diff --git a/example/glide_computer/include/geographic.h b/example/glide_computer/include/geographic.h
index eba165c3..e94f2894 100644
--- a/example/glide_computer/include/geographic.h
+++ b/example/glide_computer/include/geographic.h
@@ -62,29 +62,29 @@ std::basic_ostream<CharT, Traits>& operator<<(std::basic_ostream<CharT, Traits>&
 
 inline namespace literals {
 
-constexpr auto operator"" _N(long double v) { return latitude<long double>(latitude<long double>::rep(v)); }
-constexpr auto operator"" _S(long double v) { return latitude<long double>(latitude<long double>::rep(v)); }
-constexpr auto operator"" _E(long double v) { return longitude<long double>(longitude<long double>::rep(v)); }
-constexpr auto operator"" _W(long double v) { return longitude<long double>(longitude<long double>::rep(v)); }
-constexpr auto operator"" _N(unsigned long long v)
+constexpr latitude<long double> operator"" _N(long double v) { return v * mp_units::si::degree; }
+constexpr latitude<long double> operator"" _S(long double v) { return -v * mp_units::si::degree; }
+constexpr longitude<long double> operator"" _E(long double v) { return v * mp_units::si::degree; }
+constexpr longitude<long double> operator"" _W(long double v) { return -v * mp_units::si::degree; }
+constexpr latitude<std::int64_t> operator"" _N(unsigned long long v)
 {
   gsl_ExpectsAudit(std::in_range<std::int64_t>(v));
-  return latitude<std::int64_t>(latitude<std::int64_t>::rep(static_cast<std::int64_t>(v)));
+  return static_cast<std::int64_t>(v) * mp_units::si::degree;
 }
-constexpr auto operator"" _S(unsigned long long v)
+constexpr latitude<std::int64_t> operator"" _S(unsigned long long v)
 {
   gsl_ExpectsAudit(std::in_range<std::int64_t>(v));
-  return latitude<std::int64_t>(-latitude<std::int64_t>::rep(static_cast<std::int64_t>(v)));
+  return -static_cast<std::int64_t>(v) * mp_units::si::degree;
 }
-constexpr auto operator"" _E(unsigned long long v)
+constexpr longitude<std::int64_t> operator"" _E(unsigned long long v)
 {
   gsl_ExpectsAudit(std::in_range<std::int64_t>(v));
-  return longitude<std::int64_t>(longitude<std::int64_t>::rep(static_cast<std::int64_t>(v)));
+  return static_cast<std::int64_t>(v) * mp_units::si::degree;
 }
-constexpr auto operator"" _W(unsigned long long v)
+constexpr longitude<std::int64_t> operator"" _W(unsigned long long v)
 {
   gsl_ExpectsAudit(std::in_range<std::int64_t>(v));
-  return longitude<std::int64_t>(-longitude<std::int64_t>::rep(static_cast<std::int64_t>(v)));
+  return -static_cast<std::int64_t>(v) * mp_units::si::degree;
 }
 
 }  // namespace literals
diff --git a/example/linear_algebra.cpp b/example/linear_algebra.cpp
index ab33e043..9ec0a94b 100644
--- a/example/linear_algebra.cpp
+++ b/example/linear_algebra.cpp
@@ -89,8 +89,8 @@ void quantity_of_vector_cast()
 {
   std::cout << "\nquantity_of_vector_cast:\n";
 
-  quantity<isq::position_vector[m], vector<int>> v{vector<int>{1001, 1002, 1003}};
-  quantity<isq::position_vector[km], vector<int>> u{vector<int>{3, 2, 1}};
+  auto v = vector<int>{1001, 1002, 1003} * isq::position_vector[m];
+  auto u = vector<int>{3, 2, 1} * isq::position_vector[km];
 
   std::cout << "v = " << v << "\n";
   std::cout << "u = " << u << "\n";
@@ -103,7 +103,7 @@ void quantity_of_vector_multiply_by_scalar_value()
 {
   std::cout << "\nquantity_of_vector_multiply_scalar_value:\n";
 
