# Quaternion type

Quaternion type
You are encouraged to solve this task according to the task description, using any language you may know.

Quaternions   are an extension of the idea of   complex numbers.

A complex number has a real and complex part,   sometimes written as    a + bi,
where    a    and    b    stand for real numbers, and    i    stands for the square root of minus 1.

An example of a complex number might be    -3 + 2i,
where the real part,    a    is    -3.0    and the complex part,    b    is    +2.0.

A quaternion has one real part and three imaginary parts,    i,     j,    and    k.

A quaternion might be written as    a + bi + cj + dk.

In the quaternion numbering system:

•    i∙i = j∙j = k∙k = i∙j∙k = -1,        or more simply,
•    ii  = jj  = kk  = ijk   = -1.

The order of multiplication is important, as, in general, for two quaternions:

 q1    and    q2:       q1q2 ≠ q2q1.

An example of a quaternion might be    1 +2i +3j +4k

There is a list form of notation where just the numbers are shown and the imaginary multipliers   i,     j,    and    k    are assumed by position.

So the example above would be written as    (1, 2, 3, 4)

Given the three quaternions and their components:

   q  = (1, 2, 3, 4) = (a,  b,  c,  d)
q1 = (2, 3, 4, 5) = (a1, b1, c1, d1)
q2 = (3, 4, 5, 6) = (a2, b2, c2, d2)


And a wholly real number    r = 7.

Create functions   (or classes)   to perform simple maths with quaternions including computing:

1. The norm of a quaternion:
${\displaystyle ={\sqrt {a^{2}+b^{2}+c^{2}+d^{2}}}}$
2. The negative of a quaternion:
 = (-a, -b, -c, -d)
3. The conjugate of a quaternion:
 = ( a, -b, -c, -d)
4. Addition of a real number    r    and a quaternion    q:
 r + q = q + r = (a+r, b, c, d)
 q1 + q2 = (a1+a2, b1+b2, c1+c2, d1+d2)
6. Multiplication of a real number and a quaternion:
 qr = rq = (ar, br, cr, dr)
7. Multiplication of two quaternions    q1    and   q2    is given by:
 ( a1a2 − b1b2 − c1c2 − d1d2,
   a1b2 + b1a2 + c1d2 − d1c2,
   a1c2 − b1d2 + c1a2 + d1b2,
   a1d2 + b1c2 − c1b2 + d1a2 )
8. Show that, for the two quaternions    q1    and    q2: q1q2 ≠ q2q1

If a language has built-in support for quaternions, then use it.

C.f.
•   Vector products
•   On Quaternions;   or on a new System of Imaginaries in Algebra.   By Sir William Rowan Hamilton LL.D, P.R.I.A., F.R.A.S., Hon. M. R. Soc. Ed. and Dub., Hon. or Corr. M. of the Royal or Imperial Academies of St. Petersburgh, Berlin, Turin and Paris, Member of the American Academy of Arts and Sciences, and of other Scientific Societies at Home and Abroad, Andrews' Prof. of Astronomy in the University of Dublin, and Royal Astronomer of Ireland.

The package specification (works with any floating-point type):

generic   type Real is digits <>;package Quaternions is   type Quaternion is record      A, B, C, D : Real;   end record;   function "abs" (Left : Quaternion) return Real;   function Conj (Left : Quaternion) return Quaternion;   function "-" (Left : Quaternion) return Quaternion;   function "+" (Left, Right : Quaternion) return Quaternion;   function "-" (Left, Right : Quaternion) return Quaternion;   function "*" (Left : Quaternion; Right : Real) return Quaternion;   function "*" (Left : Real; Right : Quaternion) return Quaternion;   function "*" (Left, Right : Quaternion) return Quaternion;   function Image (Left : Quaternion) return String;end Quaternions;

The package implementation:

with Ada.Numerics.Generic_Elementary_Functions;package body Quaternions is   package Elementary_Functions is      new Ada.Numerics.Generic_Elementary_Functions (Real);   use Elementary_Functions;   function "abs" (Left : Quaternion) return Real is   begin      return Sqrt (Left.A**2 + Left.B**2 + Left.C**2 + Left.D**2);   end "abs";   function Conj (Left : Quaternion) return Quaternion is   begin      return (A => Left.A, B => -Left.B, C => -Left.C, D => -Left.D);   end Conj;   function "-" (Left : Quaternion) return Quaternion is   begin      return (A => -Left.A, B => -Left.B, C => -Left.C, D => -Left.D);   end "-";   function "+" (Left, Right : Quaternion) return Quaternion is   begin      return      (  A => Left.A + Right.A, B => Left.B + Right.B,          C => Left.C + Right.C, D => Left.D + Right.D      );   end "+";   function "-" (Left, Right : Quaternion) return Quaternion is   begin      return      (  A => Left.A - Right.A, B => Left.B - Right.B,          C => Left.C - Right.C, D => Left.D - Right.D      );   end "-";   function "*" (Left : Quaternion; Right : Real) return Quaternion is   begin      return      (  A => Left.A * Right, B => Left.B * Right,          C => Left.C * Right, D => Left.D * Right      );   end "*";   function "*" (Left : Real; Right : Quaternion) return Quaternion is   begin      return Right * Left;   end "*";   function "*" (Left, Right : Quaternion) return Quaternion is   begin      return      (  A => Left.A * Right.A - Left.B * Right.B - Left.C * Right.C - Left.D * Right.D,         B => Left.A * Right.B + Left.B * Right.A + Left.C * Right.D - Left.D * Right.C,         C => Left.A * Right.C - Left.B * Right.D + Left.C * Right.A + Left.D * Right.B,         D => Left.A * Right.D + Left.B * Right.C - Left.C * Right.B + Left.D * Right.A      );   end "*";   function Image (Left : Quaternion) return String is   begin      return Real'Image (Left.A) & " +"  &             Real'Image (Left.B) & "i +" &             Real'Image (Left.C) & "j +" &             Real'Image (Left.D) & "k";   end Image;end Quaternions;

Test program:

with Ada.Text_IO;  use Ada.Text_IO;with Quaternions;procedure Test_Quaternion is   package Float_Quaternion is new Quaternions (Float);   use Float_Quaternion;   q  : Quaternion := (1.0, 2.0, 3.0, 4.0);   q1 : Quaternion := (2.0, 3.0, 4.0, 5.0);   q2 : Quaternion := (3.0, 4.0, 5.0, 6.0);   r  : Float      := 7.0;begin   Put_Line ("q = "       & Image (q));   Put_Line ("q1 = "      & Image (q1));   Put_Line ("q2 = "      & Image (q2));   Put_Line ("r ="        & Float'Image (r));   Put_Line ("abs q ="    & Float'Image (abs q));   Put_Line ("abs q1 ="   & Float' Image (abs q1));   Put_Line ("abs q2 ="   & Float' Image (abs q2));   Put_Line ("-q = "      & Image (-q));   Put_Line ("conj q = "  & Image (Conj (q)));   Put_Line ("q1 + q2 = " & Image (q1 + q2));   Put_Line ("q2 + q1 = " & Image (q2 + q1));   Put_Line ("q * r = "   & Image (q * r));   Put_Line ("r * q = "   & Image (r * q));   Put_Line ("q1 * q2 = " & Image (q1 * q2));   Put_Line ("q2 * q1 = " & Image (q2 * q1));end Test_Quaternion;
Output:
q =  1.00000E+00 + 2.00000E+00i + 3.00000E+00j + 4.00000E+00k
q1 =  2.00000E+00 + 3.00000E+00i + 4.00000E+00j + 5.00000E+00k
q2 =  3.00000E+00 + 4.00000E+00i + 5.00000E+00j + 6.00000E+00k
r = 7.00000E+00
abs q = 5.47723E+00
abs q1 = 7.34847E+00
abs q2 = 9.27362E+00
-q = -1.00000E+00 +-2.00000E+00i +-3.00000E+00j +-4.00000E+00k
conj q =  1.00000E+00 +-2.00000E+00i +-3.00000E+00j +-4.00000E+00k
q1 + q2 =  5.00000E+00 + 7.00000E+00i + 9.00000E+00j + 1.10000E+01k
q2 + q1 =  5.00000E+00 + 7.00000E+00i + 9.00000E+00j + 1.10000E+01k
q * r =  7.00000E+00 + 1.40000E+01i + 2.10000E+01j + 2.80000E+01k
r * q =  7.00000E+00 + 1.40000E+01i + 2.10000E+01j + 2.80000E+01k
q1 * q2 = -5.60000E+01 + 1.60000E+01i + 2.40000E+01j + 2.60000E+01k
q2 * q1 = -5.60000E+01 + 1.80000E+01i + 2.00000E+01j + 2.80000E+01k


## ALGOL 68

Translation of: python
Note: This specimen retains the original python coding style.
Works with: ALGOL 68 version Revision 1 - one minor extension to language used - PRAGMA READ, similar to C's #include directive.
Works with: ALGOL 68G version Any - tested with release algol68g-2.6.
File: prelude/Quaternion.a68
# -*- coding: utf-8 -*- # COMMENT REQUIRES:  MODE QUATSCAL = REAL; # Scalar #  QUATSCAL quat small scal = small real;END COMMENT # PROVIDES: #FORMAT quat scal fmt := $g(-0, 4)$;FORMAT signed fmt = $b("+", "")f(quat scal fmt)$; FORMAT quat fmt = $f(quat scal fmt)"+"f(quat scal fmt)"i+"f(quat scal fmt)"j+"f(quat scal fmt)"k"$;FORMAT squat fmt = $f(signed fmt)f(signed fmt)"i"f(signed fmt)"j"f(signed fmt)"k"$; MODE QUAT = STRUCT(QUATSCAL r, i, j, k);QUAT i=(0, 1, 0, 0),     j=(0, 0, 1, 0),     k=(0, 0, 0, 1); MODE QUATCOSCAL = UNION(INT, SHORT REAL, SHORT INT); MODE QUATSUBSCAL = UNION(QUATCOSCAL, QUATSCAL); MODE COMPLSCAL = STRUCT(QUATSCAL r, im);# compatable but not the same #MODE ISOQUAT = UNION([]REAL, []INT, []SHORT REAL, []SHORT INT, []QUATSCAL);MODE COQUAT  = UNION(COMPLSCAL, QUATCOSCAL, ISOQUAT);MODE SUBQUAT = UNION(COQUAT, QUAT); # subset is itself # MODE QUATERNION = QUAT; PROC quat fix type error = (QUAT quat, []STRING msg)BOOL: (  putf(stand error, ($"Type error:"$,$" "g$, msg, quat fmt, quat, $l$));  stop); COMMENTFor a list of coercions expected in A68 c.f. * http://rosettacode.org/wiki/ALGOL_68#Coercion_.28casting.29 # ... Pre-Strong context: Deproceduring, dereferencing & uniting. e.g. OP arguments  * soft(deproceduring for assignment),   * weak(dereferencing for slicing and OF selection),   * meek(dereferencing for indexing, enquiries and PROC calls),  * firm(uniting of OPerators), Strong context only: widening (INT=>REAL=>COMPL), rowing (REAL=>[]REAL) & voiding  * strong(widening,rowing,voiding for identities/initialisations, arguments and casts et al)Key points:  * arguments to OPerators do not widen or row!  * UNITING is permitted in OP/String ccontext. There are 4 principle scenerios for most operators:+---------------+-------------------------------+-------------------------------+|  OP e.g. *    |  SCALar                       |  QUATernion                   |+---------------+-------------------------------+-------------------------------+|  SCALar       |  SCAL * SCAL ... inherit      |  SCAL * QUAT                  |+---------------+-------------------------------+-------------------------------+|  QUATernion   |  QUAT * SCAL                  |  QUAT * QUAT                  |+---------------+-------------------------------+-------------------------------+However this is compounded with SUBtypes of the SCALar & isomorphs the QUATernion, e.g. * SCAL may be a superset of SHORT REAL or INT - a widening coercion is required* QUAT may be a superset eg of COMPL or [4]INT* QUAT may be a structural isomorph eg of [4]REAL+---------------+---------------+---------------+---------------+---------------+|  OP e.g. *    |  SUBSCAL      |  SCALar       |  COQUAT       |  QUATernion   |+---------------+---------------+---------------+---------------+---------------+|  SUBSCAL      |                               |  inherit      |  SUBSCAT*QUAT |+---------------+           inherit             +---------------+---------------+|  SCALar       |                               |  inherit      |  SCAL * QUAT  |+---------------+---------------+---------------+---------------+---------------+|  COQUAT       |  inherit      |  inherit      |  inherit      |  COQUAT*QUAT  |+---------------+---------------+---------------+---------------+---------------+|  QUATernion   | QUAT*SUBSCAL  |  QUAT*SCAL    | QUAT * COQUAT |  QUAT * QUAT  |+---------------+---------------+---------------+---------------+---------------+Keypoint: if an EXPLICIT QUAT is not involved, then we can simple inherit, OR QUATINIT!END COMMENT MODE CLASSQUAT = STRUCT(    PROC (REF QUAT #new#, QUATSCAL #r#, QUATSCAL #i#, QUATSCAL #j#, QUATSCAL #k#)REF QUAT new,    PROC (REF QUAT #self#)QUAT conjugate,    PROC (REF QUAT #self#)QUATSCAL norm sq,    PROC (REF QUAT #self#)QUATSCAL norm,    PROC (REF QUAT #self#)QUAT reciprocal,    PROC (REF QUAT #self#)STRING repr,    PROC (REF QUAT #self#)QUAT neg,    PROC (REF QUAT #self#, SUBQUAT #other#)QUAT add,    PROC (REF QUAT #self#, SUBQUAT #other#)QUAT radd,    PROC (REF QUAT #self#, SUBQUAT #other#)QUAT sub,    PROC (REF QUAT #self#, SUBQUAT #other#)QUAT mul,    PROC (REF QUAT #self#, SUBQUAT #other#)QUAT rmul,    PROC (REF QUAT #self#, SUBQUAT #other#)QUAT div,    PROC (REF QUAT #self#, SUBQUAT #other#)QUAT rdiv,    PROC (REF QUAT #self#)QUAT exp); CLASSQUAT class quat = (   # PROC new =#(REF QUAT new, QUATSCAL r, i, j, k)REF QUAT: (        # 'Defaults all parts of quaternion to zero' #        IF new ISNT REF QUAT(NIL) THEN new ELSE HEAP QUAT FI := (r, i, j, k)    ),   # PROC conjugate =#(REF QUAT self)QUAT:        (r OF self, -i OF self, -j OF self, -k OF self),   # PROC norm sq =#(REF QUAT self)QUATSCAL:        r OF self**2 + i OF self**2 + j OF self**2 + k OF self**2,   # PROC norm =#(REF QUAT self)QUATSCAL:        sqrt((norm sq OF class quat)(self)),   # PROC reciprocal =#(REF QUAT self)QUAT:(        QUATSCAL n2 = (norm sq OF class quat)(self);        QUAT conj = (conjugate OF class quat)(self);        (r OF conj/n2, i OF conj/n2, j OF conj/n2, k OF conj/n2)    ),   # PROC repr =#(REF QUAT self)STRING: (        # 'Shorter form of Quaternion as string' #        FILE f; STRING s; associate(f, s);        putf(f, (squat fmt, r OF self>=0, r OF self,             i OF self>=0, i OF self, j OF self>=0, j OF self, k OF self>=0, k OF self));        close(f);        s    ),   # PROC neg =#(REF QUAT self)QUAT:        (-r OF self, -i OF self, -j OF self, -k OF self),   # PROC add =#(REF QUAT self, SUBQUAT other)QUAT:        CASE other IN            (QUAT other): (r OF self + r OF other, i OF self + i OF other, j OF self + j OF other, k OF self + k OF other),            (QUATSUBSCAL other): (r OF self + QUATSCALINIT other, i OF self, j OF self, k OF self)        OUT IF quat fix type error(SKIP,"in add") THEN SKIP ELSE stop FI        ESAC,   # PROC radd =#(REF QUAT self, SUBQUAT other)QUAT:        (add OF class quat)(self, other),   # PROC sub =#(REF QUAT self, SUBQUAT other)QUAT:        CASE other IN            (QUAT other): (r OF self - r OF other, i OF self - i OF other, j OF self - j OF other, k OF self - k OF other),            (QUATSCAL other): (r OF self - other, i OF self, j OF self, k OF self)        OUT IF quat fix type error(self,"in sub") THEN SKIP ELSE stop FI        ESAC,   # PROC mul =#(REF QUAT self, SUBQUAT other)QUAT:        CASE other IN            (QUAT other):(                 r OF self*r OF other - i OF self*i  OF other - j OF self*j  OF other - k OF self*k  OF other,                 r OF self*i  OF other + i OF self*r OF other + j OF self*k  OF other - k OF self*j  OF other,                 r OF self*j  OF other - i OF self*k  OF other + j OF self*r OF other + k OF self*i  OF other,                 r OF self*k  OF other + i OF self*j  OF other - j OF self*i  OF other + k OF self*r OF other            ),            (QUATSCAL other): ( r OF self * other, i OF self * other, j OF self * other, k OF self * other)        OUT IF quat fix type error(self,"in mul") THEN SKIP ELSE stop FI        ESAC,   # PROC rmul =#(REF QUAT self, SUBQUAT other)QUAT:        CASE other IN          (QUAT other): (mul OF class quat)(LOC QUAT := other, self),          (QUATSCAL other): (mul OF class quat)(self, other)        OUT IF quat fix type error(self,"in rmul") THEN SKIP ELSE stop FI        ESAC,   # PROC div =#(REF QUAT self, SUBQUAT other)QUAT:        CASE other IN            (QUAT other): (mul OF class quat)(self, (reciprocal OF class quat)(LOC QUAT := other)),            (QUATSCAL other): (mul OF class quat)(self, 1/other)        OUT IF quat fix type error(self,"in div") THEN SKIP ELSE stop FI        ESAC,   # PROC rdiv =#(REF QUAT self, SUBQUAT other)QUAT:        CASE other IN          (QUAT other): (div OF class quat)(LOC QUAT := other, self),          (QUATSCAL other): (div OF class quat)(LOC QUAT := (other, 0, 0, 0), self)        OUT IF quat fix type error(self,"in rdiv") THEN SKIP ELSE stop FI        ESAC,   # PROC exp =#(REF QUAT self)QUAT: (    QUAT fac := self;    QUAT sum := 1.0 + fac;    FOR i FROM 2 TO bits width WHILE ABS(fac + quat small scal) /= quat small scal DO      VOID(sum +:= (fac *:= self / ##QUATSCAL(i)))    OD;    sum  )); PRIO INIT = 1;OP QUATSCALINIT = (QUATSUBSCAL scal)QUATSCAL:   CASE scal IN    (INT scal): scal,    (SHORT INT scal): scal,    (SHORT REAL scal): scal    OUT IF quat fix type error(SKIP,"in QUATSCALINIT") THEN SKIP ELSE stop FI  ESAC; OP INIT = (REF QUAT new, SUBQUAT from)REF QUAT:  new :=     CASE from IN      (QUATSUBSCAL scal):(QUATSCALINIT scal, 0, 0, 0)      #(COQUAT rijk):(new OF class quat)(LOC QUAT := new, rijk[1], rijk[2], rijk[3], rijk[4]),#    OUT IF quat fix type error(SKIP,"in INIT") THEN SKIP ELSE stop FI    ESAC;  OP QUATINIT = (COQUAT lhs)REF QUAT: (HEAP QUAT)INIT lhs; OP +    = (QUAT q)QUAT:   q,   -    = (QUAT q)QUAT:   (neg  OF class quat)(LOC QUAT := q),   CONJ = (QUAT q)QUAT:   (conjugate OF class quat)(LOC QUAT := q),   ABS  = (QUAT q)QUATSCAL:   (norm OF class quat)(LOC QUAT := q),   REPR = (QUAT q)STRING: (repr OF class quat)(LOC QUAT := q);# missing: Diadic: I, J, K END # OP +:= = (REF QUAT a, QUAT b)QUAT: a:=( add OF class quat)(a, b),   +:= = (REF QUAT a, COQUAT b)QUAT: a:=( add OF class quat)(a, b),   +=: = (QUAT a, REF QUAT b)QUAT: b:=(radd OF class quat)(b, a),   +=: = (COQUAT a, REF QUAT b)QUAT: b:=(radd OF class quat)(b, a);# missing: Worthy PLUSAB, PLUSTO for SHORT/LONG INT QUATSCAL & COMPL # OP -:= = (REF QUAT a, QUAT b)QUAT: a:=( sub OF class quat)(a, b),   -:= = (REF QUAT a, COQUAT b)QUAT: a:=( sub OF class quat)(a, b);# missing: Worthy MINUSAB for SHORT/LONG INT ##COQUAT & COMPL # PRIO *=: = 1, /=: = 1;OP *:= = (REF QUAT a, QUAT b)QUAT: a:=( mul OF class quat)(a, b),   *:= = (REF QUAT a, COQUAT b)QUAT: a:=( mul OF class quat)(a, b),   *=: = (QUAT a, REF QUAT b)QUAT: b:=(rmul OF class quat)(b, a),   *=: = (COQUAT a, REF QUAT b)QUAT: b:=(rmul OF class quat)(b, a);# missing: Worthy TIMESAB, TIMESTO for SHORT/LONG INT ##COQUAT & COMPL # OP /:= = (REF QUAT a, QUAT b)QUAT: a:=( div OF class quat)(a, b),   /:= = (REF QUAT a, COQUAT b)QUAT: a:=( div OF class quat)(a, b),   /=: = (QUAT a, REF QUAT b)QUAT: b:=(rdiv OF class quat)(b, a),   /=: = (COQUAT a, REF QUAT b)QUAT: b:=(rdiv OF class quat)(b, a);# missing: Worthy OVERAB, OVERTO for SHORT/LONG INT ##COQUAT & COMPL # OP + = (QUAT a, b)QUAT:      ( add OF class quat)(LOC QUAT := a, b),   + = (QUAT a, COQUAT b)QUAT: ( add OF class quat)(LOC QUAT := a, b),   + = (COQUAT a, QUAT b)QUAT: (radd OF class quat)(LOC QUAT := b, a); OP - = (QUAT a, b)QUAT:      ( sub OF class quat)(LOC QUAT := a, b),   - = (QUAT a, COQUAT b)QUAT: ( sub OF class quat)(LOC QUAT := a, b),   - = (COQUAT a, QUAT b)QUAT:-( sub OF class quat)(LOC QUAT := b, a); OP * = (QUAT a, b)QUAT:      ( mul OF class quat)(LOC QUAT := a, b),   * = (QUAT a, COQUAT b)QUAT: ( mul OF class quat)(LOC QUAT := a, b),   * = (COQUAT a, QUAT b)QUAT: (rmul OF class quat)(LOC QUAT := b, a); OP / = (QUAT a, b)QUAT:      ( div OF class quat)(LOC QUAT := a, b),   / = (QUAT a, COQUAT b)QUAT: ( div OF class quat)(LOC QUAT := a, b),   / = (COQUAT a, QUAT b)QUAT:          ( div OF class quat)(LOC QUAT := QUATINIT 1, a); PROC quat exp = (QUAT q)QUAT:   (exp OF class quat)(LOC QUAT := q); SKIP # missing: quat arc{sin, cos, tan}h, log, exp, ln etc END #
File: test/Quaternion.a68
#!/usr/bin/a68g --script ## -*- coding: utf-8 -*- # # REQUIRES: #  MODE QUATSCAL = REAL; # Scalar #  QUATSCAL quat small scal = small real; PR READ "prelude/Quaternion.a68" PR; test:(    REAL r = 7;    QUAT q  = (1, 2, 3, 4),         q1 = (2, 3, 4, 5),         q2 = (3, 4, 5, 6);     printf((        $"r = " f(quat scal fmt)l$, r,        $"q = " f(quat fmt)l$, q,        $"q1 = " f(quat fmt)l$, q1,        $"q2 = " f(quat fmt)l$, q2,        $"ABS q = " f(quat scal fmt)", "$, ABS q,        $"ABS q1 = " f(quat scal fmt)", "$, ABS q1,        $"ABS q2 = " f(quat scal fmt)l$, ABS q2,        $"-q = " f(quat fmt)l$, -q,        $"CONJ q = " f(quat fmt)l$, CONJ q,        $"r + q = " f(quat fmt)l$, r + q,        $"q + r = " f(quat fmt)l$, q + r,        $"q1 + q2 = "f(quat fmt)l$, q1 + q2,        $"q2 + q1 = "f(quat fmt)l$, q2 + q1,        $"q * r = " f(quat fmt)l$, q * r,        $"r * q = " f(quat fmt)l$, r * q,        $"q1 * q2 = "f(quat fmt)l$, q1 * q2,        $"q2 * q1 = "f(quat fmt)l$, q2 * q1    )); CO        $"ASSERT q1 * q2 != q2 * q1 = "f(quat fmt)l$, ASSERT q1 * q2 != q2 * q1, $l$;END CO     printf((        $"i*i = " f(quat fmt)l$, i*i,        $"j*j = " f(quat fmt)l$, j*j,        $"k*k = " f(quat fmt)l$, k*k,        $"i*j*k = " f(quat fmt)l$, i*j*k,        $"q1 / q2 = " f(quat fmt)l$, q1 / q2,        $"q1 / q2 * q2 = "f(quat fmt)l$, q1 / q2 * q2,        $"q2 * q1 / q2 = "f(quat fmt)l$, q2 * q1 / q2,        $"1/q1 * q1 = " f(quat fmt)l$, 1.0/q1 * q1,        $"q1 / q1 = " f(quat fmt)l$, q1 / q1,        $"quat exp(pi * i) = " f(quat fmt)l$, quat exp(pi * i),        $"quat exp(pi * j) = " f(quat fmt)l$, quat exp(pi * j),        $"quat exp(pi * k) = " f(quat fmt)l$, quat exp(pi * k)    ));    print((REPR(-q1*q2), ", ", REPR(-q2*q1), new line)))
Output:
r = 7.0000
q = 1.0000+2.0000i+3.0000j+4.0000k
q1 = 2.0000+3.0000i+4.0000j+5.0000k
q2 = 3.0000+4.0000i+5.0000j+6.0000k
ABS q = 5.4772, ABS q1 = 7.3485, ABS q2 = 9.2736
-q = -1.0000+-2.0000i+-3.0000j+-4.0000k
CONJ q = 1.0000+-2.0000i+-3.0000j+-4.0000k
r + q = 8.0000+2.0000i+3.0000j+4.0000k
q + r = 8.0000+2.0000i+3.0000j+4.0000k
q1 + q2 = 5.0000+7.0000i+9.0000j+11.0000k
q2 + q1 = 5.0000+7.0000i+9.0000j+11.0000k
q * r = 7.0000+14.0000i+21.0000j+28.0000k
r * q = 7.0000+14.0000i+21.0000j+28.0000k
q1 * q2 = -56.0000+16.0000i+24.0000j+26.0000k
q2 * q1 = -56.0000+18.0000i+20.0000j+28.0000k
i*i = -1.0000+.0000i+.0000j+.0000k
j*j = -1.0000+.0000i+.0000j+.0000k
k*k = -1.0000+.0000i+.0000j+.0000k
i*j*k = -1.0000+.0000i+.0000j+.0000k
q1 / q2 = .7907+.0233i+-.0000j+.0465k
q1 / q2 * q2 = 2.0000+3.0000i+4.0000j+5.0000k
q2 * q1 / q2 = 2.0000+3.4651i+3.9070j+4.7674k
1/q1 * q1 = 2.0000+3.0000i+4.0000j+5.0000k
q1 / q1 = 1.0000+.0000i+.0000j+.0000k
quat exp(pi * i) = -1.0000+.0000i+.0000j+.0000k
quat exp(pi * j) = -1.0000+.0000i+.0000j+.0000k
quat exp(pi * k) = -1.0000+.0000i+.0000j+.0000k
+56.0000-16.0000i-24.0000j-26.0000k, +56.0000-18.0000i-20.0000j-28.0000k


