Unary Plus and Minus Operators

template<typename T> octonion<T> operator + (octonion<T> const & o);

This unary operator simply returns o.

template<typename T> octonion<T> operator - (octonion<T> const & o);

This unary operator returns the opposite of o.

Binary Addition Operators

template<typename T> octonion<T> operator + (T const & lhs, octonion<T> const & rhs);
template<typename T> octonion<T> operator + (octonion<T> const & lhs, T const & rhs);
template<typename T> octonion<T> operator + (::std::complex<T> const & lhs, octonion<T> const & rhs);
template<typename T> octonion<T> operator + (octonion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T> octonion<T> operator + (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
template<typename T> octonion<T> operator + (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
template<typename T> octonion<T> operator + (octonion<T> const & lhs, octonion<T> const & rhs);

These operators return octonion<T>(lhs) += rhs.

Binary Subtraction Operators

template<typename T> octonion<T> operator - (T const & lhs, octonion<T> const & rhs);
template<typename T> octonion<T> operator - (octonion<T> const & lhs, T const & rhs);
template<typename T> octonion<T> operator - (::std::complex<T> const & lhs, octonion<T> const & rhs);
template<typename T> octonion<T> operator - (octonion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T> octonion<T> operator - (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
template<typename T> octonion<T> operator - (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
template<typename T> octonion<T> operator - (octonion<T> const & lhs, octonion<T> const & rhs);

These operators return octonion<T>(lhs) -= rhs.

Binary Multiplication Operators

template<typename T> octonion<T> operator * (T const & lhs, octonion<T> const & rhs);
template<typename T> octonion<T> operator * (octonion<T> const & lhs, T const & rhs);
template<typename T> octonion<T> operator * (::std::complex<T> const & lhs, octonion<T> const & rhs);
template<typename T> octonion<T> operator * (octonion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T> octonion<T> operator * (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
template<typename T> octonion<T> operator * (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
template<typename T> octonion<T> operator * (octonion<T> const & lhs, octonion<T> const & rhs);

These operators return octonion<T>(lhs) *= rhs.

Binary Division Operators

template<typename T> octonion<T> operator / (T const & lhs, octonion<T> const & rhs);
template<typename T> octonion<T> operator / (octonion<T> const & lhs, T const & rhs);
template<typename T> octonion<T> operator / (::std::complex<T> const & lhs, octonion<T> const & rhs);
template<typename T> octonion<T> operator / (octonion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T> octonion<T> operator / (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
template<typename T> octonion<T> operator / (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
template<typename T> octonion<T> operator / (octonion<T> const & lhs, octonion<T> const & rhs);

These operators return octonion<T>(lhs) /= rhs. It is of course still an error to divide by zero...

Binary Equality Operators

template<typename T> bool operator == (T const & lhs, octonion<T> const & rhs);
template<typename T> bool operator == (octonion<T> const & lhs, T const & rhs);
template<typename T> bool operator == (::std::complex<T> const & lhs, octonion<T> const & rhs);
template<typename T> bool operator == (octonion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T> bool operator == (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
template<typename T> bool operator == (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
template<typename T> bool operator == (octonion<T> const & lhs, octonion<T> const & rhs);

These return true if and only if the four components of octonion<T>(lhs) are equal to their counterparts in octonion<T>(rhs). As with any floating-type entity, this is essentially meaningless.

Binary Inequality Operators

template<typename T> bool operator != (T const & lhs, octonion<T> const & rhs);
template<typename T> bool operator != (octonion<T> const & lhs, T const & rhs);
template<typename T> bool operator != (::std::complex<T> const & lhs, octonion<T> const & rhs);
template<typename T> bool operator != (octonion<T> const & lhs, ::std::complex<T> const & rhs);
template<typename T> bool operator != (::boost::math::quaternion<T> const & lhs, octonion<T> const & rhs);
template<typename T> bool operator != (octonion<T> const & lhs, ::boost::math::quaternion<T> const & rhs);
template<typename T> bool operator != (octonion<T> const & lhs, octonion<T> const & rhs);

These return true if and only if octonion<T>(lhs) == octonion<T>(rhs) is false. As with any floating-type entity, this is essentially meaningless.

Stream Extractor

template<typename T, typename charT, class traits>
::std::basic_istream<charT,traits> & operator >> (::std::basic_istream<charT,traits> & is, octonion<T> & o);

Extracts an octonion o. We accept any format which seems reasonable. However, since this leads to a great many ambiguities, decisions were made to lift these. In case of doubt, stick to lists of reals.

The input values must be convertible to T. If bad input is encountered, calls is.setstate(ios::failbit) (which may throw ios::failure (27.4.5.3)).

Returns is.

Stream Inserter

template<typename T, typename charT, class traits>
::std::basic_ostream<charT,traits> & operator << (::std::basic_ostream<charT,traits> & os, octonion<T> const & o);

Inserts the octonion o onto the stream os as if it were implemented as follows:

template<typename T, typename charT, class traits>
::std::basic_ostream<charT,traits> & operator << ( ::std::basic_ostream<charT,traits> & os,
octonion<T> const & o)
{
   ::std::basic_ostringstream<charT,traits> s;

   s.flags(os.flags());
   s.imbue(os.getloc());
   s.precision(os.precision());

   s << '(' << o.R_component_1() << ','
      << o.R_component_2() << ','
      << o.R_component_3() << ','
      << o.R_component_4() << ','
      << o.R_component_5() << ','
      << o.R_component_6() << ','
      << o.R_component_7() << ','
      << o.R_component_8() << ')';

   return os << s.str();
}