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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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* with significant contributions from:
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<TITLE>Complex numbers</TITLE>
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<!--End of Navigation Panel-->
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<H1><A NAME="SECTION001110000000000000000"></A>
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<A NAME="sect7.complex"></A>
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<BR>
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Complex numbers
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</H1>
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<P>
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Complex
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<A NAME="7761"></A>numbers are written as:
|
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<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
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<!-- MATH
|
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\begin{displaymath}
|
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Z = a + bi
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="72" HEIGHT="25" BORDER="0"
|
||||
SRC="img624.png"
|
||||
ALT="\begin{displaymath}
|
||||
Z = a + bi
|
||||
\end{displaymath}">
|
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</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
where <IMG
|
||||
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img4.png"
|
||||
ALT="$a$"> and <IMG
|
||||
WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$b$"> are real numbers and <IMG
|
||||
WIDTH="63" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img625.png"
|
||||
ALT="$i=\sqrt{-1}$">. (In this book we'll use
|
||||
the upper case Roman letters such as <IMG
|
||||
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img20.png"
|
||||
ALT="$Z$"> to denote complex numbers. Real
|
||||
numbers appear as lower case Roman or Greek letters, except for
|
||||
integer bounds, usually written as <IMG
|
||||
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img86.png"
|
||||
ALT="$M$"> or <IMG
|
||||
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img3.png"
|
||||
ALT="$N$">.) Since a complex
|
||||
number has two real components, we use a Cartesian plane (in place of a number
|
||||
line) to graph it, as shown in Figure <A HREF="#fig07.01">7.1</A>. The quantities <IMG
|
||||
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img4.png"
|
||||
ALT="$a$"> and
|
||||
<IMG
|
||||
WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$b$"> are called the
|
||||
<A NAME="7764"></A><I>real</I>
|
||||
and
|
||||
<A NAME="7766"></A><I>imaginary parts</I> of <IMG
|
||||
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img20.png"
|
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ALT="$Z$">, written as:
|
||||
<BR><P></P>
|
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<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
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a = \mathrm{re}(Z)
|
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\end{displaymath}
|
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-->
|
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|
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<IMG
|
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WIDTH="65" HEIGHT="28" BORDER="0"
|
||||
SRC="img626.png"
|
||||
ALT="\begin{displaymath}
|
||||
a = \mathrm{re}(Z)
|
||||
\end{displaymath}">
|
||||
</DIV>
|
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<BR CLEAR="ALL">
|
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<P></P>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
b = \mathrm{im}(Z)
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
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<IMG
|
||||
WIDTH="68" HEIGHT="28" BORDER="0"
|
||||
SRC="img627.png"
|
||||
ALT="\begin{displaymath}
|
||||
b = \mathrm{im}(Z)
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
|
||||
<P>
|
||||
|
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<DIV ALIGN="CENTER"><A NAME="fig07.01"></A><A NAME="7772"></A>
|
||||
<TABLE>
|
||||
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.1:</STRONG>
|
||||
A number, <IMG
|
||||
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img20.png"
|
||||
ALT="$Z$">, in the complex plane. The axes are for the real
|
||||
part <IMG
|
||||
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img4.png"
|
||||
ALT="$a$"> and the imaginary part <IMG
|
||||
WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$b$">.</CAPTION>
|
||||
<TR><TD><IMG
|
||||
WIDTH="254" HEIGHT="205" BORDER="0"
|
||||
SRC="img628.png"
|
||||
ALT="\begin{figure}\psfig{file=figs/fig07.01.ps}\end{figure}"></TD></TR>
|
||||
</TABLE>
|
||||
</DIV>
|
||||
|
||||
<P>
|
||||
If <IMG
|
||||
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img20.png"
|
||||
ALT="$Z$"> is a complex number, its
|
||||
<A NAME="7775"></A><A NAME="7776"></A><I>magnitude</I> (or <I>absolute value</I>),
|
||||
written as <IMG
|
||||
WIDTH="24" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img629.png"
|
||||
ALT="$\vert Z\vert$">, is just the distance in the plane from the origin to the
|
||||
point <IMG
|
||||
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img630.png"
|
||||
ALT="$(a,b)$">:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