## ALGOL W

begin    % Quaternion record type                                                 %    record Quaternion ( real a, b, c, d );     % returns the norm of the specified quaternion                           %    real procedure normQ ( reference(Quaternion) value q ) ;        sqrt( (a(q) * a(q)) + (b(q) * b(q)) + (c(q) * c(q)) + (d(q) * d(q)) );      % returns the negative of the specified quaternion                       %    reference(Quaternion) procedure negQ ( reference(Quaternion) value q ) ;        Quaternion( - a(q), - b(q), - c(q), - d(q) );     % returns the conjugate of the specified quaternion                      %    reference(Quaternion) procedure conjQ ( reference(Quaternion) value q ) ;        Quaternion(   a(q), - b(q), - c(q), - d(q) );     % returns the sum of a real and a quaternion                             %    reference(Quaternion) procedure addRQ ( real                  value r                                          ; reference(Quaternion) value q                                          ) ;        Quaternion( r + a(q), b(q), c(q), d(q) );     % returns the sum of a quaternion and a real                             %    reference(Quaternion) procedure addQR ( reference(Quaternion) value q                                          ; real                  value r                                          ) ;        Quaternion( r + a(q), b(q), c(q), d(q) );     % returns the sum of the specified quaternions                           %    reference(Quaternion) procedure addQQ ( reference(Quaternion) value q1                                          ; reference(Quaternion) value q2                                          ) ;        Quaternion( a(q1) + a(q2), b(q1) + b(q2), c(q1) + c(q2), d(q1) + d(q2) );     % returns the specified quaternion multiplied by a real                  %    reference(Quaternion) procedure mulQR ( reference(Quaternion) value q                                          ; real                  value r                                          ) ;        Quaternion( r * a(q), r * b(q), r * c(q), r * d(q) );     % returns a real multiplied by the specified quaternion                  %    reference(Quaternion) procedure mulRQ ( real                  value r                                          ; reference(Quaternion) value q                                          ) ;        mulQR( q, r );     % returns the Quaternion product of the specified quaternions            %    reference(Quaternion) procedure mulQQ( reference(Quaternion) value q1                                         ; reference(Quaternion) value q2                                         ) ;        Quaternion( (a(q1) * a(q2)) - (b(q1) * b(q2)) - (c(q1) * c(q2)) - (d(q1) * d(q2))                  , (a(q1) * b(q2)) + (b(q1) * a(q2)) + (c(q1) * d(q2)) - (d(q1) * c(q2))                  , (a(q1) * c(q2)) - (b(q1) * d(q2)) + (c(q1) * a(q2)) + (d(q1) * b(q2))                  , (a(q1) * d(q2)) + (b(q1) * c(q2)) - (c(q1) * b(q2)) + (d(q1) * a(q2))                  );     % returns true if the two quaternions are equal, false otherwise         %    logical procedure equalQ( reference(Quaternion) value q1                            ; reference(Quaternion) value q2                            ) ;        a(q1) = a(q2) and b(q1) = b(q2) and c(q1) = c(q2) and d(q1) = d(q2);     % writes a quaternion                                                    %    procedure writeonQ( reference(Quaternion) value q ) ;        writeon( "(", a(q), ", ", b(q), ", ", c(q), ", ", d(q), ")" );      % test q1q2 = q2q1                                                       %    reference(Quaternion) q, q1, q2;     q  := Quaternion( 1, 2, 3, 4 );    q1 := Quaternion( 2, 3, 4, 5 );    q2 := Quaternion( 3, 4, 5, 6 );     % set output format                                                      %    s_w := 0; r_format := "A"; r_w := 5; r_d := 1;     write( "      q:" );writeonQ( q );    write( "     q1:" );writeonQ( q1 );    write( "     q2:" );writeonQ( q2 );    write( "norm  q:" );writeon( normQ( q ) );    write( "norm q1:" );writeon( normQ( q1 ) );    write( "norm q2:" );writeon( normQ( q2 ) );     write( " conj q:" );writeonQ( conjQ( q ) );    write( "    - q:" );writeonQ( negQ( q ) );    write( "  7 + q:" );writeonQ( addRQ( 7, q ) );    write( "  q + 9:" );writeonQ( addQR( q, 9 ) );    write( " q + q1:" );writeonQ( addQQ( q, q1 ) );    write( "  3 * q:" );writeonQ( mulRQ( 3, q ) );    write( "  q * 4:" );writeonQ( mulQR( q, 4 ) );     % check that q1q2 not = q2q1                                             %    if equalQ( mulQQ( q1, q2 ), mulQQ( q2, q1 ) )    then write( "q1q2 = q2q1 ??" )    else write( "q1q2 <> q2q1" );     write( "   q1q2:" );writeonQ( mulQQ( q1, q2 ) );    write( "   q2q1:" );writeonQ( mulQQ( q2, q1 ) ); end.
Output:
      q:(  1.0,   2.0,   3.0,   4.0)
q1:(  2.0,   3.0,   4.0,   5.0)
q2:(  3.0,   4.0,   5.0,   6.0)
norm  q:  5.4
norm q1:  7.3
norm q2:  9.2
conj q:(  1.0,  -2.0,  -3.0,  -4.0)
- q:( -1.0,  -2.0,  -3.0,  -4.0)
7 + q:(  8.0,   2.0,   3.0,   4.0)
q + 9:( 10.0,   2.0,   3.0,   4.0)
q + q1:(  3.0,   5.0,   7.0,   9.0)
3 * q:(  3.0,   6.0,   9.0,  12.0)
q * 4:(  4.0,   8.0,  12.0,  16.0)
q1q2 <> q2q1
q1q2:(-56.0,  16.0,  24.0,  26.0)
q2q1:(-56.0,  18.0,  20.0,  28.0)


## AutoHotkey

Works with: AutoHotkey_L
(AutoHotkey1.1+)
q  := [1, 2, 3, 4]q1 := [2, 3, 4, 5]q2 := [3, 4, 5, 6]r := 7 MsgBox, % "q = " PrintQ(q)	. "nq1 = " PrintQ(q1)	. "nq2 = " PrintQ(q2)	. "nr = " r	. "nNorm(q) = " Norm(q)	. "nNegative(q) = " PrintQ(Negative(q))	. "nConjugate(q) = " PrintQ(Conjugate(q))	. "nq + r = " PrintQ(AddR(q, r))	. "nq1 + q2 = " PrintQ(AddQ(q1, q2))	. "nq2 + q1 = " PrintQ(AddQ(q2, q1))	. "nqr = " PrintQ(MulR(q, r))	. "nq1q2 = " PrintQ(MulQ(q1, q2))	. "nq2q1 = " PrintQ(MulQ(q2, q1)) Norm(q) {	return sqrt(q[1]**2 + q[2]**2 + q[3]**2 + q[4]**2)} Negative(q) {	a := []	for k, v in q		a[A_Index] := v * -1	return a} Conjugate(q) {	a := []	for k, v in q		a[A_Index] := v * (A_Index = 1 ? 1 : -1)	return a} AddR(q, r) {	a := []	for k, v in q 		a[A_Index] := v + (A_Index = 1 ? r : 0)	return a} AddQ(q1, q2) {	a := []	for k, v in q1		a[A_Index] := v + q2[A_Index]	return a} MulR(q, r) {	a := []	for k, v in q		a[A_Index] := v * r	return a} MulQ(q, u) {	a := []	, a[1] := q[1]*u[1] - q[2]*u[2] - q[3]*u[3] - q[4]*u[4]	, a[2] := q[1]*u[2] + q[2]*u[1] + q[3]*u[4] - q[4]*u[3]	, a[3] := q[1]*u[3] - q[2]*u[4] + q[3]*u[1] + q[4]*u[2]	, a[4] := q[1]*u[4] + q[2]*u[3] - q[3]*u[2] + q[4]*u[1]	return a} PrintQ(q, b="(") {	for k, v in q		b .= v (A_Index = q.MaxIndex() ? ")" : ", ")	return b}
Output:
q = (1, 2, 3, 4)
q1 = (2, 3, 4, 5)
q2 = (3, 4, 5, 6)
r = 7
Norm(q) = 5.477226
Negative(q) = (-1, -2, -3, -4)
Conjugate(q) = (1, -2, -3, -4)
q + r = (8, 2, 3, 4)
q1 + q2 = (5, 7, 9, 11)
q2 + q1 = (5, 7, 9, 11)
qr = (7, 14, 21, 28)
q1q2 = (-56, 16, 24, 26)
q2q1 = (-56, 18, 20, 28)

## Axiom

Axiom has built-in support for quaternions.

qi := quatern$Quaternion(Integer); Type: ((Integer,Integer,Integer,Integer) -> Quaternion(Integer))q := qi(1,2,3,4); Type: Quaternion(Integer)q1 := qi(2,3,4,5); Type: Quaternion(Integer)q2 := qi(3,4,5,6); Type: Quaternion(Integer)r : Integer := 7; Type: Integersqrt norm q +--+ (6) \|30 Type: AlgebraicNumber-q (7) - 1 - 2i - 3j - 4k Type: Quaternion(Integer)conjugate q (8) 1 - 2i - 3j - 4k Type: Quaternion(Integer)r + q (9) 8 + 2i + 3j + 4k Type: Quaternion(Integer)q1 + q2 (10) 5 + 7i + 9j + 11k Type: Quaternion(Integer)q*r (11) 7 + 14i + 21j + 28k Type: Quaternion(Integer)r*q (12) 7 + 14i + 21j + 28k Type: Quaternion(Integer)q1*q2 ~= q2*q1 (13) true Type: Boolean ## BBC BASIC Although BBC BASIC doesn't have native support for quaternions its array arithmetic provides all of the required operations either directly or very straightforwardly.  DIM q(3), q1(3), q2(3), t(3) q() = 1, 2, 3, 4 q1() = 2, 3, 4, 5 q2() = 3, 4, 5, 6 r = 7 PRINT "q = " FNq_show(q()) PRINT "q1 = " FNq_show(q1()) PRINT "q2 = " FNq_show(q2()) PRINT "r = "; r PRINT "norm(q) = "; FNq_norm(q()) t() = q() : PROCq_neg(t()) : PRINT "neg(q) = " FNq_show(t()) t() = q() : PROCq_conj(t()) : PRINT "conjugate(q) = " FNq_show(t()) t() = q() : PROCq_addreal(t(),r) : PRINT "q + r = " FNq_show(t()) t() = q1() : PROCq_add(t(),q2()) : PRINT "q1 + q2 = " FNq_show(t()) t() = q2() : PROCq_add(t(),q1()) : PRINT "q2 + q1 = " FNq_show(t()) t() = q() : PROCq_mulreal(t(),r) : PRINT "qr = " FNq_show(t()) t() = q1() : PROCq_mul(t(),q2()) : PRINT "q1q2 = " FNq_show(t()) t() = q2() : PROCq_mul(t(),q1()) : PRINT "q2q1 = " FNq_show(t()) END DEF FNq_norm(q()) = MOD(q()) DEF PROCq_neg(q()) : q() *= -1 : ENDPROC DEF PROCq_conj(q()) : q() *= -1 : q(0) *= -1 : ENDPROC DEF PROCq_addreal(q(), r) : q(0) += r : ENDPROC DEF PROCq_add(q(), r()) : q() += r() : ENDPROC DEF PROCq_mulreal(q(), r) : q() *= r : ENDPROC DEF PROCq_mul(q(), r()) : LOCAL s() : DIM s(3,3) s() = r(0), -r(1), -r(2), -r(3), r(1), r(0), r(3), -r(2), \ \ r(2), -r(3), r(0), r(1), r(3), r(2), -r(1), r(0) q() = s() . q() ENDPROC DEF FNq_show(q()) : LOCAL i%, a$ : a$= "(" FOR i% = 0 TO 3 : a$ += STR$(q(i%)) + ", " : NEXT = LEFT$(LEFT$(a$)) + ")"
Output:
q = (1, 2, 3, 4)
q1 = (2, 3, 4, 5)
q2 = (3, 4, 5, 6)
r = 7
norm(q) = 5.47722558
neg(q) = (-1, -2, -3, -4)
conjugate(q) = (1, -2, -3, -4)
q + r = (8, 2, 3, 4)
q1 + q2 = (5, 7, 9, 11)
q2 + q1 = (5, 7, 9, 11)
qr = (7, 14, 21, 28)
q1q2 = (-56, 16, 24, 26)
q2q1 = (-56, 18, 20, 28)