-  quantity<isq::position_vector[m], vector<int>> v{vector<int>{1, 2, 3}};
+  auto v = vector<int>{1, 2, 3} * isq::position_vector[m];
 
   std::cout << "v = " << v << "\n";
 
@@ -114,7 +114,7 @@ void quantity_of_vector_divide_by_scalar_value()
 {
   std::cout << "\nquantity_of_vector_divide_scalar_value:\n";
 
-  quantity<isq::position_vector[m], vector<int>> v{vector<int>{2, 4, 6}};
+  auto v = vector<int>{2, 4, 6} * isq::position_vector[m];
 
   std::cout << "v = " << v << "\n";
 
@@ -131,9 +131,9 @@ void quantity_of_vector_add()
   // auto t = vector<int>{3, 2, 1} * isq::position_vector[km];
 
   // quantity<isq::distance[m], vector<int>> v{vector<int>{1, 2, 3}};  // should not compile
-  quantity<isq::position_vector[m], vector<int>> v{vector<int>{1, 2, 3}};
-  quantity<isq::position_vector[m], vector<int>> u{vector<int>{3, 2, 1}};
-  quantity<isq::position_vector[km], vector<int>> t{vector<int>{3, 2, 1}};
+  auto v = vector<int>{1, 2, 3} * isq::position_vector[m];
+  auto u = vector<int>{3, 2, 1} * isq::position_vector[m];
+  auto t = vector<int>{3, 2, 1} * isq::position_vector[km];
 
   std::cout << "v = " << v << "\n";
   std::cout << "u = " << u << "\n";
@@ -147,9 +147,9 @@ void quantity_of_vector_subtract()
 {
   std::cout << "\nquantity_of_vector_subtract:\n";
 
-  quantity<isq::position_vector[m], vector<int>> v{vector<int>{1, 2, 3}};
-  quantity<isq::position_vector[m], vector<int>> u{vector<int>{3, 2, 1}};
-  quantity<isq::position_vector[km], vector<int>> t{vector<int>{3, 2, 1}};
+  auto v = vector<int>{1, 2, 3} * isq::position_vector[m];
+  auto u = vector<int>{3, 2, 1} * isq::position_vector[m];
+  auto t = vector<int>{3, 2, 1} * isq::position_vector[km];
 
   std::cout << "v = " << v << "\n";
   std::cout << "u = " << u << "\n";
@@ -163,7 +163,7 @@ void quantity_of_vector_divide_by_scalar_quantity()
 {
   std::cout << "\nquantity_of_vector_divide_scalar:\n";
 
-  quantity<isq::position_vector[km], vector<int>> pos{vector<int>{30, 20, 10}};
+  auto pos = vector<int>{30, 20, 10} * isq::position_vector[km];
   auto dur = 2 * isq::duration[h];
   quantity<isq::velocity[km / h], vector<int>> v = pos / dur;
 
diff --git a/src/core/include/mp_units/bits/sudo_cast.h b/src/core/include/mp_units/bits/sudo_cast.h
index d16041b4..5e01a99b 100644
--- a/src/core/include/mp_units/bits/sudo_cast.h
+++ b/src/core/include/mp_units/bits/sudo_cast.h
@@ -51,10 +51,10 @@ template<Quantity To, auto R, typename Rep>
 {
   if constexpr (get_unit(R) == To::unit) {
     // no scaling of the number needed
-    return To(static_cast<TYPENAME To::rep>(q.number()));  // this is the only (and recommended) way to do
-                                                           // a truncating conversion on a number, so we are
-                                                           // using static_cast to suppress all the compiler
-                                                           // warnings on conversions
+    return static_cast<TYPENAME To::rep>(q.number()) * To::reference;  // this is the only (and recommended) way to do
+                                                                       // a truncating conversion on a number, so we are
+                                                                       // using static_cast to suppress all the compiler
+                                                                       // warnings on conversions
   } else {
     // scale the number
     using rep_type = decltype([] {
@@ -85,7 +85,8 @@ template<Quantity To, auto R, typename Rep>
     constexpr Magnitude auto irr = c_mag * (den / num);
 