|Z| = \sqrt {({a^2} + {b^2})}
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="118" HEIGHT="28" BORDER="0"
|
||||
SRC="img631.png"
|
||||
ALT="\begin{displaymath}
|
||||
\vert Z\vert = \sqrt {({a^2} + {b^2})}
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
and its
|
||||
<A NAME="7781"></A><I>argument</I>,
|
||||
written as <IMG
|
||||
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img632.png"
|
||||
ALT="$\angle(Z)$">,
|
||||
is the angle from the positive <IMG
|
||||
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img4.png"
|
||||
ALT="$a$"> axis to the point <IMG
|
||||
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img630.png"
|
||||
ALT="$(a,b)$">:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
\angle(Z) = \mathrm{arctan}
|
||||
{ \left ( {
|
||||
b \over a
|
||||
} \right ) }
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="130" HEIGHT="45" BORDER="0"
|
||||
SRC="img633.png"
|
||||
ALT="\begin{displaymath}
|
||||
\angle(Z) = \mathrm{arctan}
|
||||
{ \left ( {
|
||||
b \over a
|
||||
} \right ) }
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
If we know the magnitude and argument of a complex number (call them <IMG
|
||||
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img467.png"
|
||||
ALT="$r$"> and
|
||||
<IMG
|
||||
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img634.png"
|
||||
ALT="$\theta$">) we can reconstruct the real and imaginary parts:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
a = r \cos (\theta)
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="79" HEIGHT="28" BORDER="0"
|
||||
SRC="img635.png"
|
||||
ALT="\begin{displaymath}
|
||||
a = r \cos (\theta)
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
b = r \sin (\theta)
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="76" HEIGHT="28" BORDER="0"
|
||||
SRC="img636.png"
|
||||
ALT="\begin{displaymath}
|
||||
b = r \sin (\theta)
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
A complex number may be written in terms of its real and imaginary parts
|
||||
<IMG
|
||||
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img4.png"
|
||||
ALT="$a$"> and <IMG
|
||||
WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$b$">, as <IMG
|
||||
WIDTH="76" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img637.png"
|
||||
ALT="$Z=a+bi$"> (this is called
|
||||
<A NAME="7785"></A><I>rectangular form</I>), or alternatively in
|
||||
<A NAME="7787"></A><I>polar form</I>,
|
||||
in terms of <IMG
|
||||
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img467.png"
|
||||
ALT="$r$"> and <IMG
|
||||
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img634.png"
|
||||
ALT="$\theta$">:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
Z = r \cdot \left[ \cos(\theta) + i \sin(\theta) \right ]
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="167" HEIGHT="28" BORDER="0"
|
||||
SRC="img638.png"
|
||||
ALT="\begin{displaymath}
|
||||
Z = r \cdot \left[ \cos(\theta) + i \sin(\theta) \right ]
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
The rectangular and polar formulations are interchangeable; the equations
|
||||
above show how to compute <IMG
|
||||
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img4.png"
|
||||
ALT="$a$"> and <IMG
|
||||
WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img21.png"
|
||||
ALT="$b$"> from <IMG
|
||||
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img467.png"
|
||||
ALT="$r$"> and <IMG
|
||||
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img634.png"
|
||||
ALT="$\theta$"> and vice versa.
|
||||
|
||||
<P>
|
||||
The main reason we use complex numbers in electronic music is because they
|
||||
magically automate trigonometric calculations. We frequently have to add
|
||||
angles together in order to talk about the changing phase of an audio signal as
|
||||
time progresses (or as it is shifted in time, as in this chapter). It turns
|
||||
out that, if you multiply two complex numbers, the argument of the product is
|
||||
the sum of the arguments of the two factors. To see how this happens, we'll
|
||||
multiply two numbers <IMG
|
||||
WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img639.png"
|
||||
ALT="$Z_1$"> and <IMG
|
||||
WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img640.png"
|
||||
ALT="$Z_2$">, written in polar form:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
{Z_1} = {r_1} \cdot \left [ \cos({\theta_1}) + i \sin({\theta_1}) \right ]
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="194" HEIGHT="28" BORDER="0"
|
||||
SRC="img641.png"
|
||||
ALT="\begin{displaymath}
|
||||
{Z_1} = {r_1} \cdot \left [ \cos({\theta_1}) + i \sin({\theta_1}) \right ]
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
{Z_2} = {r_2} \cdot \left [ \cos({\theta_2}) + i \sin({\theta_2}) \right ]
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="193" HEIGHT="28" BORDER="0"
|
||||
SRC="img642.png"
|
||||
ALT="\begin{displaymath}
|
||||
{Z_2} = {r_2} \cdot \left [ \cos({\theta_2}) + i \sin({\theta_2}) \right ]
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
giving:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
{Z_1}{Z_2} = {r_1}{r_2} \cdot
|
||||
{ \left [ {
|
||||
\cos({\theta_1}) \cos({\theta_2}) -
|
||||
\sin({\theta_1}) \sin({\theta_2})
|
||||
} \right . } \, +
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="332" HEIGHT="28" BORDER="0"
|
||||
SRC="img643.png"
|
||||
ALT="\begin{displaymath}
|
||||
{Z_1}{Z_2} = {r_1}{r_2} \cdot
|
||||
{ \left [ {
|
||||
\cos({\theta_1...