## C

#include <stdio.h>#include <stdlib.h>#include <stdbool.h>#include <math.h> typedef struct quaternion{  double q[4];} quaternion_t;  quaternion_t *quaternion_new(void){  return malloc(sizeof(quaternion_t));} quaternion_t *quaternion_new_set(double q1,				 double q2,				 double q3,				 double q4){  quaternion_t *q = malloc(sizeof(quaternion_t));  if (q != NULL) {    q->q[0] = q1; q->q[1] = q2; q->q[2] = q3; q->q[3] = q4;  }  return q;}  void quaternion_copy(quaternion_t *r, quaternion_t *q){  size_t i;   if (r == NULL || q == NULL) return;  for(i = 0; i < 4; i++) r->q[i] = q->q[i];}  double quaternion_norm(quaternion_t *q){  size_t i;  double r = 0.0;   if (q == NULL) {    fprintf(stderr, "NULL quaternion in norm\n");    return 0.0;  }   for(i = 0; i < 4; i++) r += q->q[i] * q->q[i];  return sqrt(r);}  void quaternion_neg(quaternion_t *r, quaternion_t *q){  size_t i;   if (q == NULL || r == NULL) return;  for(i = 0; i < 4; i++) r->q[i] = -q->q[i];}  void quaternion_conj(quaternion_t *r, quaternion_t *q){  size_t i;   if (q == NULL || r == NULL) return;  r->q[0] = q->q[0];  for(i = 1; i < 4; i++) r->q[i] = -q->q[i];}  void quaternion_add_d(quaternion_t *r, quaternion_t *q, double d){  if (q == NULL || r == NULL) return;  quaternion_copy(r, q);  r->q[0] += d;}  void quaternion_add(quaternion_t *r, quaternion_t *a, quaternion_t *b){  size_t i;   if (r == NULL || a == NULL || b == NULL) return;  for(i = 0; i < 4; i++) r->q[i] = a->q[i] + b->q[i];}  void quaternion_mul_d(quaternion_t *r, quaternion_t *q, double d){  size_t i;   if (r == NULL || q == NULL) return;  for(i = 0; i < 4; i++) r->q[i] = q->q[i] * d;} bool quaternion_equal(quaternion_t *a, quaternion_t *b){  size_t i;   for(i = 0; i < 4; i++) if (a->q[i] != b->q[i]) return false;  return true;}  #define A(N) (a->q[(N)])#define B(N) (b->q[(N)])#define R(N) (r->q[(N)])void quaternion_mul(quaternion_t *r, quaternion_t *a, quaternion_t *b){  size_t i;  double ri = 0.0;   if (r == NULL || a == NULL || b == NULL) return;  R(0) = A(0)*B(0) - A(1)*B(1) - A(2)*B(2) - A(3)*B(3);  R(1) = A(0)*B(1) + A(1)*B(0) + A(2)*B(3) - A(3)*B(2);  R(2) = A(0)*B(2) - A(1)*B(3) + A(2)*B(0) + A(3)*B(1);  R(3) = A(0)*B(3) + A(1)*B(2) - A(2)*B(1) + A(3)*B(0);}#undef A#undef B#undef R  void quaternion_print(quaternion_t *q){  if (q == NULL) return;  printf("(%lf, %lf, %lf, %lf)\n", 	 q->q[0], q->q[1], q->q[2], q->q[3]);}
int main(){  size_t i;  double d = 7.0;  quaternion_t *q[3];  quaternion_t *r  = quaternion_new();   quaternion_t *qd = quaternion_new_set(7.0, 0.0, 0.0, 0.0);  q[0] = quaternion_new_set(1.0, 2.0, 3.0, 4.0);  q[1] = quaternion_new_set(2.0, 3.0, 4.0, 5.0);  q[2] = quaternion_new_set(3.0, 4.0, 5.0, 6.0);   printf("r = %lf\n", d);   for(i = 0; i < 3; i++) {    printf("q[%u] = ", i);    quaternion_print(q[i]);    printf("abs q[%u] = %lf\n", i, quaternion_norm(q[i]));  }   printf("-q[0] = ");  quaternion_neg(r, q[0]);  quaternion_print(r);   printf("conj q[0] = ");  quaternion_conj(r, q[0]);  quaternion_print(r);   printf("q[1] + q[2] = ");  quaternion_add(r, q[1], q[2]);  quaternion_print(r);   printf("q[2] + q[1] = ");  quaternion_add(r, q[2], q[1]);  quaternion_print(r);    printf("q[0] * r = ");  quaternion_mul_d(r, q[0], d);  quaternion_print(r);   printf("q[0] * (r, 0, 0, 0) = ");  quaternion_mul(r, q[0], qd);  quaternion_print(r);    printf("q[1] * q[2] = ");  quaternion_mul(r, q[1], q[2]);  quaternion_print(r);   printf("q[2] * q[1] = ");  quaternion_mul(r, q[2], q[1]);  quaternion_print(r);      free(q[0]); free(q[1]); free(q[2]); free(r);  return EXIT_SUCCESS;}

## C++

This example uses templates to provide the underlying data-type, and includes several extra functions and constructors that often come up when using quaternions.

#include <iostream>using namespace std; template<class T = double>class Quaternion{public:  T w, x, y, z;   // Numerical constructor  Quaternion(const T &w, const T &x, const T &y, const T &z): w(w), x(x), y(y), z(z) {};  Quaternion(const T &x, const T &y, const T &z): w(T()), x(x), y(y), z(z) {}; // For 3-rotations  Quaternion(const T &r): w(r), x(T()), y(T()), z(T()) {};  Quaternion(): w(T()), x(T()), y(T()), z(T()) {};   // Copy constructor and assignment  Quaternion(const Quaternion &q): w(q.w), x(q.x), y(q.y), z(q.z) {};  Quaternion& operator=(const Quaternion &q) { w=q.w; x=q.x; y=q.y; z=q.z; return *this; }   // Unary operators  Quaternion operator-() const { return Quaternion(-w, -x, -y, -z); }  Quaternion operator~() const { return Quaternion(w, -x, -y, -z); } // Conjugate   // Norm-squared. SQRT would have to be made generic to be used here  T normSquared() const { return w*w + x*x + y*y + z*z; }   // In-place operators  Quaternion& operator+=(const T &r)     { w += r; return *this; }  Quaternion& operator+=(const Quaternion &q)     { w += q.w; x += q.x; y += q.y; z += q.z; return *this; }   Quaternion& operator-=(const T &r)     { w -= r; return *this; }  Quaternion& operator-=(const Quaternion &q)     { w -= q.w; x -= q.x; y -= q.y; z -= q.z; return *this; }   Quaternion& operator*=(const T &r)     { w *= r; x *= r; y *= r; z *= r; return *this; }  Quaternion& operator*=(const Quaternion &q)   {     T oldW(w), oldX(x), oldY(y), oldZ(z);    w = oldW*q.w - oldX*q.x - oldY*q.y - oldZ*q.z;    x = oldW*q.x + oldX*q.w + oldY*q.z - oldZ*q.y;    y = oldW*q.y + oldY*q.w + oldZ*q.x - oldX*q.z;    z = oldW*q.z + oldZ*q.w + oldX*q.y - oldY*q.x;    return *this;  }   Quaternion& operator/=(const T &r)     { w /= r; x /= r; y /= r; z /= r; return *this; }  Quaternion& operator/=(const Quaternion &q)   {     T oldW(w), oldX(x), oldY(y), oldZ(z), n(q.normSquared());    w = (oldW*q.w + oldX*q.x + oldY*q.y + oldZ*q.z) / n;    x = (oldX*q.w - oldW*q.x + oldY*q.z - oldZ*q.y) / n;    y = (oldY*q.w - oldW*q.y + oldZ*q.x - oldX*q.z) / n;    z = (oldZ*q.w - oldW*q.z + oldX*q.y - oldY*q.x) / n;    return *this;  }   // Binary operators based on in-place operators  Quaternion operator+(const T &r) const { return Quaternion(*this) += r; }  Quaternion operator+(const Quaternion &q) const { return Quaternion(*this) += q; }  Quaternion operator-(const T &r) const { return Quaternion(*this) -= r; }  Quaternion operator-(const Quaternion &q) const { return Quaternion(*this) -= q; }  Quaternion operator*(const T &r) const { return Quaternion(*this) *= r; }  Quaternion operator*(const Quaternion &q) const { return Quaternion(*this) *= q; }  Quaternion operator/(const T &r) const { return Quaternion(*this) /= r; }  Quaternion operator/(const Quaternion &q) const { return Quaternion(*this) /= q; }   // Comparison operators, as much as they make sense  bool operator==(const Quaternion &q) const     { return (w == q.w) && (x == q.x) && (y == q.y) && (z == q.z); }  bool operator!=(const Quaternion &q) const { return !operator==(q); }   // The operators above allow quaternion op real. These allow real op quaternion.  // Uses the above where appropriate.  template<class T> friend Quaternion<T> operator+(const T &r, const Quaternion<T> &q);  template<class T> friend Quaternion<T> operator-(const T &r, const Quaternion<T> &q);  template<class T> friend Quaternion<T> operator*(const T &r, const Quaternion<T> &q);  template<class T> friend Quaternion<T> operator/(const T &r, const Quaternion<T> &q);   // Allows cout << q   template<class T> friend ostream& operator<<(ostream &io, const Quaternion<T> &q);}; // Friend functions need to be outside the actual class definitiontemplate<class T>Quaternion<T> operator+(const T &r, const Quaternion<T> &q)   { return q+r; } template<class T>Quaternion<T> operator-(const T &r, const Quaternion<T> &q)  { return Quaternion<T>(r-q.w, q.x, q.y, q.z); } template<class T>Quaternion<T> operator*(const T &r, const Quaternion<T> &q)   { return q*r; } template<class T>Quaternion<T> operator/(const T &r, const Quaternion<T> &q){  T n(q.normSquared());  return Quaternion(r*q.w/n, -r*q.x/n, -r*q.y/n, -r*q.z/n);} template<class T>ostream& operator<<(ostream &io, const Quaternion<T> &q){   io << q.w;  (q.x < T()) ? (io << " - " << (-q.x) << "i") : (io << " + " << q.x << "i");  (q.y < T()) ? (io << " - " << (-q.y) << "j") : (io << " + " << q.y << "j");  (q.z < T()) ? (io << " - " << (-q.z) << "k") : (io << " + " << q.z << "k");  return io;}

Test program:

int main(){  Quaternion<> q0(1, 2, 3, 4);  Quaternion<> q1(2, 3, 4, 5);  Quaternion<> q2(3, 4, 5, 6);  double r = 7;   cout << "q0:      " << q0 << endl;  cout << "q1:      " << q1 << endl;  cout << "q2:      " << q2 << endl;  cout << "r:       " << r << endl;  cout << endl;  cout << "-q0:     " << -q0 << endl;  cout << "~q0:     " << ~q0 << endl;  cout << endl;  cout << "r * q0:  " << r*q0 << endl;  cout << "r + q0:  " << r+q0 << endl;  cout << "q0 / r:  " << q0/r << endl;  cout << "q0 - r:  " << q0-r << endl;  cout << endl;  cout << "q0 + q1: " << q0+q1 << endl;  cout << "q0 - q1: " << q0-q1 << endl;  cout << "q0 * q1: " << q0*q1 << endl;  cout << "q0 / q1: " << q0/q1 << endl;  cout << endl;  cout << "q0 * ~q0:     " << q0*~q0 << endl;  cout << "q0 + q1*q2:   " << q0+q1*q2 << endl;  cout << "(q0 + q1)*q2: " << (q0+q1)*q2 << endl;  cout << "q0*q1*q2:     " << q0*q1*q2 << endl;  cout << "(q0*q1)*q2:   " << (q0*q1)*q2 << endl;  cout << "q0*(q1*q2):   " << q0*(q1*q2) << endl;  cout << endl;  cout << "||q0||:  " << sqrt(q0.normSquared()) << endl;  cout << endl;  cout << "q0*q1 - q1*q0: " << (q0*q1 - q1*q0) << endl;   // Other base types  Quaternion<int> q5(2), q6(3);  cout << endl << q5*q6 << endl;}
Output:
q0:      1 + 2i + 3j + 4k
q1:      2 + 3i + 4j + 5k
q2:      3 + 4i + 5j + 6k
r:       7

-q0:     -1 - 2i - 3j - 4k
~q0:     1 - 2i - 3j - 4k

r * q0:  7 + 14i + 21j + 28k
r + q0:  8 + 2i + 3j + 4k
q0 / r:  0.142857 + 0.285714i + 0.428571j + 0.571429k
q0 - r:  -6 + 2i + 3j + 4k

q0 + q1: 3 + 5i + 7j + 9k
q0 - q1: -1 - 1i - 1j - 1k
q0 * q1: -36 + 6i + 12j + 12k
q0 / q1: 0.740741 + 0i + 0.0740741j + 0.037037k

q0 * ~q0:     30 + 0i + 0j + 0k
q0 + q1*q2:   -55 + 18i + 27j + 30k
(q0 + q1)*q2: -100 + 24i + 42j + 42k
q0*q1*q2:     -264 - 114i - 132j - 198k
(q0*q1)*q2:   -264 - 114i - 132j - 198k
q0*(q1*q2):   -264 - 114i - 132j - 198k

||q0||:  5.47723

q0*q1 - q1*q0: 0 - 2i + 4j - 2k

6 + 0i + 0j + 0k


## C#

using System; struct Quaternion : IEquatable<Quaternion>{    public readonly double A, B, C, D;     public Quaternion(double a, double b, double c, double d)    {        this.A = a;        this.B = b;        this.C = c;        this.D = d;    }     public double Norm()    {        return Math.Sqrt(A * A + B * B + C * C + D * D);    }     public static Quaternion operator -(Quaternion q)    {        return new Quaternion(-q.A, -q.B, -q.C, -q.D);    }     public Quaternion Conjugate()    {        return new Quaternion(A, -B, -C, -D);    }     // implicit conversion takes care of real*quaternion and real+quaternion    public static implicit operator Quaternion(double d)    {        return new Quaternion(d, 0, 0, 0);    }     public static Quaternion operator +(Quaternion q1, Quaternion q2)    {        return new Quaternion(q1.A + q2.A, q1.B + q2.B, q1.C + q2.C, q1.D + q2.D);    }     public static Quaternion operator *(Quaternion q1, Quaternion q2)    {        return new Quaternion(            q1.A * q2.A - q1.B * q2.B - q1.C * q2.C - q1.D * q2.D,            q1.A * q2.B + q1.B * q2.A + q1.C * q2.D - q1.D * q2.C,            q1.A * q2.C - q1.B * q2.D + q1.C * q2.A + q1.D * q2.B,            q1.A * q2.D + q1.B * q2.C - q1.C * q2.B + q1.D * q2.A);    }     public static bool operator ==(Quaternion q1, Quaternion q2)    {        return q1.A == q2.A && q1.B == q2.B && q1.C == q2.C && q1.D == q2.D;    }     public static bool operator !=(Quaternion q1, Quaternion q2)    {        return !(q1 == q2);    }     #region Object Members     public override bool Equals(object obj)    {        if (obj is Quaternion)            return Equals((Quaternion)obj);         return false;    }     public override int GetHashCode()    {        return A.GetHashCode() ^ B.GetHashCode() ^ C.GetHashCode() ^ D.GetHashCode();    }     public override string ToString()    {        return string.Format("Q({0}, {1}, {2}, {3})", A, B, C, D);    }     #endregion     #region IEquatable<Quaternion> Members     public bool Equals(Quaternion other)    {        return other == this;    }     #endregion}

Demonstration:

using System; static class Program{    static void Main(string[] args)    {        Quaternion q = new Quaternion(1, 2, 3, 4);        Quaternion q1 = new Quaternion(2, 3, 4, 5);        Quaternion q2 = new Quaternion(3, 4, 5, 6);        double r = 7;         Console.WriteLine("q = {0}", q);        Console.WriteLine("q1 = {0}", q1);        Console.WriteLine("q2 = {0}", q2);        Console.WriteLine("r = {0}", r);         Console.WriteLine("q.Norm() = {0}", q.Norm());        Console.WriteLine("q1.Norm() = {0}", q1.Norm());        Console.WriteLine("q2.Norm() = {0}", q2.Norm());         Console.WriteLine("-q = {0}", -q);        Console.WriteLine("q.Conjugate() = {0}", q.Conjugate());         Console.WriteLine("q + r = {0}", q + r);        Console.WriteLine("q1 + q2 = {0}", q1 + q2);        Console.WriteLine("q2 + q1 = {0}", q2 + q1);         Console.WriteLine("q * r = {0}", q * r);        Console.WriteLine("q1 * q2 = {0}", q1 * q2);        Console.WriteLine("q2 * q1 = {0}", q2 * q1);         Console.WriteLine("q1*q2 {0} q2*q1", (q1 * q2) == (q2 * q1) ? "==" : "!=");    }}
Output:
q = Q(1, 2, 3, 4)
q1 = Q(2, 3, 4, 5)
q2 = Q(3, 4, 5, 6)
r = 7
q.Norm() = 5.47722557505166
q1.Norm() = 7.34846922834953
q2.Norm() = 9.2736184954957
-q = Q(-1, -2, -3, -4)
q.Conjugate() = Q(1, -2, -3, -4)
q + r = Q(8, 2, 3, 4)
q1 + q2 = Q(5, 7, 9, 11)
q2 + q1 = Q(5, 7, 9, 11)
q * r = Q(7, 14, 21, 28)
q1 * q2 = Q(-56, 16, 24, 26)
q2 * q1 = Q(-56, 18, 20, 28)
q1*q2 != q2*q1

