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\newcommand{\BinaryExpansionXmpTitle}{BinaryExpansion}
\newcommand{\BinaryExpansionXmpNumber}{9.4}
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\begin{page}{BinaryExpansionXmpPage}{9.4 BinaryExpansion}
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All rational numbers have repeating binary expansions.
Operations to access the individual bits of a binary expansion can
be obtained by converting the value to \spadtype{RadixExpansion(2)}.
More examples of expansions are available in
\downlink{`DecimalExpansion'}{DecimalExpansionXmpPage}\ignore{DecimalExpansion},
\downlink{`HexadecimalExpansion'}{HexadecimalExpansionXmpPage}\ignore{HexadecimalExpansion}, and
\downlink{`RadixExpansion'}{RadixExpansionXmpPage}\ignore{RadixExpansion}.

\xtc{
The expansion (of type \spadtype{BinaryExpansion}) of a rational number
is returned by the \spadfunFrom{binary}{BinaryExpansion} operation.
}{
\spadpaste{r := binary(22/7) \bound{r}}
}
\xtc{
Arithmetic is exact.
}{
\spadpaste{r + binary(6/7) \free{r}}
}
\xtc{
The period of the expansion can be short or long \ldots
}{
\spadpaste{[binary(1/i) for i in 102..106] }
}
\xtc{
or very long.
}{
\spadpaste{binary(1/1007) }
}
\xtc{
These numbers are bona fide algebraic objects.
}{
\spadpaste{p := binary(1/4)*x^2 + binary(2/3)*x + binary(4/9)  \bound{p}}
}
\xtc{
}{
\spadpaste{q := D(p, x) \free{p}\bound{q}}
}
\xtc{
}{
\spadpaste{g := gcd(p, q)            \free{p q}\bound{g}}
}
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