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+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\ComplexXmpTitle}{Complex}
+\newcommand{\ComplexXmpNumber}{9.11}
+%
+% =====================================================================
+\begin{page}{ComplexXmpPage}{9.11 Complex}
+% =====================================================================
+\beginscroll
+%
+
+The \spadtype{Complex} constructor implements complex objects over a
+commutative ring \spad{R}.
+Typically, the ring \spad{R} is \spadtype{Integer}, \spadtype{Fraction
+Integer}, \spadtype{Float} or \spadtype{DoubleFloat}.
+\spad{R} can also be a symbolic type, like \spadtype{Polynomial Integer}.
+For more information about the numerical and graphical aspects of complex
+numbers, see \downlink{``\ugProblemNumericTitle''}{ugProblemNumericPage} in Section \ugProblemNumericNumber\ignore{ugProblemNumeric}.
+
+\xtc{
+Complex objects are created by the \spadfunFrom{complex}{Complex} operation.
+}{
+\spadpaste{a := complex(4/3,5/2) \bound{a}}
+}
+\xtc{
+}{
+\spadpaste{b := complex(4/3,-5/2) \bound{b}}
+}
+\xtc{
+The standard arithmetic operations are available.
+}{
+\spadpaste{a + b \free{a b}}
+}
+\xtc{
+}{
+\spadpaste{a - b \free{a b}}
+}
+\xtc{
+}{
+\spadpaste{a * b \free{a b}}
+}
+\xtc{
+If \spad{R} is a field, you can also divide the complex objects.
+}{
+\spadpaste{a / b \free{a b}\bound{adb}}
+}
+\xtc{
+Use a conversion (\downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} in Section \ugTypesConvertNumber\ignore{ugTypesConvert}) to view the last
+object as a fraction of complex integers.
+}{
+\spadpaste{\% :: Fraction Complex Integer \free{adb}}
+}
+\xtc{
+The predefined macro \spad{\%i} is defined to be \spad{complex(0,1)}.
+}{
+\spadpaste{3.4 + 6.7 * \%i}
+}
+\xtc{
+You can also compute the \spadfunFrom{conjugate}{Complex} and
+\spadfunFrom{norm}{Complex} of a complex number.
+}{
+\spadpaste{conjugate a \free{a}}
+}
+\xtc{
+}{
+\spadpaste{norm a \free{a}}
+}
+\xtc{
+The \spadfunFrom{real}{Complex} and \spadfunFrom{imag}{Complex} operations
+are provided to extract the real and imaginary parts, respectively.
+}{
+\spadpaste{real a \free{a}}
+}
+\xtc{
+}{
+\spadpaste{imag a \free{a}}
+}
+
+\xtc{
+The domain \spadtype{Complex Integer} is also called the Gaussian
+integers.
+%-% \HDindex{Gaussian integer}{ComplexXmpPage}{9.11}{Complex}
+If \spad{R} is the integers (or, more generally,
+a \spadtype{EuclideanDomain}), you can compute greatest common divisors.
+}{
+\spadpaste{gcd(13 - 13*\%i,31 + 27*\%i)}
+}
+\xtc{
+You can also compute least common multiples.
+}{
+\spadpaste{lcm(13 - 13*\%i,31 + 27*\%i)}
+}
+\xtc{
+You can \spadfunFrom{factor}{Complex} Gaussian integers.
+}{
+\spadpaste{factor(13 - 13*\%i)}
+}
+\xtc{
+}{
+\spadpaste{factor complex(2,0)}
+}
+\endscroll
+\autobuttons
+\end{page}
+%