+i%5Csin+n%5Cphi%29&bg=T&fg=000000&s=0 "\displaystyle z^n = [\rho(\cos\phi + i\sin\phi)]^n = \rho^n(\cos n\phi + i\sin n\phi)")
which is often called De Moivre’s theorem. You can use this to determine roots of complex numbers. For example if n is a positive integer,
![\displaystyle z^{\frac{1}{n}} = [\rho(\cos\phi + i\sin\phi)]^\frac{1}{n} = \rho^{\frac{1}{n}}\left\{\cos\left(\frac{\phi + 2k\pi}{n}\right) + i\sin\left(\frac{\phi + 2k\pi}{n}\right)\right\}\\\vspace{0.2 in} \ \ k = 0,\ 1,\ 2,\ \cdots,\ n-1](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+z%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D+%3D+%5B%5Crho%28%5Ccos%5Cphi+%2B+i%5Csin%5Cphi%29%5D%5E%5Cfrac%7B1%7D%7Bn%7D+%3D+%5Crho%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D%5Cleft%5C%7B%5Ccos%5Cleft%28%5Cfrac%7B%5Cphi+%2B+2k%5Cpi%7D%7Bn%7D%5Cright%29+%2B+i%5Csin%5Cleft%28%5Cfrac%7B%5Cphi+%2B+2k%5Cpi%7D%7Bn%7D%5Cright%29%5Cright%5C%7D%5C%5C%5Cvspace%7B0.2+in%7D++%5C+%5C+k+%3D+0%2C%5C+1%2C%5C+2%2C%5C+%5Ccdots%2C%5C+n-1&bg=T&fg=000000&s=0 "\displaystyle z^{\frac{1}{n}} = [\rho(\cos\phi + i\sin\phi)]^\frac{1}{n} = \rho^{\frac{1}{n}}\left\{\cos\left(\frac{\phi + 2k\pi}{n}\right) + i\sin\left(\frac{\phi + 2k\pi}{n}\right)\right\}\\\vspace{0.2 in} \ \ k = 0,\ 1,\ 2,\ \cdots,\ n-1")
Using the series for ex, sin x, cos x, you are led to define
which are called Euler’s formulas and which enable you to rewrite equations in terms of exponentials.
