Appendix A
Complex Numbers
— Leonhard Euler
The set of complex numbers is the set of all numbers of the form where
and
are real numbers and
. This might not be the most formal way of presenting them, but it will do for our purposes!
The way you operate with complex numbers is pretty straightforward. Let ,
,
, and
be some real numbers. We add complex numbers as
Regarding multiplication, we have
In particular, when , we can deduce that
Given any complex number , its real part, which we denote as
, is
, and its imaginary part, which we denote as
, is
. Moreover, any such number
can be represented in the two-dimensional plane as a vector
. The length of the resulting vector is said to be the module of
, and it is computed as
If is a complex number, its conjugate is
. In layman’s terms, if you want to get the conjugate of any complex number, all you have to do is flip the sign of its imaginary part. It is easy to check that, given any complex...