Appendix B - Complex Numbers

Complex numbers arise naturally from an attempt to solve the equation

$a^{2}+1=0.\,\!$

It's easy enough to write such an equation down, but how would you solve it? The answer is

$a=\pm {\sqrt {-1}}=\pm i.\,\!$

We let the symbol $i\,\!$ represent ${\sqrt {-1}}\,\!$, so that $i^{2}=-1\,\!$. Then any number of the form

$z=x+iy,\,\!$

where $x\,\!$ and $y\,\!$ are real, is called a complex number. Let's take some other complex number to be $\eta =c+id\,\!$, where $c\,\!$ and $d\,\!$ are real. Then the two complex numbers are equal,

$\eta =z\,\!$

which is to say

$x+iy=c+id,\,\!$

if and only if

$x=c,\;\;{\mbox{ and }}\;\;y=d.\,\!$

We refer to $x\,\!$ as the real part of the complex number $z\,\!$ and $y\,\!$ as the complex part. Sometimes these are written as Re($z\,\!$) and Im($z\,\!$), respectively.

We may restate the equivalence condition as $z=\eta \,\!$ if and only if the real part of $z\,\!$ is equal to the real part of $\eta \,\!$ and the imaginary part of $z\,\!$ is equal to the imaginary part of $\eta \,\!$.

Complex numbers are multiplied like any other binomial expression:

$z\eta =(x+iy)(c+id)=xc-yd+i(yc+xd),\,\!$

where we have used $i^{2}=-1\,\!$.

The complex conjugate of the complex number $z\,\!$ is denoted $z^{*}\,\!$ and is given by

$z^{*}=x-iy.\,\!$

One reason for defining this is that a number times its own complex conjugate is real,

$zz^{*}=(x+iy)(x-iy)=x^{2}+y^{2}+i(xy-yx)=x^{2}+y^{2}.\,\!$

Note that the complex conjugate of the complex conjugate is the original complex number and

$z^{*}z=(x+iy)(x-iy)=x^{2}+y^{2}+i(xy-yx)=x^{2}+y^{2}=zz^{*}.\,\!$

We also call this the modulus squared so that the modulus is

$|z|={\sqrt {(z^{*}z)}}={\sqrt {x^{2}+y^{2}}}.\,\!$

Note that the complex conjugate of a product is the product of the complex conjugates:

$(z\eta )^{*}=z^{*}\eta ^{*}.\,\!$

It is often useful to look at a graph for a complex number. The graph consists of an x-axis for the real part, and a y-axis for the complex part. This is shown in Fig. B.1. In this figure, it is easily seen that we can think of $z\,\!$ as a two-dimensional vector and that the magnitude (length) of the vector is the modulus of the complex number, $|z|={\sqrt {x^{2}+y^{2}}}\,\!$.

Figure B.1: A complex number in Cartesian coordinates.

Another useful way to represent this is with polar coordinates. We can do this by writing

$z=|z|(\cos \theta +i\sin \theta ).\,\!$

It turns out that

$e^{i\theta }=\cos \theta +i\sin \theta ,\,\!$

so we could also write

$z=|z|e^{i\theta }.\,\!$

It is often the case that people will write this as

$z=re^{i\theta },\,\!$

where $r={\sqrt {x^{2}+y^{2}}}\,\!$ as is usual for polar coordinates. Then, everything is just like polar coordinates, with the exception of the inclusion of the factor $i\,\!$. (See Fig. B.2.)

Figure B.2: A polar coordinate representation of a complex number.