Appendix B - Complex Numbers

Complex numbers arise naturally from an attempt to solve the equation

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

It's easy enough to write such an equation down, but how would you solve it? The answer 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 a = \pm \sqrt{-1} =\pm i. \,\!}

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

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, \,\!}

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

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 = z. \,\!}

This means 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 +iy = c+id, \,\!}

is true if and only if

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

We refer to $x\,\!$ as the real part 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\,\!} 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\,\!} as the imaginary part. Sometimes these are written as Re(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 Im($z\,\!$) respectively.

We may restate the equivalence condition 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=\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 $\eta \,\!$.

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

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

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

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

We also call this the modulus squared . 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, 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 $z\,\!$ as a two-dimensional vector with the magnitude (length) being equivalent to 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. 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.