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

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

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

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

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

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

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

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

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.~?. 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, $|z|={\sqrt {x^{2}+y^{2}}}\,\!$.

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = |z|(\cos\theta +i \sin\theta). \,\!}

It turns out that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{i\theta} = \cos\theta + i \sin\theta, \,\!}

so we could also write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = |z|e^{i\theta}. \,\!}

It is often the case that people will write this as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = re^{i\theta}, \,\!}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\,\!} . (See Fig.~?.)