REVIEW
complex numbers and related concepts
 By definition, the complex numbers are numbers of the form
$\,a + bi\,$,
where $\,a\,$ and $\,b\,$ are real numbers, and $\,i = \sqrt{1}\,$.
The number $\,a\,$ is called the real part of the complex number.
The number $\,b\,$ is called the imaginary part of the complex number.

Note that $\,i^2 = 1\,$.
The number $\,i\,$ is not a real number, since there is no real
number which,
when squared, equals $\,1\,$.
 The solutions to $\ x^2 = 1\ $ are $\,\pm i\,$:
 when $\,x = i\,$: $\,i^2 = 1\,$
 when $\,x = i\,$: $\,(i)^2 = i^2 = 1\,$

a (pure) imaginary number has real part equal to $\,0\,$: e.g., $\,0 + 7i = 7i\,$

a real number has imaginary part equal to $\,0\,$: e.g., $\,5 + 0i = 5\,$
 The complex numbers are graphed in a coordinate plane where:
 the real numbers lie along the horizontal axis; this is called the real axis
 the scalar multiples of $\,i\,$ lie along the vertical axis; this is called the imaginary axis
 the intersection of the real and imaginary axes is called the origin
In this scheme, the complex number $\,a + bi\,$ is found as follows:
 put a vertical line through the real number $\,a\,$ on the horizontal axis
 put a horizontal line through the imaginary number $\,bi\,$ on the vertical axis
 the number $\,a + bi\,$ is the unique intersection point of these two lines
This plane is a perfect representation of the complex numbers,
and is called the complex plane:
 every complex number corresponds to a unique point in this plane
 every point in this plane corresponds to a unique complex number

equality of complex numbers:
For all real numbers $\,a\,$, $\,b\,$,
$\,c\,$ and $\,d\,$,
$$a+bi=c+di\ \ \ \ \ \text{if and only if}\ \ \ \ \ (a=c\ \text{and}\ b = d)$$
Arithmetic with complex numbers is covered in the next section.

a piece of the complex plane,
showing the numbers $\,2 + i\,$ and $\,1 + 2i\,$
