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,

$\eta =z.\,\!$

This means that

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

is true 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 imaginary 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 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 \eta\,\!} .

To add two complex numbers, add the real parts and the imaginary parts separately:

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 + \eta = (x+iy) + (c+id) = (x + c) + i(y + d). \,\!}

Complex numbers are multiplied like any other binomial expression:

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

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

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

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^* = 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

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

We also call this the modulus squared . The modulus is

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

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

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\eta)^* = z^* \eta^*. \,\!}

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 Fig. B.1. In this way, we can think of 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\,\!} as a two-dimensional vector with the magnitude (length) being equivalent to the modulus of the complex number, 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| = \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

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). \,\!}

Clearly if we wanted to rewrite this in terms of 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 x\,\!} and 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 y\,\!} , we find 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 x=|z|\cos\theta\,\!} and 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 y = |z|\sin\theta\,\!} It turns out that

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

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. So 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. B.2.)

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