Contents
Complex number basics and examples
Background and Form
Complex numbers are written in the form $ a+bi $, where a is the "real" part of the number and b is the "imaginary" part. The variable that denotes the imaginary part of the complex number is $ i $, where
$ i=\sqrt{-1} $ and $ \,\!i^2=-1 $
The real part of the number is thought of as lying on the x axis, while the imaginary lies on the y axis, so $ \,\!a+bi $ can also be graphed as a vector. This has special pertinence to Euler's formula, which relates sinusoidal and exponential equations using the complex number plane, but this is outside our scope.
Adding
Complex numbers are added by separating their real and imaginary parts, and adding them separately. The general case is:
$ \,\!(a+bi)+(c+di)=(a+c)+i(b+d) $
An example would be:
$ \,\!(5+2i)+(3-i)=8+i $
Multiplying
Complex numbers are multiplied using the standard distributive properties, but remembering that $ \,\!i^2=-1 $.
General case:
$ \,\!(a+bi)*(c+di)=ac-bd+i(ad+bc) $
An example is: $ \,\!(1+2i)*(4-i)=4-i+8i+2=6+7i $
Complex Conjugate
Oftentimes the imaginary part of a complex number must be eliminated from a fraction. This is accomplished by multiplying both the numerator and denominator by a complex conjugate. If the complex number is of the form $ \,\!a+bi $, then the complex conjugate of that number is $ \,\!a-bi $, or the imaginary part is made negative.
General case:
$ \,\!(a+bi)(a-bi)=a^2+b^2 $
This has practical applications for graphing complex numbers.
Magnitude and Angle
Since a complex number can be displayed in the x-y complex plane, complex numbers are thought to have a magnitude consistent with with ordinary vector magnitude rules: $ |\overrightarrow{a b}|=\sqrt{a^2+b^2} $
Finding the angle $ \boldsymbol{\theta} $ is just:
$ \boldsymbol{\theta}=\arctan(b/a) $
Example:
$ |4+3i|=\sqrt{4^2+3^2}=\sqrt{25}=5 $