     constexpr auto val = [](Magnitude auto m) { return get_value<multiplier_type>(m); };
-    return To(static_cast<TYPENAME To::rep>(static_cast<rep_type>(q.number()) * val(num) / val(den) * val(irr)));
+    return static_cast<TYPENAME To::rep>(static_cast<rep_type>(q.number()) * val(num) / val(den) * val(irr)) *
+           To::reference;
   }
 }
 
diff --git a/src/core/include/mp_units/quantity.h b/src/core/include/mp_units/quantity.h
index 5e0dfe8d..445bb464 100644
--- a/src/core/include/mp_units/quantity.h
+++ b/src/core/include/mp_units/quantity.h
@@ -121,12 +121,6 @@ public:
   quantity(const quantity&) = default;
   quantity(quantity&&) = default;
 
-  template<typename Value>
-    requires detail::RepSafeConstructibleFrom<rep, Value>
-  constexpr explicit(!detail::QuantityOne<quantity>) quantity(Value&& v) : number_(std::forward<Value>(v))
-  {
-  }
-
   template<detail::QuantityConvertibleTo<quantity> Q>
   constexpr explicit(false) quantity(const Q& q) : number_(detail::sudo_cast<quantity>(q).number())
   {
@@ -439,6 +433,17 @@ public:
 #endif
 
   [[nodiscard]] friend constexpr bool operator==(const quantity& lhs, const quantity& rhs) = default;
+
+private:
+  template<Reference R2, typename Rep2>
+    requires RepresentationOf<std::remove_cvref_t<Rep2>, get_quantity_spec(R2{}).character>
+  friend constexpr quantity<R2{}, std::remove_cvref_t<Rep2>> operator*(Rep2&& lhs, R2);
+
+  template<typename Value>
+    requires detail::RepSafeConstructibleFrom<rep, Value>
+  constexpr explicit quantity(Value&& v) : number_(std::forward<Value>(v))
+  {
+  }
 };
 
 // CTAD
@@ -465,7 +470,7 @@ template<Quantity Q1, Quantity Q2>
 {
   constexpr auto ref = common_reference(Q1::reference, Q2::reference);
   using ret = quantity<ref, decltype(lhs.number() + rhs.number())>;
-  return ret(ret(lhs).number() + ret(rhs).number());
+  return (ret(lhs).number() + ret(rhs).number()) * ref;
 }
 
 template<Quantity Q1, Quantity Q2>
@@ -476,7 +481,7 @@ template<Quantity Q1, Quantity Q2>
 {
   constexpr auto ref = common_reference(Q1::reference, Q2::reference);
   using ret = quantity<ref, decltype(lhs.number() - rhs.number())>;
-  return ret(ret(lhs).number() - ret(rhs).number());
+  return (ret(lhs).number() - ret(rhs).number()) * ref;
 }
 
 template<Quantity Q1, Quantity Q2>
diff --git a/src/utility/include/mp_units/math.h b/src/utility/include/mp_units/math.h
index aeacf88a..4f671fd9 100644
--- a/src/utility/include/mp_units/math.h
+++ b/src/utility/include/mp_units/math.h
@@ -61,8 +61,8 @@ template<std::intmax_t Num, std::intmax_t Den = 1, Quantity Q>
     return q;
   } else {
     using std::pow;
-    return quantity<reference<pow<Num, Den>(Q::quantity_spec), pow<Num, Den>(Q::unit)>{}, rep>(
-      static_cast<rep>(pow(q.number(), static_cast<double>(Num) / static_cast<double>(Den))));
+    return static_cast<rep>(pow(q.number(), static_cast<double>(Num) / static_cast<double>(Den))) *
+           reference<pow<Num, Den>(Q::quantity_spec), pow<Num, Den>(Q::unit)>{};
   }
 }
 