## Common Lisp

 (defclass quaternion () ((a :accessor q-a :initarg :a :type real)                         (b :accessor q-b :initarg :b :type real)                         (c :accessor q-c :initarg :c :type real)                         (d :accessor q-d :initarg :d :type real))  (:default-initargs :a 0 :b 0 :c 0 :d 0))  (defun make-q (&optional (a 0) (b 0) (c 0) (d 0))  (make-instance 'quaternion :a a :b b :c c :d d)) (defgeneric sum (x y)) (defmethod sum ((x quaternion) (y quaternion))  (make-q  (+ (q-a x) (q-a y))           (+ (q-b x) (q-b y))           (+ (q-c x) (q-c y))           (+ (q-d x) (q-d y)))) (defmethod sum ((x quaternion) (y real))  (make-q  (+ (q-a x) y) (q-b x) (q-c x) (q-d x))) (defmethod sum ((x real) (y quaternion))  (make-q  (+ (q-a y) x) (q-b y) (q-c y) (q-d y))) (defgeneric sub (x y)) (defmethod sub ((x quaternion) (y quaternion))  (make-q  (- (q-a x) (q-a y))           (- (q-b x) (q-b y))           (- (q-c x) (q-c y))           (- (q-d x) (q-d y)))) (defmethod sub ((x quaternion) (y real))  (make-q  (- (q-a x) y)           (q-b x)           (q-c x)           (q-d x))) (defmethod sub ((x real) (y quaternion))  (make-q  (- (q-a y) x)           (q-b y)           (q-c y)           (q-d y))) (defgeneric mul (x y)) (defmethod mul ((x quaternion) (y real))  (make-q  (* (q-a x) y)           (* (q-b x) y)           (* (q-c x) y)           (* (q-d x) y))) (defmethod mul ((x real) (y quaternion))  (make-q  (* (q-a y) x)           (* (q-b y) x)           (* (q-c y) x)           (* (q-d y) x))) (defmethod mul ((x quaternion) (y quaternion))  (make-q  (- (* (q-a x) (q-a y)) (* (q-b x) (q-b y)) (* (q-c x) (q-c y)) (* (q-d x) (q-d y)))           (- (+ (* (q-a x) (q-b y)) (* (q-b x) (q-a y)) (* (q-c x) (q-d y))) (* (q-d x) (q-c y)))           (- (+ (* (q-a x) (q-c y)) (* (q-c x) (q-a y)) (* (q-d x) (q-b y))) (* (q-b x) (q-d y)))           (- (+ (* (q-a x) (q-d y)) (* (q-b x) (q-c y)) (* (q-d x) (q-a y))) (* (q-c x) (q-b y))))) (defmethod norm ((x quaternion))  (+ (sqrt (q-a x)) (sqrt (q-b x)) (sqrt (q-c x)) (sqrt (q-d x)))) (defmethod print-object ((x quaternion) stream)  (format stream "[email protected][email protected][email protected][email protected]" (q-a x) (q-b x) (q-c x) (q-d x))) (defvar q (make-q 0 1 0 0))(defvar q1 (make-q 0 0 1 0))(defvar q2 (make-q 0 0 0 1))(defvar r 7)(format t "q+q1+q2 = ~a~&" (reduce #'sum (list q q1 q2)))(format t "r*(q+q1+q2) = ~a~&" (mul r (reduce #'sum (list q q1 q2))))(format t "q*q1*q2 = ~a~&" (reduce #'mul (list q q1 q2)))(format t "q-q1-q2 = ~a~&" (reduce #'sub (list q q1 q2)))
Output:
q+q1+q2 = +0.0+1.0i+1.0j+1.0k
r*(q+q1+q2) = +0.0+7.0i+7.0j+7.0k
q*q1*q2 = -1.0+0.0i+0.0j+0.0k
q-q1-q2 = +0.0+1.0i-1.0j-1.0k


## D

import std.math, std.numeric, std.traits, std.conv, std.complex;  struct Quat(T) if (isFloatingPoint!T) {    alias CT = Complex!T;     union {        struct { T re, i, j, k; } // Default init to NaN.        struct { CT x, y; }        struct { T[4] vector; }    }     string toString() const pure /*nothrow*/ @safe {        return vector.text;    }     @property T norm2() const pure nothrow @safe @nogc { /// Norm squared.        return re ^^ 2 + i ^^ 2 + j ^^ 2 + k ^^ 2;    }     @property T abs() const pure nothrow @safe @nogc { /// Norm.        return sqrt(norm2);    }     @property T arg() const pure nothrow @safe @nogc { /// Theta.        return acos(re / abs); // this may be incorrect...    }     @property Quat!T conj() const pure nothrow @safe @nogc { /// Conjugate.        return Quat!T(re, -i, -j, -k);    }     @property Quat!T recip() const pure nothrow @safe @nogc {  /// Reciprocal.        return Quat!T(re / norm2, -i / norm2, -j / norm2, -k / norm2);    }     @property Quat!T pureim() const pure nothrow @safe @nogc { /// Pure imagery.        return Quat!T(0, i, j, k);    }     @property Quat!T versor() const pure nothrow @safe @nogc { /// Unit versor.        return this / abs;    }     /// Unit versor of imagery part.    @property Quat!T iversor() const pure nothrow @safe @nogc {        return pureim / pureim.abs;    }     /// Assignment.    Quat!T opAssign(U : T)(Quat!U z) pure nothrow @safe @nogc {        x = z.x;  y = z.y;        return this;    }     Quat!T opAssign(U : T)(Complex!U c) pure nothrow @safe @nogc {        x = c;  y = 0;        return this;    }     Quat!T opAssign(U : T)(U r) pure nothrow @safe @nogc    if (isNumeric!U) {        re = r; i = 0; y = 0;        return this;    }     /// Test for equal, not ordered so no opCmp.    bool opEquals(U : T)(Quat!U z) const pure nothrow @safe @nogc {        return re == z.re && i == z.i && j == z.j && k == z.k;    }     bool opEquals(U : T)(Complex!U c) const pure nothrow @safe @nogc {        return re == c.re && i == c.im && j == 0 && k == 0;    }     bool opEquals(U : T)(U r) const pure nothrow @safe @nogc    if (isNumeric!U) {        return re == r && i == 0 && j == 0 && k == 0;    }     /// Unary op.    Quat!T opUnary(string op)() const pure nothrow @safe @nogc    if (op == "+") {        return this;    }     Quat!T opUnary(string op)() const pure nothrow @safe @nogc    if (op == "-") {        return Quat!T(-re, -i, -j, -k);    }     /// Binary op, Quaternion on left of op.    Quat!(CommonType!(T,U)) opBinary(string op, U)(Quat!U z)    const pure nothrow @safe @nogc {        alias typeof(return) C;         static if (op == "+" ) {            return C(re + z.re, i + z.i, j + z.j, k + z.k);        } else static if (op == "-") {            return C(re - z.re, i - z.i, j - z.j, k - z.k);        } else static if (op == "*") {            return C(re * z.re - i * z.i  - j * z.j  - k * z.k,                     re * z.i  + i * z.re + j * z.k  - k * z.j,                     re * z.j  - i * z.k  + j * z.re + k * z.i,                     re * z.k  + i * z.j  - j * z.i  + k * z.re);        } else static if (op == "/") {            return this * z.recip;        }    }     /// Extend complex to quaternion.    Quat!(CommonType!(T,U)) opBinary(string op, U)(Complex!U c)    const pure nothrow @safe @nogc {        return opBinary!op(typeof(return)(c.re, c.im, 0, 0));    }     /// For scalar.    Quat!(CommonType!(T,U)) opBinary(string op, U)(U r)    const pure nothrow @safe @nogc    if (isNumeric!U) {        alias typeof(return) C;         static if (op == "+" ) {            return C(re + r, i, j, k);        } else static if (op == "-") {            return C(re - r, i, j, k);        } else static if (op == "*") {            return C(re * r, i * r, j * r, k * r);        } else static if (op == "/") {            return C(re / r, i / r, j / r, k / r);        } else static if (op == "^^") {            return pow(r);        }    }     /// Power function.    Quat!(CommonType!(T,U)) pow(U)(U r)    const pure nothrow @safe @nogc    if (isNumeric!U) {        return (abs^^r) * exp(r * iversor * arg);    }     /// Handle binary op if Quaternion on right of op and left is    /// not quaternion.    Quat!(CommonType!(T,U)) opBinaryRight(string op, U)(Complex!U c)    const pure nothrow @safe @nogc {        alias typeof(return) C;        auto w = C(c.re, c.im, 0, 0);        return w.opBinary!(op)(this);    }     Quat!(CommonType!(T,U)) opBinaryRight(string op, U)(U r)    const pure nothrow @safe @nogc    if (isNumeric!U) {        alias typeof(return) C;         static if (op == "+" || op == "*") {            return opBinary!op(r);        } else static if (op == "-") {            return C(r - re , -i, -j, -k);        } else static if (op == "/") {            auto w = C(re, i, j, k);            return w.recip * r;        }    }}  HT exp(HT)(HT z) pure nothrow @safe @nogcif (is(HT T == Quat!T)) {    immutable inorm = z.pureim.abs;    return std.math.exp(z.re) * (cos(inorm) + z.iversor * sin(inorm));} HT log(HT)(HT z) pure nothrow @safe @nogcif (is(HT T == Quat!T)) {    return std.math.log(z.abs) + z.iversor * acos(z.re / z.abs);}  void main() @safe { // Demo code.    import std.stdio;     alias QR = Quat!real;    enum real r = 7.0;     immutable QR q  = QR(2, 3, 4, 5),                 q1 = QR(2, 3, 4, 5),                 q2 = QR(3, 4, 5, 6);     writeln("1.             q - norm: ", q.abs);    writeln("2.         q - negative: ", -q);    writeln("3.        q - conjugate: ", q.conj);    writeln("4.                r + q: ", r + q);    writeln("                  q + r: ", q + r);    writeln("5.              q1 + q2: ", q1 + q2);    writeln("6.                r * q: ", r * q);    writeln("                  q * r: ", q * r);    writeln("7.              q1 * q2: ", q1 * q2);    writeln("                q2 * q1: ", q2 * q1);    writeln("8.  q1 * q2 != q2 * Q1 ? ", q1 * q2 != q2 * q1);     immutable QR i = QR(0, 1, 0, 0),                 j = QR(0, 0, 1, 0),                 k = QR(0, 0, 0, 1);    writeln("9.1               i * i: ", i * i);    writeln("                  J * j: ", j * j);    writeln("                  k * k: ", k * k);    writeln("              i * j * k: ", i * j * k);    writeln("9.2             q1 / q2: ", q1 / q2);    writeln("9.3        q1 / q2 * q2: ", q1 / q2 * q2);    writeln("           q2 * q1 / q2: ", q2 * q1 / q2);    writeln("9.4         exp(pi * i): ", exp(PI * i));    writeln("            exp(pi * j): ", exp(PI * j));    writeln("            exp(pi * k): ", exp(PI * k));    writeln("                 exp(q): ", exp(q));    writeln("                 log(q): ", log(q));    writeln("            exp(log(q)): ", exp(log(q)));    writeln("            log(exp(q)): ", log(exp(q)));    immutable s = q.exp.log;    writeln("9.5 let s = log(exp(q)): ", s);    writeln("                 exp(s): ", exp(s));    writeln("                 log(s): ", log(s));    writeln("            exp(log(s)): ", exp(log(s)));    writeln("            log(exp(s)): ", log(exp(s)));}
Output:
1.             q - norm: 7.34847
2.         q - negative: [-2, -3, -4, -5]
3.        q - conjugate: [2, -3, -4, -5]
4.                r + q: [9, 3, 4, 5]
q + r: [9, 3, 4, 5]
5.              q1 + q2: [5, 7, 9, 11]
6.                r * q: [14, 21, 28, 35]
q * r: [14, 21, 28, 35]
7.              q1 * q2: [-56, 16, 24, 26]
q2 * q1: [-56, 18, 20, 28]
8.  q1 * q2 != q2 * Q1 ? true
9.1               i * i: [-1, 0, 0, 0]
J * j: [-1, 0, 0, 0]
k * k: [-1, 0, 0, 0]
i * j * k: [-1, 0, 0, 0]
9.2             q1 / q2: [0.790698, 0.0232558, -1.35525e-20, 0.0465116]
9.3        q1 / q2 * q2: [2, 3, 4, 5]
q2 * q1 / q2: [2, 3.46512, 3.90698, 4.76744]
9.4         exp(pi * i): [-1, -5.42101e-20, -0, -0]
exp(pi * j): [-1, -0, -5.42101e-20, -0]
exp(pi * k): [-1, -0, -0, -5.42101e-20]
exp(q): [5.21186, 2.22222, 2.96296, 3.7037]
log(q): [1.99449, 0.549487, 0.732649, 0.915812]
exp(log(q)): [2, 3, 4, 5]
log(exp(q)): [2, 0.33427, 0.445694, 0.557117]
9.5 let s = log(exp(q)): [2, 0.33427, 0.445694, 0.557117]
exp(s): [5.21186, 2.22222, 2.96296, 3.7037]
log(s): [0.765279, 0.159215, 0.212286, 0.265358]
exp(log(s)): [2, 0.33427, 0.445694, 0.557117]
log(exp(s)): [2, 0.33427, 0.445694, 0.557117]

## Delphi

unit Quaternions; interface type   TQuaternion = record    A, B, C, D: double;     function  Init          (aA, aB, aC, aD : double): TQuaternion;    function  Norm          : double;    function  Conjugate     : TQuaternion;    function  ToString      : string;     class operator Negative (Left : TQuaternion): TQuaternion;    class operator Positive (Left : TQuaternion): TQuaternion;    class operator Add      (Left, Right : TQuaternion): TQuaternion;    class operator Add      (Left : TQuaternion; Right : double): TQuaternion; overload;    class operator Add      (Left : double; Right : TQuaternion): TQuaternion; overload;    class operator Subtract (Left, Right : TQuaternion): TQuaternion;    class operator Multiply (Left, Right : TQuaternion): TQuaternion;    class operator Multiply (Left : TQuaternion; Right : double): TQuaternion; overload;    class operator Multiply (Left : double; Right : TQuaternion): TQuaternion; overload;  end; implementation uses  SysUtils; { TQuaternion } function TQuaternion.Init(aA, aB, aC, aD: double): TQuaternion;begin  A := aA;  B := aB;  C := aC;  D := aD;   result := Self;end; function TQuaternion.Norm: double;begin  result := sqrt(sqr(A) + sqr(B) + sqr(C) + sqr(D));end; function TQuaternion.Conjugate: TQuaternion;begin  result.B := -B;  result.C := -C;  result.D := -D;end; class operator TQuaternion.Negative(Left: TQuaternion): TQuaternion;begin  result.A := -Left.A;  result.B := -Left.B;  result.C := -Left.C;  result.D := -Left.D;end; class operator TQuaternion.Positive(Left: TQuaternion): TQuaternion;begin  result := Left;end; class operator TQuaternion.Add(Left, Right: TQuaternion): TQuaternion;begin  result.A := Left.A + Right.A;  result.B := Left.B + Right.B;  result.C := Left.C + Right.C;  result.D := Left.D + Right.D;end; class operator TQuaternion.Add(Left: TQuaternion; Right: double): TQuaternion;begin  result.A := Left.A + Right;  result.B := Left.B;  result.C := Left.C;  result.D := Left.D;end; class operator TQuaternion.Add(Left: double; Right: TQuaternion): TQuaternion;begin  result.A := Left + Right.A;  result.B := Right.B;  result.C := Right.C;  result.D := Right.D;end; class operator TQuaternion.Subtract(Left, Right: TQuaternion): TQuaternion;begin  result.A := Left.A - Right.A;  result.B := Left.B - Right.B;  result.C := Left.C - Right.C;  result.D := Left.D - Right.D;end; class operator TQuaternion.Multiply(Left, Right: TQuaternion): TQuaternion;begin  result.A := Left.A * Right.A - Left.B * Right.B - Left.C * Right.C - Left.D * Right.D;  result.B := Left.A * Right.B + Left.B * Right.A + Left.C * Right.D - Left.D * Right.C;  result.C := Left.A * Right.C - Left.B * Right.D + Left.C * Right.A + Left.D * Right.B;  result.D := Left.A * Right.D + Left.B * Right.C - Left.C * Right.B + Left.D * Right.A;end; class operator TQuaternion.Multiply(Left: double; Right: TQuaternion): TQuaternion;begin  result.A := Left * Right.A;  result.B := Left * Right.B;  result.C := Left * Right.C;  result.D := Left * Right.D;end; class operator TQuaternion.Multiply(Left: TQuaternion; Right: double): TQuaternion;begin  result.A := Left.A * Right;  result.B := Left.B * Right;  result.C := Left.C * Right;  result.D := Left.D * Right;end; function TQuaternion.ToString: string;begin  result := Format('%f + %fi + %fj + %fk', [A, B, C, D]);end; end.

Test program

## Euphoria

function norm(sequence q)    return sqrt(power(q[1],2)+power(q[2],2)+power(q[3],2)+power(q[4],2))end function function conj(sequence q)    q[2..4] = -q[2..4]    return qend function function add(object q1, object q2)    if atom(q1) != atom(q2) then        if atom(q1) then            q1 = {q1,0,0,0}        else            q2 = {q2,0,0,0}        end if    end if    return q1+q2end function function mul(object q1, object q2)    if sequence(q1) and sequence(q2) then        return { q1[1]*q2[1] - q1[2]*q2[2] - q1[3]*q2[3] - q1[4]*q2[4],                 q1[1]*q2[2] + q1[2]*q2[1] + q1[3]*q2[4] - q1[4]*q2[3],                 q1[1]*q2[3] - q1[2]*q2[4] + q1[3]*q2[1] + q1[4]*q2[2],                 q1[1]*q2[4] + q1[2]*q2[3] - q1[3]*q2[2] + q1[4]*q2[1] }    else        return q1*q2    end ifend function function quats(sequence q)    return sprintf("%g + %gi + %gj + %gk",q)end function constant    q  = {1, 2, 3, 4},    q1 = {2, 3, 4, 5},    q2 = {5, 6, 7, 8},    r  = 7 printf(1, "norm(q) = %g\n", norm(q))printf(1, "-q = %s\n", {quats(-q)})printf(1, "conj(q) = %s\n", {quats(conj(q))})printf(1, "q + r = %s\n", {quats(add(q,r))})printf(1, "q1 + q2 = %s\n", {quats(add(q1,q2))})printf(1, "q1 * q2 = %s\n", {quats(mul(q1,q2))})printf(1, "q2 * q1 = %s\n", {quats(mul(q2,q1))})
Output:
norm(q) = 5.47723
-q = -1 + -2i + -3j + -4k
conj(q) = 1 + -2i + -3j + -4k
q + r = 8 + 2i + 3j + 4k
q1 + q2 = 7 + 9i + 11j + 13k
q1 * q2 = -76 + 24i + 40j + 38k
q2 * q1 = -76 + 30i + 28j + 44k

## F#

Mainly a
Translation of: C#
On the minus side we have no way to define a conversion to Quaternion from any suitable (numeric) type.

On the plus side we can avoid the stuff to make the equality structual (from the referential equality default) by just declaring it as an attribute to the type and let the compiler handle the details.

open System [<Struct; StructuralEquality; NoComparison>]type Quaternion(r : float, i : float, j : float, k : float) =    member this.A = r    member this.B = i    member this.C = j    member this.D = k     new (f : float) = Quaternion(f, 0., 0., 0.)     static member (~-) (q : Quaternion) = Quaternion(-q.A, -q.B, -q.C, -q.D)     static member (+) (q1 : Quaternion, q2 : Quaternion) =        Quaternion(q1.A + q2.A, q1.B + q2.B, q1.C + q2.C, q1.D + q2.D)    static member (+) (q : Quaternion, r : float) = q + Quaternion(r)    static member (+) (r : float, q: Quaternion) = Quaternion(r) + q     static member (*) (q1 : Quaternion, q2 : Quaternion) =        Quaternion(            q1.A * q2.A - q1.B * q2.B - q1.C * q2.C - q1.D * q2.D,            q1.A * q2.B + q1.B * q2.A + q1.C * q2.D - q1.D * q2.C,            q1.A * q2.C - q1.B * q2.D + q1.C * q2.A + q1.D * q2.B,            q1.A * q2.D + q1.B * q2.C - q1.C * q2.B + q1.D * q2.A)    static member (*) (q : Quaternion, r : float) = q * Quaternion(r)    static member (*) (r : float, q: Quaternion) = Quaternion(r) * q     member this.Norm = Math.Sqrt(r * r + i * i + j * j + k * k)     member this.Conjugate = Quaternion(r, -i, -j, -k)     override this.ToString() = sprintf "Q(%f, %f, %f, %f)" r i j k [<EntryPoint>]let main argv =    let q = Quaternion(1., 2., 3., 4.)    let q1 = Quaternion(2., 3., 4., 5.)    let q2 = Quaternion(3., 4., 5., 6.)    let r = 7.     printfn "q = %A" q    printfn "q1 = %A" q1    printfn "q2 = %A" q2    printfn "r = %A" r     printfn "q.Norm = %A" q.Norm    printfn "q1.Norm = %A" q1.Norm    printfn "q2.Norm = %A" q2.Norm     printfn "-q = %A" -q    printfn "q.Conjugate = %A" q.Conjugate     printfn "q + r = %A" (q + (Quaternion r))    printfn "q1 + q2 = %A" (q1 + q2)    printfn "q2 + q1 = %A" (q2 + q1)     printfn "q * r = %A" (q * r)    printfn "q1 * q2 = %A" (q1 * q2)    printfn "q2 * q1 = %A" (q2 * q1)     printfn "q1*q2 %s q2*q1" (if (q1 * q2) = (q2 * q1) then "=" else "<>")    printfn "q %s Q(1.,2.,3.,4.)" (if q = Quaternion(1., 2., 3., 4.) then "=" else "<>")    0
Output:
q = Q(1.000000, 2.000000, 3.000000, 4.000000)
q1 = Q(2.000000, 3.000000, 4.000000, 5.000000)
q2 = Q(3.000000, 4.000000, 5.000000, 6.000000)
r = 7.0
q.Norm = 5.477225575
q1.Norm = 7.348469228
q2.Norm = 9.273618495
-q = Q(-1.000000, -2.000000, -3.000000, -4.000000)
q.Conjugate = Q(1.000000, -2.000000, -3.000000, -4.000000)
q + r = Q(8.000000, 2.000000, 3.000000, 4.000000)
q1 + q2 = Q(5.000000, 7.000000, 9.000000, 11.000000)
q2 + q1 = Q(5.000000, 7.000000, 9.000000, 11.000000)
q * r = Q(7.000000, 14.000000, 21.000000, 28.000000)
q1 * q2 = Q(-56.000000, 16.000000, 24.000000, 26.000000)
q2 * q1 = Q(-56.000000, 18.000000, 20.000000, 28.000000)
q1*q2 <> q2*q1
q = Q(1.,2.,3.,4.)