|
||||
...a_2}) -
|
||||
\sin({\theta_1}) \sin({\theta_2})
|
||||
} \right . } \, +
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
{ \left . {
|
||||
+ i \left (
|
||||
\sin({\theta_1}) \cos({\theta_2}) +
|
||||
\cos({\theta_1}) \sin({\theta_2})
|
||||
\right )
|
||||
} \right ] }
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="249" HEIGHT="28" BORDER="0"
|
||||
SRC="img644.png"
|
||||
ALT="\begin{displaymath}
|
||||
{ \left . {
|
||||
+ i \left (
|
||||
\sin({\theta_1}) \cos({\theta_2}) +
|
||||
\cos({\theta_1}) \sin({\theta_2})
|
||||
\right )
|
||||
} \right ] }
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
Here the minus sign in front of the <!-- MATH
|
||||
$\sin({\theta_1}) \sin({\theta_2})$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="99" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img645.png"
|
||||
ALT="$\sin({\theta_1}) \sin({\theta_2})$"> term
|
||||
comes from multiplying <IMG
|
||||
WIDTH="9" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||||
SRC="img646.png"
|
||||
ALT="$i$"> by itself, which gives <IMG
|
||||
WIDTH="23" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img401.png"
|
||||
ALT="$-1$">. We can spot the
|
||||
cosine and sine summation formulas in the above expression, and so it simplifies
|
||||
to:
|
||||
<BR><P></P>
|
||||
<DIV ALIGN="CENTER">
|
||||
<!-- MATH
|
||||
\begin{displaymath}
|
||||
{Z_1}{Z_2} = {r_1}{r_2}
|
||||
\cdot \left[ \cos({\theta_1}+{\theta_2})
|
||||
+ i \sin({\theta_1}+{\theta_2}) \right ]
|
||||
\end{displaymath}
|
||||
-->
|
||||
|
||||
<IMG
|
||||
WIDTH="294" HEIGHT="28" BORDER="0"
|
||||
SRC="img647.png"
|
||||
ALT="\begin{displaymath}
|
||||
{Z_1}{Z_2} = {r_1}{r_2}
|
||||
\cdot \left[ \cos({\theta_1}+{\theta_2})
|
||||
+ i \sin({\theta_1}+{\theta_2}) \right ]
|
||||
\end{displaymath}">
|
||||
</DIV>
|
||||
<BR CLEAR="ALL">
|
||||
<P></P>
|
||||
By inspection, it follows that the product <IMG
|
||||
WIDTH="39" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img648.png"
|
||||
ALT="${Z_1}{Z_2}$">
|
||||
has magnitude
|
||||
<IMG
|
||||
WIDTH="32" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img649.png"
|
||||
ALT="${r_1}{r_2}$"> and argument <!-- MATH
|
||||
${\theta_1}+{\theta_2}$
|
||||
-->
|
||||
<IMG
|
||||
WIDTH="52" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||||
SRC="img650.png"
|
||||
ALT="${\theta_1}+{\theta_2}$">.
|
||||
|
||||
<P>
|
||||
We can use this property of complex numbers to add and subtract angles (by
|
||||
multiplying and dividing complex numbers with the appropriate arguments) and
|
||||
then to take the cosine and sine of the result by extracting the real and
|
||||
imaginary parts.
|
||||
|
||||
<P>
|
||||
<BR><HR>
|
||||
<!--Table of Child-Links-->
|
||||
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
||||
|
||||
<UL>
|
||||
<LI><A NAME="tex2html2132"
|
||||
HREF="node106.html">Complex sinusoids</A>
|
||||
</UL>
|
||||
<!--End of Table of Child-Links-->
|
||||
<HR>
|
||||
<!--Navigation Panel-->
|
||||
<A NAME="tex2html2130"
|
||||
HREF="node106.html">
|
||||
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
||||
SRC="file:/usr/local/share/lib/latex2html/icons/next.png"></A>
|
||||
<A NAME="tex2html2124"
|
||||
HREF="node104.html">
|
||||
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
||||
SRC="file:/usr/local/share/lib/latex2html/icons/up.png"></A>
|
||||
<A NAME="tex2html2118"
|
||||
HREF="node104.html">
|
||||
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
||||
SRC="file:/usr/local/share/lib/latex2html/icons/prev.png"></A>
|
||||
<A NAME="tex2html2126"
|
||||
HREF="node4.html">
|
||||
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
||||
SRC="file:/usr/local/share/lib/latex2html/icons/contents.png"></A>
|
||||
<A NAME="tex2html2128"
|
||||
HREF="node201.html">
|
||||
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
||||
SRC="file:/usr/local/share/lib/latex2html/icons/index.png"></A>
|
||||
<BR>
|
||||
<B> Next:</B> <A NAME="tex2html2131"
|
||||
HREF="node106.html">Complex sinusoids</A>
|
||||
<B> Up:</B> <A NAME="tex2html2125"
|
||||
HREF="node104.html">Time shifts and delays</A>
|
||||
<B> Previous:</B> <A NAME="tex2html2119"
|
||||
HREF="node104.html">Time shifts and delays</A>
|
||||
<B> <A NAME="tex2html2127"
|
||||
HREF="node4.html">Contents</A></B>
|
||||
<B> <A NAME="tex2html2129"
|
||||
HREF="node201.html">Index</A></B>
|
||||
<!--End of Navigation Panel-->
|
||||
<ADDRESS>
|
||||
Miller Puckette
|
||||
2006-12-30
|
||||
</ADDRESS>
|
||||
</BODY>
|
||||
</HTML>
|
||||
Reference in New Issue
Block a user