@@ -80,8 +80,7 @@ template<Quantity Q>
 {
   using rep = TYPENAME Q::rep;
   using std::sqrt;
-  return quantity<reference<pow<1, 2>(Q::quantity_spec), pow<1, 2>(Q::unit)>{}, rep>(
-    static_cast<rep>(sqrt(q.number())));
+  return static_cast<rep>(sqrt(q.number())) * reference<pow<1, 2>(Q::quantity_spec), pow<1, 2>(Q::unit)>{};
 }
 
 /**
@@ -98,8 +97,7 @@ template<Quantity Q>
 {
   using rep = TYPENAME Q::rep;
   using std::cbrt;
-  return quantity<reference<pow<1, 3>(Q::quantity_spec), pow<1, 3>(Q::unit)>{}, rep>(
-    static_cast<rep>(cbrt(q.number())));
+  return static_cast<rep>(cbrt(q.number())) * reference<pow<1, 3>(Q::quantity_spec), pow<1, 3>(Q::unit)>{};
 }
 
 /**
@@ -171,10 +169,9 @@ template<Unit auto To, auto R, typename Rep>
   if constexpr (treat_as_floating_point<Rep>) {
     using std::floor;
     if constexpr (To == get_unit(R)) {
-      return quantity<reference<get_quantity_spec(R), To>{}, Rep>(floor(q.number()));
+      return floor(q.number()) * reference<get_quantity_spec(R), To>{};
     } else {
-      return handle_signed_results(
-        quantity<reference<get_quantity_spec(R), To>{}, Rep>(floor(value_cast<To>(q).number())));
+      return handle_signed_results(floor(value_cast<To>(q).number()) * reference<get_quantity_spec(R), To>{});
     }
   } else {
     if constexpr (To == get_unit(R)) {
@@ -208,10 +205,9 @@ template<Unit auto To, auto R, typename Rep>
   if constexpr (treat_as_floating_point<Rep>) {
     using std::ceil;
     if constexpr (To == get_unit(R)) {
-      return quantity<reference<get_quantity_spec(R), To>{}, Rep>(ceil(q.number()));
+      return ceil(q.number()) * reference<get_quantity_spec(R), To>{};
     } else {
-      return handle_signed_results(
-        quantity<reference<get_quantity_spec(R), To>{}, Rep>(ceil(value_cast<To>(q).number())));
+      return handle_signed_results(ceil(value_cast<To>(q).number()) * reference<get_quantity_spec(R), To>{});
     }
   } else {
     if constexpr (To == get_unit(R)) {
@@ -242,7 +238,7 @@ template<Unit auto To, auto R, typename Rep>
   if constexpr (To == get_unit(R)) {
     if constexpr (treat_as_floating_point<Rep>) {
       using std::round;
-      return quantity<reference<get_quantity_spec(R), To>{}, Rep>(round(q.number()));
+      return round(q.number()) * reference<get_quantity_spec(R), To>{};
     } else {
       return value_cast<To>(q);
     }
@@ -306,7 +302,7 @@ template<QuantityOf<angular_measure> Q>
   requires requires { sin(q.number()); } || requires { std::sin(q.number()); }
 {
   using std::sin;
-  return quantity{sin(q[si::radian].number())};
+  return sin(q[si::radian].number()) * one;
 }
 
 template<QuantityOf<angular_measure> Q>
@@ -315,7 +311,7 @@ template<QuantityOf<angular_measure> Q>
   requires requires { cos(q.number()); } || requires { std::cos(q.number()); }
 {
   using std::cos;
-  return quantity{cos(q[si::radian].number())};
+  return cos(q[si::radian].number()) * one;
 }
 
 template<QuantityOf<angular_measure> Q>
@@ -324,7 +320,7 @@ template<QuantityOf<angular_measure> Q>
   requires requires { tan(q.number()); } || requires { std::tan(q.number()); }
 {
   using std::tan;
-  return quantity{tan(q[si::radian].number())};
+  return tan(q[si::radian].number()) * one;
 }
 
 template<QuantityOf<dimension_one> Q>