## Factor

The math.quaternions vocabulary provides words for treating sequences like quaternions. norm and vneg come from the math.vectors vocabulary. Oddly, I wasn't able to find a word for adding a real to a quaternion, so I wrote one.

USING: generalizations io kernel locals math.quaternionsmath.vectors prettyprint sequences ;IN: rosetta-code.quaternion-type : show ( quot -- )    [ unparse 2 tail but-last "= " append write ] [ call . ] bi    ; inline : 2show ( quots -- )    [ 2curry show ] map-compose [ call ] each ; inline : q+n ( q n -- q+n ) n>q q+ ; [let    { 1 2 3 4 } 7 { 2 3 4 5 } { 3 4 5 6 } :> ( q r q1 q2 )    q [ norm ]    q [ vneg ]    q [ qconjugate ]    [ curry show ] [email protected]    {        [ q  r  [ q+n ] ]        [ q  r  [ q*n ] ]        [ q1 q2 [ q+  ] ]        [ q1 q2 [ q*  ] ]        [ q2 q1 [ q*  ] ]    } 2show]
Output:
{ 1 2 3 4 } norm = 5.477225575051661
{ 1 2 3 4 } vneg = { -1 -2 -3 -4 }
{ 1 2 3 4 } qconjugate = { 1 -2 -3 -4 }
{ 1 2 3 4 } 7 q+n = { 8 2 3 4 }
{ 1 2 3 4 } 7 q*n = { 7 14 21 28 }
{ 2 3 4 5 } { 3 4 5 6 } q+ = { 5 7 9 11 }
{ 2 3 4 5 } { 3 4 5 6 } q* = { -56 16 24 26 }
{ 3 4 5 6 } { 2 3 4 5 } q* = { -56 18 20 28 }


## Forth

: quaternions  4 * floats ; : qvariable create 1 quaternions allot ; : q! ( a b c d q -- )  dup 3 floats + f!  dup 2 floats + f!  dup float+ f!  f! ; : qcopy ( src dest -- ) 1 quaternions move ; : qnorm  ( q -- f )  0e 4 0 do  dup [email protected] fdup f* f+  float+ loop drop fsqrt ; : qf* ( q f -- )  4 0 do dup [email protected] fover f* dup f!  float+ loop fdrop drop ; : qnegate ( q -- )  -1e qf* ; : qconj ( q -- )  float+ 3 0 do dup [email protected] fnegate dup f!  float+ loop drop ; : qf+ ( q f -- ) dup [email protected] f+ f! ; : q+ ( q1 q2 -- )  4 0 do over [email protected] dup [email protected] f+ dup f!  float+ swap float+ swap loop 2drop ; \ access: q.a             [email protected] ;: q.b      float+ [email protected] ;: q.c  2 floats + [email protected] ;: q.d  3 floats + [email protected] ; : q* ( dest q1 q2 -- )  over q.a dup q.d f*  over q.b dup q.c f* f+  over q.c dup q.b f* f-  over q.d dup q.a f* f+  over q.a dup q.c f*  over q.b dup q.d f* f-  over q.c dup q.a f* f+  over q.d dup q.b f* f+  over q.a dup q.b f*  over q.b dup q.a f* f+  over q.c dup q.d f* f+  over q.d dup q.c f* f-  over q.a dup q.a f*  over q.b dup q.b f* f-  over q.c dup q.c f* f-  over q.d dup q.d f* f-  2drop  4 0 do dup f!  float+ loop  drop ; : q= ( q1 q2 -- ? )  4 0 do    over [email protected] dup [email protected] f<> if 2drop false unloop exit then    float+ swap float+  loop  2drop true ; \ testing : q. ( q -- )  [char] ( emit space  4 0 do dup [email protected] f.  float+ loop drop  [char] ) emit space ; qvariable q   1e 2e 3e 4e q  q!qvariable q1  2e 3e 4e 5e q1 q!create q2     3e f, 4e f, 5e f, 6e f,	\ by hand qvariable tmpqvariable m1qvariable m2 q qnorm f.				\ 5.47722557505166q tmp qcopy  tmp qnegate  tmp q.	\ ( -1. -2. -3. -4. )q tmp qcopy  tmp qconj    tmp q.	\ ( 1. -2. -3. -4. ) q m1 qcopy  m1 7e qf+   m1 q.		\ ( 8. 2. 3. 4. )q m2 qcopy  7e m2 qf+   m2 q.		\ ( 8. 2. 3. 4. )m1 m2 q= .				\ -1  (true) q2 tmp qcopy  q1 tmp q+   tmp q.	\ ( 5. 7. 9. 11. ) q m1 qcopy  m1 7e qf*     m1 q.		\ ( 7. 14. 21. 28. )q m2 qcopy  7e m2 qf*     m2 q.		\ ( 7. 14. 21. 28. )m1 m2 q= .				\ -1  (true) m1 q1 q2 q*  m1 q.			\ ( -56. 16. 24. 26. )m2 q2 q1 q*  m2 q.			\ ( -56. 18. 20. 28. )m1 m2 q= .				\ 0  (false)

## Fortran

Works with: Fortran version 90 and later
module Q_mod  implicit none   type quaternion    real :: a, b, c, d  end type   public :: norm, neg, conj  public :: operator (+)  public :: operator (*)   private ::  q_plus_q, q_plus_r, r_plus_q, &              q_mult_q, q_mult_r, r_mult_q, &              norm_q, neg_q, conj_q   interface norm    module procedure norm_q  end interface   interface neg    module procedure neg_q  end interface   interface conj    module procedure conj_q  end interface   interface operator (+)    module procedure q_plus_q, q_plus_r, r_plus_q  end interface   interface operator (*)    module procedure q_mult_q, q_mult_r, r_mult_q  end interface contains function norm_q(x) result(res)  real :: res  type (quaternion), intent (in) :: x   res = sqrt(x%a*x%a + x%b*x%b + x%c*x%c + x%d*x%d) end function norm_q function neg_q(x) result(res)  type (quaternion) :: res  type (quaternion), intent (in) :: x   res%a = -x%a  res%b = -x%b  res%c = -x%c  res%d = -x%d end function neg_q function conj_q(x) result(res)  type (quaternion) :: res  type (quaternion), intent (in) :: x   res%a = x%a  res%b = -x%b  res%c = -x%c  res%d = -x%d end function conj_q function q_plus_q(x, y) result (res)  type (quaternion) :: res  type (quaternion), intent (in) :: x, y   res%a = x%a + y%a  res%b = x%b + y%b  res%c = x%c + y%c  res%d = x%d + y%d end function q_plus_q function q_plus_r(x, r) result (res)  type (quaternion) :: res   type (quaternion), intent (in) :: x  real, intent(in) :: r    res = x   res%a = x%a + r end function q_plus_r function r_plus_q(r, x) result (res)  type (quaternion) :: res   type (quaternion), intent (in) :: x  real, intent(in) :: r    res = x   res%a = x%a + r end function r_plus_q function q_mult_q(x, y) result (res)  type (quaternion) :: res   type (quaternion), intent (in) :: x, y    res%a = x%a*y%a - x%b*y%b - x%c*y%c - x%d*y%d   res%b = x%a*y%b + x%b*y%a + x%c*y%d - x%d*y%c   res%c = x%a*y%c - x%b*y%d + x%c*y%a + x%d*y%b   res%d = x%a*y%d + x%b*y%c - x%c*y%b + x%d*y%a end function q_mult_q function q_mult_r(x, r) result (res)  type (quaternion) :: res   type (quaternion), intent (in) :: x  real, intent(in) ::  r    res%a = x%a*r   res%b = x%b*r   res%c = x%c*r     res%d = x%d*r  end function q_mult_r function r_mult_q(r, x) result (res)  type (quaternion) :: res   type (quaternion), intent (in) :: x  real, intent(in) ::  r    res%a = x%a*r   res%b = x%b*r   res%c = x%c*r     res%d = x%d*r  end function r_mult_qend module Q_mod program Quaternions  use Q_mod  implicit none   real :: r = 7.0  type(quaternion) :: q, q1, q2   q  = quaternion(1, 2, 3, 4)  q1 = quaternion(2, 3, 4, 5)  q2 = quaternion(3, 4, 5, 6)   write(*, "(a, 4f8.3)") "             q = ", q  write(*, "(a, 4f8.3)") "            q1 = ", q1  write(*, "(a, 4f8.3)") "            q2 = ", q2  write(*, "(a, f8.3)")  "             r = ", r  write(*, "(a, f8.3)")  "     Norm of q = ", norm(q)   write(*, "(a, 4f8.3)") " Negative of q = ", neg(q)  write(*, "(a, 4f8.3)") "Conjugate of q = ", conj(q)   write(*, "(a, 4f8.3)") "         q + r = ", q + r  write(*, "(a, 4f8.3)") "         r + q = ", r + q  write(*, "(a, 4f8.3)") "       q1 + q2 = ", q1 + q2  write(*, "(a, 4f8.3)") "         q * r = ", q * r    write(*, "(a, 4f8.3)") "         r * q = ", r * q   write(*, "(a, 4f8.3)") "       q1 * q2 = ", q1 * q2   write(*, "(a, 4f8.3)") "       q2 * q1 = ", q2 * q1  end program
Output:
             q =    1.000   2.000   3.000   4.000
q1 =    2.000   3.000   4.000   5.000
q2 =    3.000   4.000   5.000   6.000
r =    7.000
Norm of q =    5.477
Negative of q =   -1.000  -2.000  -3.000  -4.000
Conjugate of q =    1.000  -2.000  -3.000  -4.000
q + r =    8.000   2.000   3.000   4.000
r + q =    8.000   2.000   3.000   4.000
q1 + q2 =    5.000   7.000   9.000  11.000
q * r =    7.000  14.000  21.000  28.000
r * q =    7.000  14.000  21.000  28.000
q1 * q2 =  -56.000  16.000  24.000  26.000
q2 * q1 =  -56.000  18.000  20.000  28.000

## GAP

# GAP has built-in support for quaternions A := QuaternionAlgebra(Rationals);# <algebra-with-one of dimension 4 over Rationals> b := BasisVectors(Basis(A));# [ e, i, j, k ] q := [1, 2, 3, 4]*b;# e+(2)*i+(3)*j+(4)*k # ConjugateComplexConjugate(q);# e+(-2)*i+(-3)*j+(-4)*k # Division1/q;# (1/30)*e+(-1/15)*i+(-1/10)*j+(-2/15)*k # Computing norm may be difficult, since the result would be in a quadratic field.# Sqrt exists in GAP, but it is quite unusual: see ?E in GAP documentation, and the following exampleSqrt(5/3);# 1/3*E(60)^7+1/3*E(60)^11-1/3*E(60)^19-1/3*E(60)^23-1/3*E(60)^31+1/3*E(60)^43-1/3*E(60)^47+1/3*E(60)^59 # However, the square of the norm is easy to computeq*ComplexConjugate(q);# (30)*e q1 := [2, 3, 4, 5]*b;# (2)*e+(3)*i+(4)*j+(5)*k q2 := [3, 4, 5, 6]*b;# (3)*e+(4)*i+(5)*j+(6)*k q1*q2 - q2*q1;# (-2)*i+(4)*j+(-2)*k # Can't add directly to a rational, one must make a quaternion of itr := 5/3*b[1];# (5/3)*er + q;# (8/3)*e+(2)*i+(3)*j+(4)*k # For multiplication, no problem (we are in an algebra over rationals !)r*q;# (5/3)*e+(10/3)*i+(5)*j+(20/3)*k5/3*q;# (5/3)*e+(10/3)*i+(5)*j+(20/3)*k # Negative-q;(-1)*e+(-2)*i+(-3)*j+(-4)*k  # While quaternions are built-in, you can define an algebra in GAP by specifying it's multiplication table.# See tutorial, p. 60, and reference of the functions used below. # A multiplication table of dimension 4. T := EmptySCTable(4, 0);SetEntrySCTable(T, 1, 1, [1, 1]);SetEntrySCTable(T, 1, 2, [1, 2]);SetEntrySCTable(T, 1, 3, [1, 3]);SetEntrySCTable(T, 1, 4, [1, 4]);SetEntrySCTable(T, 2, 1, [1, 2]);SetEntrySCTable(T, 2, 2, [-1, 1]);SetEntrySCTable(T, 2, 3, [1, 4]);SetEntrySCTable(T, 2, 4, [-1, 3]);SetEntrySCTable(T, 3, 1, [1, 3]);SetEntrySCTable(T, 3, 2, [-1, 4]);SetEntrySCTable(T, 3, 3, [-1, 1]);SetEntrySCTable(T, 3, 4, [1, 2]);SetEntrySCTable(T, 4, 1, [1, 4]);SetEntrySCTable(T, 4, 2, [1, 3]);SetEntrySCTable(T, 4, 3, [-1, 2]);SetEntrySCTable(T, 4, 4, [-1, 1]); A := AlgebraByStructureConstants(Rationals, T, ["e", "i", "j", "k"]);b := GeneratorsOfAlgebra(A); IsAssociative(A);# true IsCommutative(A);# false # Then, like above q := [1, 2, 3, 4]*b;# e+(2)*i+(3)*j+(4)*k # However, as is, GAP does not know division or conjugate on this algebra.# QuaternionAlgebra is useful as well for extensions of rationals,# and this one _has_ conjugate and division, as seen previously. # Try this on Q[z] where z is the square root of 5 (in GAP it's ER(5))F := FieldByGenerators([ER(5)]);A := QuaternionAlgebra(F);b := GeneratorsOfAlgebra(A); q := [1, 2, 3, 4]*b;# e+(2)*i+(3)*j+(4)*k # Conjugate and division ComplexConjugate(q);# e+(-2)*i+(-3)*j+(-4)*k 1/q;# (1/30)*e+(-1/15)*i+(-1/10)*j+(-2/15)*k

## Go

Conventions for method receiver, parameter, and return values modeled after Go's big number package. It provides flexibility without requiring unnecessary object creation. The test program creates only four quaternion objects, the three inputs and one more for an output. The three inputs are reused repeatedly without being modified. The output is also reused repeatedly, being overwritten for each operation.

package main import (    "fmt"    "math") type qtn struct {    r, i, j, k float64} var (    q  = &qtn{1, 2, 3, 4}    q1 = &qtn{2, 3, 4, 5}    q2 = &qtn{3, 4, 5, 6}     r  float64 = 7) func main() {    fmt.Println("Inputs")    fmt.Println("q:", q)    fmt.Println("q1:", q1)    fmt.Println("q2:", q2)    fmt.Println("r:", r)     var qr qtn    fmt.Println("\nFunctions")    fmt.Println("q.norm():", q.norm())    fmt.Println("neg(q):", qr.neg(q))    fmt.Println("conj(q):", qr.conj(q))    fmt.Println("addF(q, r):", qr.addF(q, r))    fmt.Println("addQ(q1, q2):", qr.addQ(q1, q2))    fmt.Println("mulF(q, r):", qr.mulF(q, r))    fmt.Println("mulQ(q1, q2):", qr.mulQ(q1, q2))    fmt.Println("mulQ(q2, q1):", qr.mulQ(q2, q1))} func (q *qtn) String() string {    return fmt.Sprintf("(%g, %g, %g, %g)", q.r, q.i, q.j, q.k)} func (q *qtn) norm() float64 {    return math.Sqrt(q.r*q.r + q.i*q.i + q.j*q.j + q.k*q.k)} func (z *qtn) neg(q *qtn) *qtn {    z.r, z.i, z.j, z.k = -q.r, -q.i, -q.j, -q.k    return z} func (z *qtn) conj(q *qtn) *qtn {    z.r, z.i, z.j, z.k = q.r, -q.i, -q.j, -q.k    return z} func (z *qtn) addF(q *qtn, r float64) *qtn {    z.r, z.i, z.j, z.k = q.r+r, q.i, q.j, q.k    return z} func (z *qtn) addQ(q1, q2 *qtn) *qtn {    z.r, z.i, z.j, z.k = q1.r+q2.r, q1.i+q2.i, q1.j+q2.j, q1.k+q2.k    return z} func (z *qtn) mulF(q *qtn, r float64) *qtn {    z.r, z.i, z.j, z.k = q.r*r, q.i*r, q.j*r, q.k*r    return z} func (z *qtn) mulQ(q1, q2 *qtn) *qtn {    z.r, z.i, z.j, z.k =        q1.r*q2.r-q1.i*q2.i-q1.j*q2.j-q1.k*q2.k,        q1.r*q2.i+q1.i*q2.r+q1.j*q2.k-q1.k*q2.j,        q1.r*q2.j-q1.i*q2.k+q1.j*q2.r+q1.k*q2.i,        q1.r*q2.k+q1.i*q2.j-q1.j*q2.i+q1.k*q2.r    return z}
Output:
Inputs
q: (1, 2, 3, 4)
q1: (2, 3, 4, 5)
q2: (3, 4, 5, 6)
r: 7

Functions
q.norm(): 5.477225575051661
neg(q): (-1, -2, -3, -4)
conj(q): (1, -2, -3, -4)
addF(q, r): (8, 2, 3, 4)
addQ(q1, q2): (5, 7, 9, 11)
mulF(q, r): (7, 14, 21, 28)
mulQ(q1, q2): (-56, 16, 24, 26)
mulQ(q2, q1): (-56, 18, 20, 28)


## Julia

https://github.com/andrioni/Quaternions.jl/blob/master/src/Quaternions.jl has a more complete implementation. This is derived from the quaternion example file included with Julia 0.2, which implements a quaternion type complete with arithmetic, type conversions / promotion rules, polymorphism over arbitrary real numeric types, and pretty-printing.

