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\newcommand{\HexadecimalExpansionXmpTitle}{HexadecimalExpansion}
\newcommand{\HexadecimalExpansionXmpNumber}{9.37}
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\begin{page}{HexadecimalExpansionXmpPage}{9.37 HexadecimalExpansion}
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All rationals have repeating hexadecimal expansions.
The operation \spadfunFrom{hex}{HexadecimalExpansion} returns these
expansions of type \spadtype{HexadecimalExpansion}.
Operations to access the individual numerals of a hexadecimal expansion can
be obtained by converting the value to \spadtype{RadixExpansion(16)}.
More examples of expansions are available in the
\downlink{`DecimalExpansion'}{DecimalExpansionXmpPage}\ignore{DecimalExpansion},
\downlink{`BinaryExpansion'}{BinaryExpansionXmpPage}\ignore{BinaryExpansion}, and
\downlink{`RadixExpansion'}{RadixExpansionXmpPage}\ignore{RadixExpansion}.

\showBlurb{HexadecimalExpansion}

\xtc{
This is a hexadecimal expansion of a rational number.
}{
\spadpaste{r := hex(22/7) \bound{r}}
}
\xtc{
Arithmetic is exact.
}{
\spadpaste{r + hex(6/7) \free{r}}
}
\xtc{
The period of the expansion can be short or long \ldots
}{
\spadpaste{[hex(1/i) for i in 350..354] }
}
\xtc{
or very long!
}{
\spadpaste{hex(1/1007) }
}
\xtc{
These numbers are bona fide algebraic objects.
}{
\spadpaste{p := hex(1/4)*x^2 + hex(2/3)*x + hex(4/9)  \bound{p}}
}
\xtc{
}{
\spadpaste{q := D(p, x) \free{p}\bound{q}}
}
\xtc{
}{
\spadpaste{g := gcd(p, q)            \free{p}\free{q}}
}
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