View source for Appendix B - Complex Numbers

Complex numbers arise naturally from an attempt to solve the equation 
<center><math>
a^2 + 1 = 0.
\,\!</math></center>
It's easy enough to write such an equation down, but how would you
solve it?  The answer is
<center><math>
a = \pm \sqrt{-1} =\pm i.
\,\!</math></center>
We let the symbol <math>i\,\!</math> represent <math>\sqrt{-1}\,\!</math>, so that <math>i^2=-1\,\!</math>.  Then
any number of the form 
{{Equation|<math>
z = x + iy, 
\,\!</math>|B.1}}
where <math>x\,\!</math> and <math>y\,\!</math> are real is called a ''complex'' number<!-- \index{complex number} -->.  Complex numbers arise in a variety of applications, including quantum mechanics.  
Suppose we have two complex numbers to be <math>z_1 = x_1+ iy_1\,\!</math> and <math>z_2 = x_2+ iy_2\,\!</math>.  Then the two complex numbers are equal, 
<center><math>
z_1 = z_2. 
\,\!</math></center>
This means that
<center><math>
x_1 +iy_1 = x_2+iy_2, 
\,\!</math></center>
is true if and only if
<center><math>
x_1 = x_2, \;\;\mbox{ and } \;\; y_1 = y_2.
\,\!</math></center>

We refer to <math>x\,\!</math> as the ''real part'' of the complex number <math>z\,\!</math> and
<math>y\,\!</math> as the ''imaginary part''.  Sometimes these are written as Re(<math>z\,\!</math>)
and Im(<math>z\,\!</math>) respectively.  That is, <math>x=\,\!</math>Re(<math>z\,\!</math>)
and <math>y=\,\!</math>Im(<math>z\,\!</math>).

We may restate the equivalence condition as <math>z_1=z_2\,\!</math> if and only if
the real part of <math>z_1\,\!</math> is equal to the real part of
<math>z_2\,\!</math> ''and'' the imaginary part of <math>z_1\,\!</math> is equal to the imaginary part of <math>z_2\,\!</math>.  

To add two complex numbers, add the real parts and the imaginary parts separately:
<center><math>
z_1 + z_2 = (x_1+iy_1) + (x_2+iy_2) = (x_1 + x_2) + i(y_1 + y_2).
\,\!</math></center> 

Complex numbers are multiplied like any other binomial expression (recall '''FOIL''': '''F'''irst two, '''O'''utside two, '''O'''nside two, '''L'''ast two):
<center><math>\begin{align}
z_1z_2 &= (x_1+iy_1)(x_2+iy_2) \\ 
       &= x_1x_2 + iy_1x_2 +  ix_1y_2 - y_1y_2 \\
       &= x_1x_2 - y_1y_2 +i(y_1x_2 + x_1y_2),  
\end{align}\,\!</math></center>
where we have used <math>i^2 = -1\,\!</math>.  

The ''complex conjugate'' <!-- \index{complex conjugate} --> of the complex
number <math>z\,\!</math> is denoted <math>z^*\,\!</math> and is given by
{{Equation|<math>
z^* = x-iy.  
\,\!</math>|B.2}}
Taking the complex conjugate of the number means just changing the sign in front of the <math>i\,\!</math>.  
One reason for defining this is that a number times its own complex conjugate is real,
<center><math>
zz^* = (x+iy)(x-iy) = x^2 + y^2 +i(xy - yx) = x^2 +y^2.  
\,\!</math></center>
The complex conjugate of the complex conjugate is the original complex number, and 
<center><math>
z^*z = (x-iy)(x+iy) = x^2 + y^2 +i(yx - xy) = x^2 +y^2 = zz^*.  
\,\!</math></center>