import Base: convert, promote_rule, show, conj, abs, +, -, * immutable Quaternion{T<:Real} <: Number    q0::T    q1::T    q2::T    q3::Tend Quaternion(q0::Real,q1::Real,q2::Real,q3::Real) = Quaternion(promote(q0,q1,q2,q3)...) convert{T}(::Type{Quaternion{T}}, x::Real) =    Quaternion(convert(T,x), zero(T), zero(T), zero(T))convert{T}(::Type{Quaternion{T}}, z::Complex) =    Quaternion(convert(T,real(z)), convert(T,imag(z)), zero(T), zero(T))convert{T}(::Type{Quaternion{T}}, z::Quaternion) =    Quaternion(convert(T,z.q0), convert(T,z.q1), convert(T,z.q2), convert(T,z.q3)) promote_rule{T,S}(::Type{Complex{T}}, ::Type{Quaternion{S}}) = Quaternion{promote_type(T,S)}promote_rule{T<:Real,S}(::Type{T}, ::Type{Quaternion{S}}) = Quaternion{promote_type(T,S)}promote_rule{T,S}(::Type{Quaternion{T}}, ::Type{Quaternion{S}}) = Quaternion{promote_type(T,S)} function show(io::IO, z::Quaternion)    pm(x) = x <	0 ? " - $(-x)" : " +$x"    print(io, z.q0, pm(z.q1), "i", pm(z.q2), "j", pm(z.q3), "k")end conj(z::Quaternion) = Quaternion(z.q0, -z.q1, -z.q2, -z.q3)abs(z::Quaternion) = sqrt(z.q0*z.q0 + z.q1*z.q1 + z.q2*z.q2 + z.q3*z.q3) (-)(z::Quaternion) = Quaternion(-z.q0, -z.q1, -z.q2, -z.q3) (+)(z::Quaternion, w::Quaternion) = Quaternion(z.q0 + w.q0, z.q1 + w.q1,                                               z.q2 + w.q2, z.q3 + w.q3)(-)(z::Quaternion, w::Quaternion) = Quaternion(z.q0 - w.q0, z.q1 - w.q1,                                               z.q2 - w.q2, z.q3 - w.q3)(*)(z::Quaternion, w::Quaternion) = Quaternion(z.q0*w.q0 - z.q1*w.q1 - z.q2*w.q2 - z.q3*w.q3,                                               z.q0*w.q1 + z.q1*w.q0 + z.q2*w.q3 - z.q3*w.q2,                                               z.q0*w.q2 - z.q1*w.q3 + z.q2*w.q0 + z.q3*w.q1,                                               z.q0*w.q3 + z.q1*w.q2 - z.q2*w.q1 + z.q3*w.q0)

Example usage and output:

julia> q = Quaternion(1,0,0,0)julia> q  = Quaternion (1, 2, 3, 4)       q1 = Quaternion(2, 3, 4, 5)       q2 = Quaternion(3, 4, 5, 6)       r = 7. julia> norm(q)5.477225575051661 julia> -q-1 - 2i - 3j - 4k julia> conj(q)1 - 2i - 3j - 4k julia> r + q, q + r(8.0 + 2.0i + 3.0j + 4.0k,8.0 + 2.0i + 3.0j + 4.0k) julia> q1 + q25 + 7i + 9j + 11k julia> r*q, q*r(7.0 + 14.0i + 21.0j + 28.0k,7.0 + 14.0i + 21.0j + 28.0k) julia> q1*q2, q2*q1, q1*q2 != q2*q1(-56 + 16i + 24j + 26k,-56 + 18i + 20j + 28k,true)

// version 1.1.2 data class Quaternion(val a: Double, val b: Double, val c: Double, val d: Double) {    operator fun plus(other: Quaternion): Quaternion {        return Quaternion (this.a + other.a, this.b + other.b,                           this.c + other.c, this.d + other.d)    }     operator fun plus(r: Double) = Quaternion(a + r, b, c, d)     operator fun times(other: Quaternion): Quaternion {        return Quaternion(            this.a * other.a - this.b * other.b - this.c * other.c - this.d * other.d,            this.a * other.b + this.b * other.a + this.c * other.d - this.d * other.c,            this.a * other.c - this.b * other.d + this.c * other.a + this.d * other.b,            this.a * other.d + this.b * other.c - this.c * other.b + this.d * other.a        )    }     operator fun times(r: Double) = Quaternion(a * r, b * r, c * r, d * r)     operator fun unaryMinus() =  Quaternion(-a, -b, -c, -d)     fun conj() = Quaternion(a, -b, -c, -d)     fun norm() = Math.sqrt(a * a + b * b + c * c + d * d)     override fun toString() = "($a,$b, $c,$d)"} // extension functions for Double typeoperator fun Double.plus(q: Quaternion) = q + thisoperator fun Double.times(q: Quaternion) = q * this fun main(args: Array<String>) {    val q  = Quaternion(1.0, 2.0, 3.0, 4.0)    val q1 = Quaternion(2.0, 3.0, 4.0, 5.0)    val q2 = Quaternion(3.0, 4.0, 5.0, 6.0)    val r  = 7.0    println("q  = $q") println("q1 =$q1")    println("q2 = $q2") println("r =$r\n")    println("norm(q) = ${"%f".format(q.norm())}") println("-q =${-q}")    println("conj(q) = ${q.conj()}\n") println("r + q =${r + q}")    println("q  + r  = ${q + r}") println("q1 + q2 =${q1 + q2}\n")    println("r  * q  = ${r * q}") println("q * r =${q * r}")    val q3 = q1 * q2    val q4 = q2 * q1    println("q1 * q2 = $q3") println("q2 * q1 =$q4\n")    println("q1 * q2 != q2 * q1 = ${q3 != q4}")} Output: q = (1.0, 2.0, 3.0, 4.0) q1 = (2.0, 3.0, 4.0, 5.0) q2 = (3.0, 4.0, 5.0, 6.0) r = 7.0 norm(q) = 5.477226 -q = (-1.0, -2.0, -3.0, -4.0) conj(q) = (1.0, -2.0, -3.0, -4.0) r + q = (8.0, 2.0, 3.0, 4.0) q + r = (8.0, 2.0, 3.0, 4.0) q1 + q2 = (5.0, 7.0, 9.0, 11.0) r * q = (7.0, 14.0, 21.0, 28.0) q * r = (7.0, 14.0, 21.0, 28.0) q1 * q2 = (-56.0, 16.0, 24.0, 26.0) q2 * q1 = (-56.0, 18.0, 20.0, 28.0) q1 * q2 != q2 * q1 = true  ## Liberty BASIC Quaternions saved as a space-separated string of four numbers.  q$ = q$( 1 , 2 , 3 , 4 )q1$ = q$( 2 , 3 , 4 , 5 )q2$ = q$( 3 , 4 , 5 , 6 ) real = 7 print "q = " ; q$print "q1 = " ; q1$print "q2 = " ; q2$  print "real = " ; real print "length /norm q  = " ; length( q$) ' =norm norm of qprint "negative (-q1) = " ; negative$( q1$) ' =negative negated q1print "conjugate q = " ; conjugate$( q$) ' conjugate conjugate qprint "real + q = " ; add1$( q$, real ) ' real +quaternion real +qprint "q + q2 = " ; add2$( q$, q2$ )          '   sum two quaternions          q +q2print "real * q        = " ; multiply1$( q$ , real )    '   real *quaternion             real *qprint "q1 * q2         = " ; multiply2$( q1$ , q2$) ' product of two quaternions q1 & q2print "q2 * q1 = " ; multiply2$( q2$, q1$ )    '   show q1 *q2 <> q2 *q1 end function q$( r , i , j , k ) q$ = str$( r); " "; str$( i); " "; str$( j); " "; str$( k)end function function length( q$) r = val( word$( q$, 1 ) ) i = val( word$( q$, 2 ) ) j = val( word$( q$, 3 ) ) k = val( word$( q$, 4 ) ) length =sqr( r^2 +i^2 +j^2 +k^2)end function function multiply1$( q$, d ) r = val( word$( q$, 1 ) ) i = val( word$( q$, 2 ) ) j = val( word$( q$, 3 ) ) k = val( word$( q$, 4 ) ) multiply1$ =q$( r*d, i*d, j*d, k*d)end function function multiply2$( q$, b$ )  ar = val( word$( q$ , 1 ) )   'a1  ai = val( word$( q$ , 2 ) )   'b1  aj = val( word$( q$ , 3 ) )   'c1  ak = val( word$( q$ , 4 ) )   'd1   br = val( word$( b$ , 1 ) )   'a2  bi = val( word$( b$ , 2 ) )   'b2  bj = val( word$( b$ , 3 ) )   'c2  bk = val( word$( b$ , 4 ) )   'd2   multiply2$=q$( _  ar *br_  +( 0 -ai) *bi_  +( 0 -aj) *bj_  +( 0 -ak) *bk _  ,_  ar *bi_  +ai *br_  +aj *bk_  +( 0 -ak) *bj_  ,_  ar *bj_  +( 0 -ai) *bk_  +aj *br_  +ak *bi_  ,_  ar *bk_  +ai *bj_  +( 0 -aj) *bi_  +ak *br )end function function negative$( q$ )  r = val( word$( q$ , 1 ) )  i = val( word$( q$ , 2 ) )  j = val( word$( q$ , 3 ) )  k = val( word$( q$ , 4 ) )  negative$=q$( 0-r, 0-i, 0-j, 0-k)end function function conjugate$( q$ )  r = val( word$( q$ , 1 ) )  i = val( word$( q$ , 2 ) )  j = val( word$( q$ , 3 ) )  k = val( word$( q$ , 4 ) )  conjugate$=q$( r, 0-i, 0-j, 0-k)end function function add1$( q$ , real )  r = val( word$( q$ , 1 ) )  i = val( word$( q$ , 2 ) )  j = val( word$( q$ , 3 ) )  k = val( word$( q$ , 4 ) )  add1$=q$( r +real, i, j, k)end function function add2$( q$ , b$) ar = val( word$( q$, 1 ) ) ai = val( word$( q$, 2 ) ) aj = val( word$( q$, 3 ) ) ak = val( word$( q$, 4 ) ) br = val( word$( b$, 1 ) ) bi = val( word$( b$, 2 ) ) bj = val( word$( b$, 3 ) ) bk = val( word$( b$, 4 ) ) add2$ =q$( ar +br, ai +bi, aj +bj, ak +bk)end function  ## Lua Quaternion = {} function Quaternion.new( a, b, c, d ) local q = { a = a or 1, b = b or 0, c = c or 0, d = d or 0 } local metatab = {} setmetatable( q, metatab ) metatab.__add = Quaternion.add metatab.__sub = Quaternion.sub metatab.__unm = Quaternion.unm metatab.__mul = Quaternion.mul return qend function Quaternion.add( p, q ) if type( p ) == "number" then return Quaternion.new( p+q.a, q.b, q.c, q.d ) elseif type( q ) == "number" then return Quaternion.new( p.a+q, p.b, p.c, p.d ) else return Quaternion.new( p.a+q.a, p.b+q.b, p.c+q.c, p.d+q.d ) endend function Quaternion.sub( p, q ) if type( p ) == "number" then return Quaternion.new( p-q.a, q.b, q.c, q.d ) elseif type( q ) == "number" then return Quaternion.new( p.a-q, p.b, p.c, p.d ) else return Quaternion.new( p.a-q.a, p.b-q.b, p.c-q.c, p.d-q.d ) endend function Quaternion.unm( p ) return Quaternion.new( -p.a, -p.b, -p.c, -p.d )end function Quaternion.mul( p, q ) if type( p ) == "number" then return Quaternion.new( p*q.a, p*q.b, p*q.c, p*q.d ) elseif type( q ) == "number" then return Quaternion.new( p.a*q, p.b*q, p.c*q, p.d*q ) else return Quaternion.new( p.a*q.a - p.b*q.b - p.c*q.c - p.d*q.d, p.a*q.b + p.b*q.a + p.c*q.d - p.d*q.c, p.a*q.c - p.b*q.d + p.c*q.a + p.d*q.b, p.a*q.d + p.b*q.c - p.c*q.b + p.d*q.a ) endend function Quaternion.conj( p ) return Quaternion.new( p.a, -p.b, -p.c, -p.d )end function Quaternion.norm( p ) return math.sqrt( p.a^2 + p.b^2 + p.c^2 + p.d^2 )end function Quaternion.print( p ) print( string.format( "%f + %fi + %fj + %fk\n", p.a, p.b, p.c, p.d ) )end Examples: q1 = Quaternion.new( 1, 2, 3, 4 )q2 = Quaternion.new( 5, 6, 7, 8 )r = 12 print( "norm(q1) = ", Quaternion.norm( q1 ) )io.write( "-q1 = " ); Quaternion.print( -q1 )io.write( "conj(q1) = " ); Quaternion.print( Quaternion.conj( q1 ) )io.write( "r+q1 = " ); Quaternion.print( r+q1 )io.write( "q1+r = " ); Quaternion.print( q1+r )io.write( "r*q1 = " ); Quaternion.print( r*q1 )io.write( "q1*r = " ); Quaternion.print( q1*r )io.write( "q1*q2 = " ); Quaternion.print( q1*q2 )io.write( "q2*q1 = " ); Quaternion.print( q2*q1 ){{out}}norm(q1) = 5.4772255750517-q1 = -1.000000 -2.000000i -3.000000j -4.000000kconj(q1) = 1.000000 -2.000000i -3.000000j -4.000000kr+q1 = 13.000000 + 2.000000i + 3.000000j + 4.000000kq1+r = 13.000000 + 2.000000i + 3.000000j + 4.000000kr*q1 = 12.000000 + 24.000000i + 36.000000j + 48.000000kq1*r = 12.000000 + 24.000000i + 36.000000j + 48.000000kq1*q2 = -60.000000 + 12.000000i + 30.000000j + 24.000000kq2*q1 = -60.000000 + 20.000000i + 14.000000j + 32.000000k ## M2000 Interpreter We can define Quaternions using a class, using operators for specific tasks, as negate, add, multiplication and equality with rounding to 13 decimal place (thats what doing "==" operator for doubles)  Module CheckIt { class Quaternion { \\ by default are double a,b,c,d Property ToString$ {                  Value {                      link parent a,b,c, d to a,b,c,d                         value$=format$("{0} + {1}i + {2}j + {3}k",a,b,c,d)                  }            }            Property Norm { Value}            Operator "==" {                  read n                  push .a==n.a and .b==n.b and .c==n.c and .d==n.d            }            Module CalcNorm {                  .[Norm]<=sqrt(.a**2+.b**2+.c**2+.d**2)             }            Operator Unary {                  .a-! : .b-! : .c-! :.d-!            }            Function Conj {                  q=this                   for q {                         .b-! : .c-! :.d-!                  }                  =q            }            Function Add {                  q=this                  for q {                         .a+=Number : .CalcNorm                  }                  =q            }            Operator "+"  {                  Read q2                  For this, q2 {                        .a+=..a :.b+=..b:.c+=..c:.d+=..d                        .CalcNorm                  }            }            Function Mul(r)  {                  q=this                   for q {                        .a*=r:.b*=r:.c*=r:.d*=r:.CalcNorm                  }                  =q            }            Operator "*" {                  Read q2                  For This, q2 {                        Push .a*..a-.b*..b-.c*..c-.d*..d                        Push .a*..b+.b*..a+.c*..d-.d*..c                        Push .a*..c-.b*..d+.c*..a+.d*..b                        .d<=.a*..d+.b*..c-.c*..b+.d*..a                        Read .c, .b, .a                        .CalcNorm                   }            }                  class:            module Quaternion {                  if match("NNNN") then {                        Read .a,.b,.c,.d                       .CalcNorm                  }            }      }      \\ variables      r=7      q=Quaternion(1,2,3,4)      q1=Quaternion(2,3,4,5)      q2=Quaternion(3,4,5,6)       \\ perform negate, conjugate, multiply by real, add a real, multiply quanterions, multiply in reverse order      qneg=-q      qconj=q.conj()      qmul=q.Mul(r)      qadd=q.Add(r)      q1q2=q1*q2      q2q1=q2*q1       Print "q = ";q.ToString$Print "Normal q = ";q.Norm Print "Neg q = ";qneg.ToString$      Print "Conj q = ";qconj.ToString$Print "Mul q 7 = ";qmul.ToString$      Print "Add q 7 = ";qadd.ToString$Print "q1 = ";q1.ToString$      Print "q2 = ";q2.ToString$Print "q1 * q2 = ";q1q2.ToString$      Print "q2 * q1 = ";q2q1.ToString$Print q1==q1 ' true Print q1q2==q2q1 ' false \\ multiplication and equality in one expression Print (q1 * q2 == q2 * q1)=false Print (q1 * q2 == q1 * q2)=True }CheckIt  Output: q = 1 + 2i + 3j + 4k Normal q = 5.47722557505166 Neg q = -1 + -2i + -3j + -4k Conj q = 1 + -2i + -3j + -4k Mul q 7 = 7 + 14i + 21j + 28k Add q 7 = 8 + 2i + 3j + 4k q1 = 2 + 3i + 4j + 5k q2 = 3 + 4i + 5j + 6k q1 * q2 = -56 + 16i + 24j + 26k q2 * q1 = -56 + 18i + 20j + 28k True False True True ## Mathematica <<Quaternionsq=Quaternion[1,2,3,4]q1=Quaternion[2,3,4,5]q2=Quaternion[3,4,5,6]r=7->Quaternion[1,2,3,4]->Quaternion[2,3,4,5]->Quaternion[3,4,5,6]->7 Abs[q]->√30-q->Quaternion[-1,-2,-3,-4]Conjugate[q]->Quaternion[1,-2,-3,-4]r+q->Quaternion[8,2,3,4]q+r->Quaternion[8,2,3,4]q1+q2->Quaternion[5,7,9,11]q*r->Quaternion[7,14,21,28]r*q->Quaternion[7,14,21,28]q1**q2->Quaternion[-56,16,24,26]q2**q1->Quaternion[-56,18,20,28]  ## Mercury A possible implementation of quaternions in Mercury (the simplest representation) would look like this. Note that this is a full module implementation, complete with boilerplate, and that it works by giving an explicit conversion function for floats, converting a float into a quaternion representation of that float. Thus the float value 7.0 gets turned into the quaternion representation q(7.0, 0.0, 0.0, 0.0) through the function call r(7.0). :- module quaternion. :- interface. :- import_module float. :- type quaternion ---> q( w :: float, i :: float, j :: float, k :: float ). % conversion:- func r(float) = quaternion is det. % operations:- func norm(quaternion) = float is det.:- func -quaternion = quaternion is det.:- func conjugate(quaternion) = quaternion is det.:- func quaternion + quaternion = quaternion is det.:- func quaternion * quaternion = quaternion is det. :- implementation. :- import_module math. % conversionr(W) = q(W, 0.0, 0.0, 0.0). % operationsnorm(q(W, I, J, K)) = math.sqrt(W*W + I*I + J*J + K*K).-q(W, I, J, K) = q(-W, -I, -J, -K).conjugate(q(W, I, J, K)) = q(W, -I, -J, -K).q(W0, I0, J0, K0) + q(W1, I1, J1, K1) = q(W0+W1, I0+I1, J0+J1, K0+K1).q(W0, I0, J0, K0) * q(W1, I1, J1, K1) = q(W0*W1 - I0*I1 - J0*J1 - K0*K1, W0*I1 + I0*W1 + J0*K1 - K0*J1, W0*J1 - I0*K1 + J0*W1 + K0*I1, W0*K1 + I0*J1 - J0*I1 + K0*W1 ). The following test module puts the module through its paces. :- module test_quaternion. :- interface. :- import_module io. :- pred main(io::di, io::uo) is det. :- implementation. :- import_module quaternion. :- import_module exception.:- import_module float.:- import_module list.:- import_module string. :- func to_string(quaternion) = string is det. main(!IO) :- Q = q(1.0, 2.0, 3.0, 4.0), Q1 = q(2.0, 3.0, 4.0, 5.0), Q2 = q(3.0, 4.0, 5.0, 6.0), R = 7.0, QR = r(R), io.print("Q = ", !IO), io.print(to_string(Q), !IO), io.nl(!IO), io.print("Q1 = ", !IO), io.print(to_string(Q1), !IO), io.nl(!IO), io.print("Q2 = ", !IO), io.print(to_string(Q2), !IO), io.nl(!IO), io.print("R = ", !IO), io.print(R, !IO), io.nl(!IO), io.nl(!IO), io.print("1. The norm of a quaternion.\n", !IO), io.print("norm(Q) = ", !IO), io.print(norm(Q), !IO), io.nl(!IO), io.nl(!IO), io.print("2. The negative of a quaternion.\n", !IO), io.print("-Q = ", !IO), io.print(to_string(-Q), !IO), io.nl(!IO), io.nl(!IO), io.print("3. The conjugate of a quaternion.\n", !IO), io.print("conjugate(Q) = ", !IO), io.print(to_string(conjugate(Q)), !IO), io.nl(!IO), io.nl(!IO), io.print("4. Addition of a real number and a quaternion.\n", !IO), ( Q + QR = QR + Q -> io.print("Addition is commutative.\n", !IO) ; io.print("Addition is not commutative.\n", !IO) ), io.print("Q + R = ", !IO), io.print(to_string(Q + QR), !IO), io.nl(!IO), io.print("R + Q = ", !IO), io.print(to_string(QR + Q), !IO), io.nl(!IO), io.nl(!IO), io.print("5. Addition of two quaternions.\n", !IO), ( Q1 + Q2 = Q2 + Q1 -> io.print("Addition is commutative.\n", !IO) ; io.print("Addition is not commutative.\n", !IO) ), io.print("Q1 + Q2 = ", !IO), io.print(to_string(Q1 + Q2), !IO), io.nl(!IO), io.print("Q2 + Q1 = ", !IO), io.print(to_string(Q2 + Q1), !IO), io.nl(!IO), io.nl(!IO), io.print("6. Multiplication of a real number and a quaternion.\n", !IO), ( Q * QR = QR * Q -> io.print("Multiplication is commutative.\n", !IO) ; io.print("Multiplication is not commutative.\n", !IO) ), io.print("Q * R = ", !IO), io.print(to_string(Q * QR), !IO), io.nl(!IO), io.print("R * Q = ", !IO), io.print(to_string(QR * Q), !IO), io.nl(!IO), io.nl(!IO), io.print("7. Multiplication of two quaternions.\n", !IO), ( Q1 * Q2 = Q2 * Q1 -> io.print("Multiplication is commutative.\n", !IO) ; io.print("Multiplication is not commutative.\n", !IO) ), io.print("Q1 * Q2 = ", !IO), io.print(to_string(Q1 * Q2), !IO), io.nl(!IO), io.print("Q2 * Q1 = ", !IO), io.print(to_string(Q2 * Q1), !IO), io.nl(!IO), io.nl(!IO). to_string(q(I, J, K, W)) = string.format("q(%f, %f, %f, %f)", [f(I), f(J), f(K), f(W)]).:- end_module test_quaternion. The output of the above code follows: % ./test_quaternion Q = q(1.000000, 2.000000, 3.000000, 4.000000) Q1 = q(2.000000, 3.000000, 4.000000, 5.000000) Q2 = q(3.000000, 4.000000, 5.000000, 6.000000) R = 7.0 1. The norm of a quaternion. norm(Q) = 5.477225575051661 2. The negative of a quaternion. -Q = q(-1.000000, -2.000000, -3.000000, -4.000000) 3. The conjugate of a quaternion. conjugate(Q) = q(1.000000, -2.000000, -3.000000, -4.000000) 4. Addition of a real number and a quaternion. Addition is commutative. Q + R = q(8.000000, 2.000000, 3.000000, 4.000000) R + Q = q(8.000000, 2.000000, 3.000000, 4.000000) 5. Addition of two quaternions. Addition is commutative. Q1 + Q2 = q(5.000000, 7.000000, 9.000000, 11.000000) Q2 + Q1 = q(5.000000, 7.000000, 9.000000, 11.000000) 6. Multiplication of a real number and a quaternion. Multiplication is commutative. Q * R = q(7.000000, 14.000000, 21.000000, 28.000000) R * Q = q(7.000000, 14.000000, 21.000000, 28.000000) 7. Multiplication of two quaternions. Multiplication is not commutative. Q1 * Q2 = q(-56.000000, 16.000000, 24.000000, 26.000000) Q2 * Q1 = q(-56.000000, 18.000000, 20.000000, 28.000000)  ## OCaml This implementation was build strictly to the specs without looking (too much) at other implementations. The implementation as a record type with only floats is said (on the ocaml mailing list) to be especially efficient. Put this into a file quaternion.ml:  type quaternion = {a: float; b: float; c: float; d: float} let norm q = sqrt (q.a**2.0 +. q.b**2.0 +. q.c**2.0 +. q.d**2.0 ) let floatneg r = ~-. r (* readability *) let negative q = {a = floatneg q.a; b = floatneg q.b; c = floatneg q.c; d = floatneg q.d } let conjugate q = {a = q.a; b = floatneg q.b; c = floatneg q.c; d = floatneg q.d } let addrq r q = {q with a = q.a +. r} let addq q1 q2 = {a = q1.a +. q2.a; b = q1.b +. q2.b; c = q1.c +. q2.c; d = q1.d +. q2.d } let multrq r q = {a = q.a *. r; b = q.b *. r; c = q.c *. r; d = q.d *. r } let multq q1 q2 = {a = q1.a*.q2.a -. q1.b*.q2.b -. q1.c*.q2.c -. q1.d*.q2.d; b = q1.a*.q2.b +. q1.b*.q2.a +. q1.c*.q2.d -. q1.d*.q2.c; c = q1.a*.q2.c -. q1.b*.q2.d +. q1.c*.q2.a +. q1.d*.q2.b; d = q1.a*.q2.d +. q1.b*.q2.c -. q1.c*.q2.b +. q1.d*.q2.a } let qmake a b c d = {a;b;c;d} (* readability omitting a= b=... *) let qstring q = Printf.sprintf "(%g, %g, %g, %g)" q.a q.b q.c q.d ;; (* test data *) let q = qmake 1.0 2.0 3.0 4.0let q1 = qmake 2.0 3.0 4.0 5.0let q2 = qmake 3.0 4.0 5.0 6.0let r = 7.0 let () = (* written strictly to spec *) let pf = Printf.printf in pf "starting with data q=%s, q1=%s, q2=%s, r=%g\n" (qstring q) (qstring q1) (qstring q2) r; pf "1. norm of q = %g \n" (norm q) ; pf "2. negative of q = %s \n" (qstring (negative q)); pf "3. conjugate of q = %s \n" (qstring (conjugate q)); pf "4. adding r to q = %s \n" (qstring (addrq r q)); pf "5. adding q1 and q2 = %s \n" (qstring (addq q1 q2)); pf "6. multiply r and q = %s \n" (qstring (multrq r q)); pf "7. multiply q1 and q2 = %s \n" (qstring (multq q1 q2)); pf "8. instead q2 * q1 = %s \n" (qstring (multq q2 q1)); pf "\n";  using this file on the command line will produce: $ ocaml quaternion.ml
starting with data q=(1, 2, 3, 4), q1=(2, 3, 4, 5),  q2=(3, 4, 5, 6), r=7
1. norm of      q     = 5.47723
2. negative of  q     = (-1, -2, -3, -4)
3. conjugate of q     = (1, -2, -3, -4)
4. adding r to q      = (8, 2, 3, 4)
5. adding q1 and q2   = (5, 7, 9, 11)
6. multiply r and q   = (7, 14, 21, 28)
7. multiply q1 and q2 = (-56, 16, 24, 26)
8. instead q2 * q1    = (-56, 18, 20, 28)


