add type_parameter.md

docs/make.jl

@@ -19,6 +19,7 @@ makedocs(;
         "APIs" => "api.md",
         "Examples" => [
             "examples/basics.md",
+            "examples/type_parameter.md",
             "examples/rotations.md",
             "examples/dual_quaternions.md"
         ],

docs/src/examples/basics.md

@@ -65,67 +65,6 @@ Quaternion(Int)  # Similar to `Complex(Int)`.
 quat(Int)  # Similar to `complex(Int)`.
 ```
 
-## The type parameter `T` in `Quaternion{T}`
-
-The type parameter `T <: Real` in `Quaternion{T}` represents the type of real and imaginary parts of a quaternion.
-
-### Lipschitz quaternions
-By using this type parameter, some special quaternions such as [**Lipschitz quaternions**](https://en.wikipedia.org/wiki/Hurwitz_quaternion) ``L`` can be represented.
-
-```math
-L = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z}\right\}
-```
-
-```@setup LipschitzHurwitz
-using Quaternions
-```
-
-```@repl LipschitzHurwitz
-q1 = Quaternion{Int}(1,2,3,4)
-q2 = Quaternion{Int}(5,6,7,8)
-islipschitz(q::Quaternion) = isinteger(q.s) & isinteger(q.v1) & isinteger(q.v2) & isinteger(q.v3)
-islipschitz(q1)
-islipschitz(q2)
-islipschitz(q1 + q2)
-islipschitz(q1 * q2)
-islipschitz(q1 / q2)  # Division is not defined on L.
-q1 * q2 == q2 * q1  # non-commutative
-```
-
-### Hurwitz quaternions
-If all coefficients of a quaternion are integers or half-integers, the quaternion is called a [**Hurwitz quaternion**](https://en.wikipedia.org/wiki/Hurwitz_quaternion).
-The set of Hurwitz quaternions is defined by
-
-```math
-H = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z} \ \text{or} \ a,b,c,d \in \mathbb{Z} + \tfrac{1}{2}\right\}.
-```
-
-Hurwitz quaternions can be implemented with [HalfIntegers.jl](https://github.com/sostock/HalfIntegers.jl) package.
-
-```@repl LipschitzHurwitz
-using HalfIntegers
-q1 = Quaternion{HalfInt}(1, 2, 3, 4)
-q2 = Quaternion{HalfInt}(5.5, 6.5, 7.5, 8.5)
-q3 = Quaternion{HalfInt}(1, 2, 3, 4.5)  # not Hurwitz quaternion
-ishurwitz(q::Quaternion) = (isinteger(q.s) & isinteger(q.v1) & isinteger(q.v2) & isinteger(q.v3)) | (ishalfinteger(q.s) & ishalfinteger(q.v1) & ishalfinteger(q.v2) & ishalfinteger(q.v3))
-ishurwitz(q1)
-ishurwitz(q2)
-ishurwitz(q3)
-ishurwitz(q1 + q2)
-ishurwitz(q1 * q2)
-ishurwitz(q1 / q2)  # Division is not defined on H.
-q1 * q2 == q2 * q1  # non-commucative
-abs2(q1)  # Squared norm is always an integer.
-abs2(q2)  # Squared norm is always an integer.
-abs2(q3)  # Squared norm is not an integer because `q3` is not Hurwitz quaternion.
-```
-
-### Biquaternions
-If all coefficients of a quaternion are complex numbers, the quaternion is called a [**Biquaternion**](https://en.wikipedia.org/wiki/Biquaternion).
-However, the type parameter `T` is restricted to `<:Real`, so biquaternions are not supported in this package.
-Note that `Base.Complex` has the same type parameter restriction, and [bicomplex numbers](https://en.wikipedia.org/wiki/Bicomplex_number) are not supported in Base.
-See [issue#79](https://github.com/JuliaGeometry/Quaternions.jl/issues/79) for more discussion.
-
 ## Compatibility with `Complex`
 There is no natural embedding ``\mathbb{C}\to\mathbb{H}``.
 Thus, `quat(w,x,0,0)` is not equal to `complex(w,x)`, i.e.

docs/src/examples/type_parameter.md

@@ -0,0 +1,60 @@
+# The type parameter `T` in `Quaternion{T}`
+
+The type parameter `T <: Real` in `Quaternion{T}` represents the type of real and imaginary parts of a quaternion.
+
+## Lipschitz quaternions
+By using this type parameter, some special quaternions such as [**Lipschitz quaternions**](https://en.wikipedia.org/wiki/Hurwitz_quaternion) ``L`` can be represented.
+
+```math
+L = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z}\right\}
+```
+
+```@setup LipschitzHurwitz
+using Quaternions
+```
+
+```@repl LipschitzHurwitz
+q1 = Quaternion{Int}(1,2,3,4)
+q2 = Quaternion{Int}(5,6,7,8)
+islipschitz(q::Quaternion) = isinteger(q.s) & isinteger(q.v1) & isinteger(q.v2) & isinteger(q.v3)
+islipschitz(q1)
+islipschitz(q2)
+islipschitz(q1 + q2)
+islipschitz(q1 * q2)
+islipschitz(q1 / q2)  # Division is not defined on L.
+q1 * q2 == q2 * q1  # non-commutative
+```
+
+## Hurwitz quaternions
+If all coefficients of a quaternion are integers or half-integers, the quaternion is called a [**Hurwitz quaternion**](https://en.wikipedia.org/wiki/Hurwitz_quaternion).
+The set of Hurwitz quaternions is defined by
+
+```math
+H = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z} \ \text{or} \ a,b,c,d \in \mathbb{Z} + \tfrac{1}{2}\right\}.
+```
+
+Hurwitz quaternions can be implemented with [HalfIntegers.jl](https://github.com/sostock/HalfIntegers.jl) package.
+
+```@repl LipschitzHurwitz
+using HalfIntegers
+q1 = Quaternion{HalfInt}(1, 2, 3, 4)
+q2 = Quaternion{HalfInt}(5.5, 6.5, 7.5, 8.5)
+q3 = Quaternion{HalfInt}(1, 2, 3, 4.5)  # not Hurwitz quaternion
+ishurwitz(q::Quaternion) = (isinteger(q.s) & isinteger(q.v1) & isinteger(q.v2) & isinteger(q.v3)) | (ishalfinteger(q.s) & ishalfinteger(q.v1) & ishalfinteger(q.v2) & ishalfinteger(q.v3))
+ishurwitz(q1)
+ishurwitz(q2)
+ishurwitz(q3)
+ishurwitz(q1 + q2)
+ishurwitz(q1 * q2)
+ishurwitz(q1 / q2)  # Division is not defined on H.
+q1 * q2 == q2 * q1  # non-commucative
+abs2(q1)  # Squared norm is always an integer.
+abs2(q2)  # Squared norm is always an integer.
+abs2(q3)  # Squared norm is not an integer because `q3` is not Hurwitz quaternion.
+```
+
+## Biquaternions
+If all coefficients of a quaternion are complex numbers, the quaternion is called a [**Biquaternion**](https://en.wikipedia.org/wiki/Biquaternion).
+However, the type parameter `T` is restricted to `<:Real`, so biquaternions are not supported in this package.
+Note that `Base.Complex` has the same type parameter restriction, and [bicomplex numbers](https://en.wikipedia.org/wiki/Bicomplex_number) are not supported in Base.
+See [issue#79](https://github.com/JuliaGeometry/Quaternions.jl/issues/79) for more discussion.