All numbers are broken into sets: natural numbers \((1,2,3,4,...\); whole numbers \((0,1,2,3,4,...\); integers \(...-3,-2,-1,0,1,2,3,...\); rational numbers (fractions); irrational numbers (like \(\pi\) and \(\sqrt{5}\). These sets together make up the real numbers.
In this topic we discuss another set of numbers called the complex numbers. This set of numbers is built upon the number that is a solution to the equation: \(x^2+1=0\). Performing the algebra, leads to the solution: \(x=\sqrt{-1}\). The previous unit, Radical Things, required the radicand to be positive. Here the radicand is the number \(-1\). This number builds the complex numbers and is represented by \(i=\sqrt{-1}\).
We can now formally define complex numbers. A complex number in standard form, \(z\), is any number that can be written in the form:
$$z=a+bi$$
where \(a\) and \(b\) are any real valued numbers. A couple of examples:
$$3+6i,\;\frac{1}{5}-\frac{2}{3}i,\;\sqrt{5}i,\;\pi+.3333333i$$
Any real valued number is also a complex number. For example, \(4\) is a real number and a complex number since \(4=4+0i\).
To begin, it is important to write into \(i\)-form first, otherwise you might create an incorrect answer.
Example Multiply: \(\sqrt{-5}\cdot\sqrt{-10}\).
The process here is very simple, just treat the \(i\)-term like a variable and add or subtract accordingly. The following example shows adding two complex numbers and then subtracting a third.
Example \(\left(2+3i\right)+\left(6+4i\right)-\left(5-9i\right)\)
Before we multiply, recall from the solution of the first example that \(i^2=-1\). This occurs frequently when multiplying so be on the lookout for it!
Example Mulitply \((6-4i)(2+5i)\).
In order to divide, we need something called the conjugate of a complex number. The conjugate is defined to be the "opposite sign of the \(i\)-part": $$z=a+bi\iff\overline{z}=a-bi$$ For example, the conjugate of \(8-5i\) becomes \(8+5i\). Only the opposite sign of the imaginary part is changed. This helps to divide complex numbers by multiplying by the conjugate of the denominator.
Example Divide: \(\Large\frac{8+9i}{5+2i}\).