We also call this the ''modulus squared'' <!-- \index{modulus squared}-->.  The ''modulus'' is
{{Equation|<math>
|z| = \sqrt{(z^*z)} = \sqrt{x^2 + y^2}.
\,\!</math>|B.3}}
Note that the complex conjugate of a product is the product of the complex
conjugates: 
<center><math>
(z_1z_2)^* = z_1^* z_2^*.
\,\!</math></center>

It is often useful to look at a graph for a complex number, consisting of an x-axis for the real part and a y-axis for the complex part.  This is shown in [[#Figure B.1|Fig. B.1]].  In this
way, we can think of <math>z\,\!</math> as a two-dimensional vector with the magnitude (length) being equivalent to the modulus of the complex number, <math>|z| = \sqrt{x^2 + y^2}\,\!</math>.  In some sense (like looking at the complex number as a vector), the modulus of the complex number is the length, or magnitude, of the complex number.  
<div id="Figure B.1"></div>
<center>
[[File:complexgraph1.jpeg|500px]]
Figure B.1: A complex number in Cartesian coordinates.</center>
<br />

<!-- \begin{figure}
\begin{center}
\includegraphics[scale=0.5]{/home/mbyrd/tex/books/qcomp/figures/Complexgraph1.jpeg}
\caption{\label{fig:compg1} A Cartesian coordinate representation of
  a complex number <math>z\,\!</math>.}
\end{center}
\end{figure} -->


Another useful way to represent this is with polar coordinates.  We
can do this by writing 
<center><math>
z = |z|(\cos\theta +i \sin\theta).
\,\!</math></center>
Clearly if we wanted to rewrite this in terms of <math>x\,\!</math> and <math>y\,\!</math>, we find that <math>x=|z|\cos\theta\,\!</math> and <math>y = |z|\sin\theta\,\!</math>.  

A very famous identity exists between the exponential function and sine and cosine, 
{{Equation|<math>
e^{i\theta} = \cos\theta + i \sin\theta,
\,\!</math>|B.4}}
where <math>e\,\!</math> is the exponential function, which is a number that is approximately 2.71828.  
Using this equation, we could also write
<center><math>
z = |z|e^{i\theta}.
\,\!</math></center>
This is really just a short-hand notation for us and most of the time, when we calculate something, we will use the <math>\sin\theta, \cos\theta\,\!</math> functions.  But is often the case that people will write this as 
{{Equation|<math>
z = re^{i\theta},
\,\!</math>|B.5}}
where <math>r = \sqrt{x^2+y^2}\,\!</math> as is usual for polar coordinates.  So
everything is just like polar coordinates with the exception of the
inclusion of the factor <math>i\,\!</math>.  (See [[#Figure B.2|Fig. B.2]].)  
<div id="Figure B.2"></div>
<center>
[[File:Complexgraph2.jpeg|500px]]
Figure B.2: A polar coordinate representation of a complex number.</center>
<br />

===Exercises===

Let <math> z_1 = 5+2i \,\!</math> and <math> z_2 = 3+i \,\!</math>.  For 2-6, write the answer in terms of the real and complex components, i.e., in the form <math> x+iy \,\!</math>, where <math> x \,\!</math> and <math> y \,\!</math> are real numbers.

# Find <math>x</math> using the quadratic equation:   <math> \;\; 
\frac{1}{2}x^2-x+1=0 \,\!</math>
# What is <math> z_1+z_2 \,\!</math>?
# What is <math> z_1-z_2 \,\!</math>?
# Calculate <math> z_1z_2^* \,\!</math>
# What is <math> 3z_1 \,\!</math>?
# Find <math> |z_1| \,\!</math>.
# Find <math> |z_2|^2 \,\!</math>.
# Write <math> z_1 \,\!</math> as Equation [[#eqB.5|(B.5)]] using Equation [[#eqB.4|(B.4)]].

Further Problems:
# Show that <math> |z_1z_2|=|z_1||z_2| \,\!</math>
# Show that <math>(z_1z_2)^*=z_1^* z_2^* \,\!</math>