For completeness, and since data types are of utmost importance in OCaml, here the types produced by pasting the code into the toplevel (ocaml is the toplevel):

 type quaternion = { a : float; b : float; c : float; d : float; }val norm : quaternion -> float = <fun>val floatneg : float -> float = <fun>val negative : quaternion -> quaternion = <fun>val conjugate : quaternion -> quaternion = <fun>val addrq : float -> quaternion -> quaternion = <fun>val addq : quaternion -> quaternion -> quaternion = <fun>val multrq : float -> quaternion -> quaternion = <fun>val multq : quaternion -> quaternion -> quaternion = <fun>val qmake : float -> float -> float -> float -> quaternion = <fun>val qstring : quaternion -> string = <fun> 

## Octave

There is an add-on package (toolbox) to Octave available from http://octave.sourceforge.net/quaternion/

Such a package can be install with the command:

pkg install -forge quaternion

Here is a sample interactive session solving the task:

> q = quaternion (1, 2, 3, 4)q = 1 + 2i + 3j + 4k>  q1 = quaternion (2, 3, 4, 5)q1 = 2 + 3i + 4j + 5k> q2 = quaternion (3, 4, 5, 6)q2 = 3 + 4i + 5j + 6k> r = 7r =  7> norm(q)ans =  5.4772> -qans = -1 - 2i - 3j - 4k> conj(q)ans = 1 - 2i - 3j - 4k> q + rans = 8 + 2i + 3j + 4k> q1 + q2ans = 5 + 7i + 9j + 11k> q * rans = 7 + 14i + 21j + 28k> q1 * q2ans = -56 + 16i + 24j + 26k> q1 == q2ans = 0

## Oforth

Setting a priority (here 160) to Quaternion class and defining #asQuaternion, integers and floats can be fully mixed with quaternions. neg is defined as "0 self -" into Number class, so no need to define it (if #- is defined).

160 Number Class newPriority: Quaternion(a, b, c, d) Quaternion method: _a  @a ;Quaternion method: _b  @b ;Quaternion method: _c  @c ;Quaternion method: _d  @d ; Quaternion method: initialize  := d := c := b := a ;Quaternion method: <<  '(' <<c @a << ',' <<c @b << ',' <<c @c << ',' <<c @d << ')' <<c ; Integer method: asQuaternion  self 0 0 0 Quaternion new ;Float   method: asQuaternion  self 0 0 0 Quaternion new ; Quaternion method: ==(q)  q _a @a == q _b @b == and q _c @c == and q _d @d == and ;Quaternion method: norm   @a sq @b sq + @c sq + @d sq + sqrt ;Quaternion method: conj   @a  @b neg  @c neg  @d neg Quaternion new ;Quaternion method: +(q)   Quaternion new(q _a @a +, q _b @b +, q _c @c +, q _d @d +) ;Quaternion method: -(q)   Quaternion new(q _a @a -, q _b @b -, q _c @c -, q _d @d -) ; Quaternion method: *(q)    Quaternion new(q _a @a * q _b @b * - q _c @c * - q _d @d * -,                  q _a @b * q _b @a * + q _c @d * + q _d @c * -,                  q _a @c * q _b @d * - q _c @a * + q _d @b * +,                  q _a @d * q _b @c * + q _c @b * - q _d @a * + ) ;

Usage :

: test| q q1 q2 r |    Quaternion new(1, 2, 3, 4) ->q   Quaternion new(2, 3, 4, 5) ->q1   Quaternion new(3, 4, 5, 6) ->q2   7.0 -> r    System.Out "q       = " << q << cr   System.Out "q1      = " << q1 << cr   System.Out "q2      = " << q2 << cr    System.Out "norm q  = " << q norm << cr    System.Out "neg q   = " << q neg << cr    System.Out "conj q  = " << q conj << cr   System.Out "q +r    = " << q r + << cr   System.Out "q1 + q2 = " << q1 q2 + << cr   System.Out "q * r   = " << q r * << cr   System.Out "q1 * q2 = " << q1 q2 * << cr   q1 q2 * q2 q1 * == ifFalse: [ "q1q2 and q2q1 are different quaternions" println ] ;
Output:
q       = (1,2,3,4)
q1      = (2,3,4,5)
q2      = (3,4,5,6)
norm q  = 5.47722557505166
neg q   = (-1,-2,-3,-4)
conj q  = (1,-2,-3,-4)
q +r    = (8,2,3,4)
q1 + q2 = (5,7,9,11)
q * r   = (7,14,21,28)
q1 * q2 = (-56,16,24,26)
q1q2 and q2q1 are different quaternions


## ooRexx

Note, this example uses operator overloads to perform the math operation. The operator overloads only work if the left-hand-side of the operation is a quaterion instance. Thus something like "7 + q1" would not work because this would get passed to the "+" of the string class. For those situations, the best solution would be an addition method on the .Quaternion class itself that took the appropriate action. I've chosen not to implement those to keep the example shorter.

 q = .quaternion~new(1, 2, 3, 4)q1 = .quaternion~new(2, 3, 4, 5)q2 = .quaternion~new(3, 4, 5, 6)r = 7 say "q            =" qsay "q1           =" q1say "q2           =" q2say "r            =" rsay "norm(q)      =" q~normsay "-q           =" (-q)say "q*           =" q~conjugatesay "q + r        =" q + rsay "q1 + q2      =" q1 + q2say "q * r        =" q * rq1q2 = q1 * q2q2q1 = q2 * q1say "q1 * q2      =" q1q2say "q2 * q1      =" q2q1say "q1 == q1     =" (q1 == q1)say "q1q2 == q2q1 =" (q1q2 == q2q1)  ::class quaternion::method init  expose r i j k  use strict arg r, i = 0, j = 0, k = 0 -- quaternion instances are immutable, so these are-- read only attributes::attribute r GET::attribute i GET::attribute j GET::attribute k GET ::method norm  expose r i j k  return rxcalcsqrt(r * r + i * i + j * j + k * k) ::method invert  expose r i j k  norm = self~norm  return self~class~new(r / norm, i / norm, j / norm, k / norm) ::method negative  expose r i j k  return self~class~new(-r, -i, -j, -k) ::method conjugate  expose r i j k  return self~class~new(r, -i, -j, -k) ::method add  expose r i j k  use strict arg other  if other~isa(.quaternion) then     return self~class~new(r + other~r, i + other~i, j + other~j, k + other~k)  else return self~class~new(r + other, i, j, k) ::method subtract  expose r i j k  use strict arg other  if other~isa(.quaternion) then     return self~class~new(r - other~r, i - other~i, j - other~j, k - other~k)  else return self~class~new(r - other, i, j, k) ::method times  expose r i j k  use strict arg other  if other~isa(.quaternion) then     return self~class~new(r * other~r - i * other~i - j * other~j - k * other~k, -                           r * other~i + i * other~r + j * other~k - k * other~j, -                           r * other~j - i * other~k + j * other~r + k * other~i, -                           r * other~k + i * other~j - j * other~i + k * other~r)  else return self~class~new(r * other, i * other, j * other, k * other) ::method divide  use strict arg other  -- this is easier if everything is a quaternion  if \other~isA(.quaternion) then other = .quaternion~new(other)  -- division is multiplication with the inversion  return self * other~invert ::method "=="  expose r i j k  use strict arg other   if \other~isa(.quaternion) then return .false  -- Note:  these are numeric comparisons, so we're using the "="  -- method so those are handled correctly  return r = other~r & i = other~i & j = other~j & k = other~k ::method "\=="  use strict arg other  return \self~"\=="(other) ::method "="  -- this is equivalent of "=="  forward message("==") ::method "\="  -- this is equivalent of "\=="  forward message("\==") ::method "<>"  -- this is equivalent of "\=="  forward message("\==") ::method "><"  -- this is equivalent of "\=="  forward message("\==") -- some operator overrides -- these only work if the left-hand-side of the-- subexpression is a quaternion::method "*"  forward message("TIMES") ::method "/"  forward message("DIVIDE") ::method "-"  -- need to check if this is a prefix minus or a subtract  if arg() == 0 then      forward message("NEGATIVE")  else      forward message("SUBTRACT") ::method "+"  -- need to check if this is a prefix plus or an addition  if arg() == 0 then      return self  -- we can return this copy since it is immutable  else      forward message("ADD") ::method string  expose r i j k  return r self~formatnumber(i)"i" self~formatnumber(j)"j" self~formatnumber(k)"k" ::method formatnumber private  use arg value  if value > 0 then return "+" value  else return "-" value~abs -- override hashcode for collection class hash uses::method hashCode  expose r i j k  return r~hashcode~bitxor(i~hashcode)~bitxor(j~hashcode)~bitxor(k~hashcode)  ::requires rxmath LIBRARY  
Output:
q            = 1 + 2i + 3j + 4k
q1           = 2 + 3i + 4j + 5k
q2           = 3 + 4i + 5j + 6k
r            = 7
norm(q)      = 5.47722558
-q           = -1 - 2i - 3j - 4k
q*           = 1 - 2i - 3j - 4k
q + r        = 8 + 2i + 3j + 4k
q1 + q2      = 5 + 7i + 9j + 11k
q * r        = 7 + 14i + 21j + 28k
q1 * q2      = -56 + 16i + 24j + 26k
q2 * q1      = -56 + 18i + 20j + 28k
q1 == q1     = 1
q1q2 == q2q1 = 0


## PARI/GP

Works with: PARI/GP version version 2.4.2 and above

Here is a simple solution in GP. I think it's possible to implement this type directly in Pari by abusing t_COMPLEX, but I haven't attempted this.

q.norm={	if(type(q) != "t_VEC" || #q != 4, error("incorrect type"));	sqrt(q[1]^2+q[2]^2+q[3]^2+q[4]^2)};q.conj={	if(type(q) != "t_VEC" || #q != 4, error("incorrect type"));	-[-q[1],q[2],q[3],q[4]]};q.add={	if(type(q) != "t_VEC" || #q != 4, error("incorrect type"));	x->if(type(x) == "t_INT" || type(x) == t_REAL,		[q[1]+x,q[2],q[3],q[4]]	,		if(type(x) == "t_VEC" && #x == 4,			q+x		,			error("incorrect type")		)	)};q.mult={	if(type(q) != "t_VEC" || #q != 4, error("incorrect type"));	x->if(type(x) == "t_INT" || type(x) == t_REAL,		x*q	,		if(type(x) == "t_VEC" && #x == 4,			[q[1]*x[1] - q[2]*x[2] - q[3]*x[3] - q[4]*x[4],			q[1]*x[2] + q[2]*x[1] + q[3]*x[4] - q[4]*x[3],			q[1]*x[3] - q[2]*x[4] + q[3]*x[1] + q[4]*x[2],			q[1]*x[4] + q[2]*x[3] - q[3]*x[2] + q[4]*x[1]]		,			error("incorrect type")		)	)};

Usage:

r=7;q=[1,2,3,4];q1=[2,3,4,5];q2=[3,4,5,6];q.norm-qq.conjq.add(r)q1.add(q2)q1.add(q2)	\\ or q1+q2q.mult(r)	\\ or r*q or q*rq1.mult(q2)q1.mult(q2) != q2.mult(q1)

## Pascal

The Delphi example also works with FreePascal.

