Contents
Basics of Complex Numbers
Definition of a Complex Number
A complex number z takes the form of z = a + bi, where a and b are real and $ i = \sqrt-1 $. (i, sometimes written as j, is an imaginary number.) Essentially, what this means is that complex numbers are numbers having both a real and imaginary part. (It is possible for a = 0 or b = 0 and the number to still be considered complex, since real numbers and imaginary numbers are simply considered special cases of complex numbers.)
Properties of i
Since $ i = \sqrt-1 $, $ i^2 = -1 $. Furthermore, $ i^3 = i^2*i = -i $ and $ i^4 = i^2*i^2 = 1 $. Once $ i^5 $ is reached, a pattern can be seen: $ i^5 = i * i^4 = i * 1 = i $. So, in general:
$ i^{4n} = 1 $
$ i^{4n+1} = i $
$ i^{4n+2} = -1 $
$ i^{4n+3} = -i $
where n is an integer.
Visualization of a Complex Number
Complex numbers can be plotted on a 2-dimensional plane, called the Argand plane. One axis represents the real part of the complex number, and the other axis represents the imaginary part of the complex number. Traditionally, the real axis is horizontal, and the imaginary axis is vertical. The number z = a + bi, then, is represented as a point (a,b) on the plane.
Absolute Value of z
$ |z| = \sqrt{a^2+b^2} $ can be visualized easily on the complex plane as the distance of the line connecting (0,0) to the (a,b). From this, one can see that the formula above is essentially an application of the Pythagorean Theorem.
Basic Functions
Consider complex numbers $ z_1 = a + bi $ and $ z_2 = c + di $.
Addition
$ z_1 + z_2 = (a+c)+(b+d)i $
Subtraction
$ z_1 - z_2 = (a-c)+(b-d)i $
Multiplication
$ z_1 * z_2 = (a+bi)(c+di) = (ac + bd*i^2) + (ad*i+bc*i) = (ac-bd) + (ad+bc)i $
Division
$ {z_1\over{z_2}} = {a+bi\over{c+di}} = {(a+bi)(c-di)\over{(c+di)(c-di)}} = {(ac+bd)+(bc-ad)i\over{c^2+d^2}} =({ac+bd\over{c^2+d^2}}) + ({bc-ad\over{c^2+d^2}})i $
Note: c-di is called the complex conjugate of c+di. (And vice versa: c+di is called the complex conjugate of c-di.) Multiplying a complex number by its complex conjugate will always result in a real number.
Examples
- $ (1+2i) + (3+4i) = (1+3) + (2+4)i = 4+6i $
- $ (2 + 3i)^2 = (2+3i)(2+3i) = (4-9)+(6+6)i = -5+12i $
- $ |3+4i|=\sqrt{3^2+4^2}=5 $