package Quaternion;use List::Util 'reduce';use List::MoreUtils 'pairwise'; sub make {        my $cls = shift; if (@_ == 1) { return bless [ @_, 0, 0, 0 ] } elsif (@_ == 4) { return bless [ @_ ] } else { die "Bad number of components: @_" }} sub _abs { sqrt reduce {$a + $b *$b } @{ +shift } }sub _neg { bless [ map(-$_, @{+shift}) ] }sub _str { "(@{+shift})" } sub _add { my ($x, $y) = @_;$y = [ $y, 0, 0, 0 ] unless ref$y;        bless [ pairwise { $a +$b } @$x, @$y ]} sub _sub {        my ($x,$y, $swap) = @_;$y = [ $y, 0, 0, 0 ] unless ref$y;        my @x = pairwise { $a -$b } @$x, @$y;        if ($swap) {$_ = -$_ for @x } bless \@x;} sub _mul { my ($x, $y) = @_; if (!ref$y) { return bless [ map($_ *$y, @$x) ] } my ($a1, $b1,$c1, $d1) = @$x;        my ($a2,$b2, $c2,$d2) = @$y; bless [$a1 * $a2 -$b1 * $b2 -$c1 * $c2 -$d1 * $d2,$a1 * $b2 +$b1 * $a2 +$c1 * $d2 -$d1 * $c2,$a1 * $c2 -$b1 * $d2 +$c1 * $a2 +$d1 * $b2,$a1 * $d2 +$b1 * $c2 -$c1 * $b2 +$d1 * $a2]} sub conjugate { my @a = map { -$_ } @{$_[0]};$a[0] = $_[0][0]; bless \@a} use overload ( '""' => \&_str, '+' => \&_add, '-' => \&_sub, '*' => \&_mul, 'neg' => \&_neg, 'abs' => \&_abs,); package main; my$a = Quaternion->make(1, 2, 3, 4);my $b = Quaternion->make(1, 1, 1, 1); print "a =$a\n";print "b = $b\n";print "|a| = ", abs($a), "\n";print "-a = ", -$a, "\n";print "a + 1 = ",$a + 1, "\n";print "a + b = ", $a +$b, "\n";print "a - b = ", $a -$b, "\n";print "a conjugate is ", $a->conjugate, "\n";print "a * b = ",$a * $b, "\n";print "b * a = ",$b * $a, "\n"; ## Perl 6 class Quaternion { has Real ($.r, $.i,$.j, $.k ); multi method new ( Real$r, Real $i, Real$j, Real $k ) { self.bless: :$r, :$i, :$j, :$k; } multi qu(*@r) is export { Quaternion.new: |@r } sub postfix:<j>(Real$x) is export { qu 0, 0, $x, 0 } sub postfix:<k>(Real$x) is export { qu 0, 0, 0, $x } method Str () { "$.r + {$.i}i + {$.j}j + {$.k}k" } method reals () {$.r, $.i,$.j, $.k } method conj () { qu$.r, -$.i, -$.j, -$.k } method norm () { sqrt [+] self.reals X** 2 } multi infix:<eqv> ( Quaternion$a, Quaternion $b ) is export {$a.reals eqv $b.reals } multi infix:<+> ( Quaternion$a,       Real $b ) is export { qu$b+$a.r,$a.i, $a.j,$a.k }    multi infix:<+> (       Real $a, Quaternion$b ) is export { qu $a+$b.r, $b.i,$b.j, $b.k } multi infix:<+> ( Quaternion$a,    Complex $b ) is export { qu$b.re + $a.r,$b.im + $a.i,$a.j, $a.k } multi infix:<+> ( Complex$a, Quaternion $b ) is export { qu$a.re + $b.r,$a.im + $b.i,$b.j, $b.k } multi infix:<+> ( Quaternion$a, Quaternion $b ) is export { qu$a.reals Z+ $b.reals } multi prefix:<-> ( Quaternion$a ) is export { qu $a.reals X* -1 } multi infix:<*> ( Quaternion$a,       Real $b ) is export { qu$a.reals X* $b } multi infix:<*> ( Real$a, Quaternion $b ) is export { qu$b.reals X* $a } multi infix:<*> ( Quaternion$a,    Complex $b ) is export {$a * qu $b.reals, 0, 0 } multi infix:<*> ( Complex$a,    Quaternion $b ) is export {$b R* qu $a.reals, 0, 0 } multi infix:<*> ( Quaternion$a, Quaternion $b ) is export { my @a_rijk =$a.reals;	my ( $r,$i, $j,$k ) = $b.reals; return qu [+]( @a_rijk Z*$r, -$i, -$j, -$k ), # real [+]( @a_rijk Z*$i,  $r,$k, -$j ), # i [+]( @a_rijk Z*$j, -$k,$r,  $i ), # j [+]( @a_rijk Z*$k,  $j, -$i,  $r ); # k }}import Quaternion; my$q  = 1 + 2i + 3j + 4k;my $q1 = 2 + 3i + 4j + 5k;my$q2 = 3 + 4i + 5j + 6k;my $r = 7; say "1) q norm = {$q.norm}";say "2) -q      = {-$q}";say "3) q conj = {$q.conj}";say "4) q  + r  = {$q +$r}";say "5) q1 + q2 = {$q1 +$q2}";say "6) q  * r  = {$q *$r}";say "7) q1 * q2 = {$q1 *$q2}";say "8) q1q2 { $q1 *$q2 eqv $q2 *$q1 ?? '==' !! '!=' } q2q1";
Output:
1) q norm  = 5.47722557505166
2) -q      = -1 + -2i + -3j + -4k
3) q conj  = 1 + -2i + -3j + -4k
4) q  + r  = 8 + 2i + 3j + 4k
5) q1 + q2 = 5 + 7i + 9j + 11k
6) q  * r  = 7 + 14i + 21j + 28k
7) q1 * q2 = -56 + 16i + 24j + 26k
8) q1q2 != q2q1

## Phix

Translation of: Euphoria
function norm(sequence q)    return sqrt(sum(sq_power(q,2)))end function function conj(sequence q)    q[2..4] = sq_uminus(q[2..4])    return qend function function add(object q1, object q2)    if atom(q1)!=atom(q2) then        if atom(q1) then            q1 = {q1,0,0,0}        else            q2 = {q2,0,0,0}        end if    end if    return sq_add(q1,q2)end function function mul(object q1, object q2)    if sequence(q1) and sequence(q2) then        return { q1[1]*q2[1] - q1[2]*q2[2] - q1[3]*q2[3] - q1[4]*q2[4],                 q1[1]*q2[2] + q1[2]*q2[1] + q1[3]*q2[4] - q1[4]*q2[3],                 q1[1]*q2[3] - q1[2]*q2[4] + q1[3]*q2[1] + q1[4]*q2[2],                 q1[1]*q2[4] + q1[2]*q2[3] - q1[3]*q2[2] + q1[4]*q2[1] }    else        return sq_mul(q1,q2)    end ifend function function quats(sequence q)    return sprintf("%g + %gi + %gj + %gk",q)end function constant    q  = {1, 2, 3, 4},    q1 = {2, 3, 4, 5},    q2 = {3, 4, 5, 6},    r  = 7 printf(1, "q = %s\n", {quats(q)})printf(1, "r = %g\n", r)printf(1, "norm(q) = %g\n", norm(q))printf(1, "-q = %s\n", {quats(-q)})printf(1, "conj(q) = %s\n", {quats(conj(q))})printf(1, "q + r = %s\n", {quats(add(q,r))})printf(1, "q * r = %s\n", {quats(mul(q,r))})printf(1, "q1 = %s\n", {quats(q1)})printf(1, "q2 = %s\n", {quats(q2)})printf(1, "q1 + q2 = %s\n", {quats(add(q1,q2))})printf(1, "q2 + q1 = %s\n", {quats(add(q2,q1))})printf(1, "q1 * q2 = %s\n", {quats(mul(q1,q2))})printf(1, "q2 * q1 = %s\n", {quats(mul(q2,q1))})
Output:
q = 1 + 2i + 3j + 4k
r = 7
norm(q) = 5.47723
-q = -1 + -2i + -3j + -4k
conj(q) = 1 + -2i + -3j + -4k
q + r = 8 + 2i + 3j + 4k
q * r = 7 + 14i + 21j + 28k
q1 = 2 + 3i + 4j + 5k
q2 = 3 + 4i + 5j + 6k
q1 + q2 = 5 + 7i + 9j + 11k
q2 + q1 = 5 + 7i + 9j + 11k
q1 * q2 = -56 + 16i + 24j + 26k
q2 * q1 = -56 + 18i + 20j + 28k


## PicoLisp

(scl 6) (def 'quatCopy copy) (de quatNorm (Q)   (sqrt (sum * Q Q)) ) (de quatNeg (Q)   (mapcar - Q) ) (de quatConj (Q)   (cons (car Q) (mapcar - (cdr Q))) ) (de quatAddR (Q R)   (cons (+ R (car Q)) (cdr Q)) ) (de quatAdd (Q1 Q2)   (mapcar + Q1 Q2) ) (de quatMulR (Q R)   (mapcar */ (mapcar * Q (circ R)) (1.0 .)) ) (de quatMul (Q1 Q2)   (mapcar      '((Ops I)         (sum '((Op R I) (Op (*/ R (get Q2 I) 1.0))) Ops Q1 I) )      '((+ - - -) (+ + + -) (+ - + +) (+ + - +))      '((1 2 3 4) (2 1 4 3) (3 4 1 2) (4 3 2 1)) ) ) (de quatFmt (Q)   (mapcar '((R S) (pack (format R *Scl) S))      Q      '(" + " "i + " "j + " "k") ) )

Test:

(setq   Q (1.0 2.0 3.0 4.0)   Q1 (2.0 3.0 4.0 5.0)   Q2 (3.0 4.0 5.0 6.0)   R 7.0 ) (prinl "R  = " (format R *Scl))(prinl "Q  = " (quatFmt Q))(prinl "Q1 = " (quatFmt Q1))(prinl "Q2 = " (quatFmt Q2))(prinl)(prinl "norm(Q)  = " (format (quatNorm Q) *Scl))(prinl "norm(Q1) = " (format (quatNorm Q1) *Scl))(prinl "norm(Q2) = " (format (quatNorm Q2) *Scl))(prinl "neg(Q)   = " (quatFmt (quatNeg Q)))(prinl "conj(Q)  = " (quatFmt (quatConj Q)))(prinl "Q + R    = " (quatFmt (quatAddR Q R)))(prinl "Q1 + Q2  = " (quatFmt (quatAdd Q1 Q2)))(prinl "Q * R    = " (quatFmt (quatMulR Q R)))(prinl "Q1 * Q2  = " (quatFmt (quatMul Q1 Q2)))(prinl "Q2 * Q1  = " (quatFmt (quatMul Q2 Q1)))(prinl (if (= (quatMul Q1 Q2) (quatMul Q2 Q1)) "Equal" "Not equal"))
Output:
R  = 7.000000
Q  = 1.000000 + 2.000000i + 3.000000j + 4.000000k
Q1 = 2.000000 + 3.000000i + 4.000000j + 5.000000k
Q2 = 3.000000 + 4.000000i + 5.000000j + 6.000000k

norm(Q)  = 5.477225
norm(Q1) = 7.348469
norm(Q2) = 9.273618
neg(Q)   = -1.000000 + -2.000000i + -3.000000j + -4.000000k
conj(Q)  = 1.000000 + -2.000000i + -3.000000j + -4.000000k
Q + R    = 8.000000 + 2.000000i + 3.000000j + 4.000000k
Q1 + Q2  = 5.000000 + 7.000000i + 9.000000j + 11.000000k
Q * R    = 7.000000 + 14.000000i + 21.000000j + 28.000000k
Q1 * Q2  = -56.000000 + 16.000000i + 24.000000j + 26.000000k
Q2 * Q1  = -56.000000 + 18.000000i + 20.000000j + 28.000000k
Not equal

## PL/I

*process source attributes xref or(!); qu: Proc Options(main); /********************************************************************** * 06.09.2013 Walter Pachl translated from REXX *            added tasks 9 and A **********************************************************************/ dcl v(4) Char(1) Var Init('','i','j','k'); define structure 1 quat, 2 x(4) Dec Float(15); Dcl q  type quat; Call quat_init(q, 1,2,3,4); Dcl q1 type quat; Call quat_init(q1,2,3,4,5); Dcl q2 type quat; Call quat_init(q2,3,4,5,6); Dcl q3 type quat; Call quat_init(q3,-2,3,-4,-5); Dcl r  Dec Float(15)Init(7);  call showq('       ','q'                    ,q); call showq('       ','q1'                   ,q1); call showq('       ','q2'                   ,q2); call showq('       ','q3'                   ,q3); call shows('       ','r'                    ,r); Call shows('task 1:','norm q'               ,norm(q)); Call showq('task 2:','quatneg q'            ,quatneg(q)); Call showq('task 3:','conjugate q'          ,quatConj(q)); Call showq('task 4:','addition r+q'         ,quatAddsq(r,q)); Call showq('task 5:','addition q1+q2'       ,quatAdd(q1,q2)); Call showq('task 6:','multiplication q*r'   ,quatMulqs(q,r)); Call showq('task 7:','multiplication q1*q2' ,quatMul(q1,q2)); Call showq('task 8:','multiplication q2*q1' ,quatMul(q2,q1)); Call showq('task 9:','quatsub q1-q1'        ,quatAdd(q1,quatneg(q1))); Call showq('task A:','addition q1+q3'       ,quatAdd(q1,q3)); Call showt('task B:','equal'                ,quatEqual(quatMul(q1,q2),                                                       quatMul(q2,q1))); Call showt('task C:','q1=q1'                ,quatEqual(q1,q1));  quatNeg: procedure(qp) Returns(type quat); Dcl (qp,qr) type quat; qr.x(*)=-qp.x(*); Return (qr); End;  quatAdd: procedure(qp,qq) Returns(type quat); Dcl (qp,qq,qr) type quat; qr.x(*)=qp.x(*)+qq.x(*); Return (qr); End;  quatAddsq: procedure(v,qp) Returns(type quat); Dcl v Dec Float(15); Dcl (qp,qr) type quat; qr.x(*)=qp.x(*); qr.x(1)=qp.x(1)+v; Return (qr); End;  quatConj: procedure(qp) Returns(type quat); Dcl (qp,qr) type quat; qr.x(*)=-qp.x(*); qr.x(1)= qp.x(1); Return (qr); End;  quatMul: procedure(qp,qq) Returns(type quat); Dcl (qp,qq,qr) type quat; qr.x(1)=        qp.x(1)*qq.x(1)-qp.x(2)*qq.x(2)-qp.x(3)*qq.x(3)-qp.x(4)*qq.x(4); qr.x(2)=        qp.x(1)*qq.x(2)+qp.x(2)*qq.x(1)+qp.x(3)*qq.x(4)-qp.x(4)*qq.x(3); qr.x(3)=        qp.x(1)*qq.x(3)-qp.x(2)*qq.x(4)+qp.x(3)*qq.x(1)+qp.x(4)*qq.x(2); qr.x(4)=        qp.x(1)*qq.x(4)+qp.x(2)*qq.x(3)-qp.x(3)*qq.x(2)+qp.x(4)*qq.x(1); Return (qr); End;  quatMulqs: procedure(qp,v) Returns(type quat); Dcl (qp,qr) type quat; Dcl v Dec Float(15); qr.x(*)=qp.x(*)*v; Return (qr); End;  shows: Procedure(t1,t2,v); Dcl (t1,t2) Char(*); Dcl v Dec Float(15); Put Edit(t1,right(t2,24),'  --> ',v)(Skip,a,a,a,f(15,13)); End;  showt: Procedure(t1,t2,v); Dcl (t1,t2) Char(*); Dcl v Char(*) Var); Put Edit(t1,right(t2,24),'  --> ',v)(Skip,a,a,a,a); End;  showq: Procedure(t1,t2,qp); Dcl qp type quat; Dcl (t1,t2) Char(*); Dcl (s,s2,p) Char(100) Var Init(''); Dcl i Bin Fixed(31); Put String(s) Edit(t1,right(t2,24),'  --> ')(a,a,a); Do i=1 To 4;   Put String(p) Edit(abs(qp.x(i)))(p'ZZZ9');   p=trim(p);   Select;     When(qp.x(i)<0) p='-'!!p!!v(i);     When(p=0) p='';     Otherwise Do       If s2^='' Then p='+'!!p;       If i>1 Then p=p!!v(i);       End;     End;   s2=s2!!p   End; If s2='' Then   s2='0'; Put Edit(s!!s2)(Skip,a); End;  norm: Procedure(qp) Returns(Dec Float(15)); Dcl qp type quat; Dcl r  Dec Float(15) Init(0); Dcl i  Bin Fixed(31); Do i=1 To 4;   r=r+qp.x(i)**2;   End; Return (sqrt(r)); End;  quat_init: Proc(qp,x,y,z,u); Dcl qp type quat; Dcl (x,y,z,u) Dec Float(15); qp.x(1)=x; qp.x(2)=y; qp.x(3)=z; qp.x(4)=u; End;  End;
Output:
                              q  --> 1+2i+3j+4k
q1  --> 2+3i+4j+5k
q2  --> 3+4i+5j+6k
q3  --> -2+3i-4j-5k
r  --> 7.0000000000000
task 1:                  norm q  --> 5.4772255750517
task 2:               quatneg q  --> -1-2i-3j-4k
task 3:             conjugate q  --> 1-2i-3j-4k
task 6:      multiplication q*r  --> 7+14i+21j+28k
task 7:    multiplication q1*q2  --> -56+16i+24j+26k
task 8:    multiplication q2*q1  --> -56+18i+20j+28k
task 9:           quatsub q1-q1  --> 0
task B:                   equal  --> not equal


## PowerShell

### Implementation

 class Quaternion {  [Double]$w [Double]$x  [Double]$y [Double]$z  Quaternion() {      $this.w = 0$this.x = 0      $this.y = 0$this.z = 0  }  Quaternion([Double]$a, [Double]$b, [Double]$c, [Double]$d) {      $this.w =$a      $this.x =$b      $this.y =$c      $this.z =$d  }  [Double]abs2() {return $this.w*$this.w + $this.x*$this.x + $this.y*$this.y + $this.z*$this.z}  [Double]abs() {return [math]::sqrt($this.wbs2())} static [Quaternion]real([Double]$r) {return [Quaternion]::new($r, 0, 0, 0)} static [Quaternion]add([Quaternion]$m,[Quaternion]$n) {return [Quaternion]::new($m.w+$n.w,$m.x+$n.x,$m.y+$n.y,$m.z+$n.z)} [Quaternion]addreal([Double]$r) {return [Quaternion]::add($this,[Quaternion]::real($r))}  static [Quaternion]mul([Quaternion]$m,[Quaternion]$n) {    return [Quaternion]::new(    ($m.w*$n.w) - ($m.x*$n.x) - ($m.y*$n.y) - ($m.z*$n.z),    ($m.w*$n.x) + ($m.x*$n.w) + ($m.y*$n.z) - ($m.z*$n.y),    ($m.w*$n.y) - ($m.x*$n.z) + ($m.y*$n.w) + ($m.z*$n.x),    ($m.w*$n.z) + ($m.x*$n.y) - ($m.y*$n.x) + ($m.z*$n.w))  }   [Quaternion]mul([Double]$r) {return [Quaternion]::new($r*$this.w,$r*$this.x,$r*$this.y,$r*$this.z)} [Quaternion]negate() {return$this.mul(-1)}  [Quaternion]conjugate() {return [Quaternion]::new($this.w, -$this.x, -$this.y, -$this.z)}  static [String]st([Double]$r) { if(0 -le$r) {return "+$r"} else {return "$r"}  }  [String]show() {return "$($this.w)$([Quaternion]::st($this.x))i$([Quaternion]::st($this.y))j$([Quaternion]::st($this.z))k"}  static [String]show([Quaternion]$other) {return$other.show()}}  $q = [Quaternion]::new(1, 2, 3, 4)$q1 = [Quaternion]::new(2, 3, 4, 5)$q2 = [Quaternion]::new(3, 4, 5, 6)$r = 7"$q:$($q.show())""$q1: $($q1.show())""$q2:$($q2.show())""$r: $r""""norm $q: $($q.wbs())""negate $q:$($q.negate().show())""conjugate $q: $($q.yonjugate().show())""$q + $r: $($q.wddreal($r).show())""$q1 + $q2:$([Quaternion]::show([Quaternion]::add($q1,$q2)))""$q * $r: $($q.mul($r).show())""$q1 * $q2:$([Quaternion]::show([Quaternion]::mul($q1,$q2)))""$q2 * $q1: $([Quaternion]::show([Quaternion]::mul($q2,$q1)))"  Output: norm$q: 5.47722557505166
negate $q: -1-2i-3j-4k conjugate$q: 1-2i-3j-4k
$q +$r: 8+2i+3j+4k
$q1 +$q2: 5+7i+9j+11k
$q *$r: 7+14i+21j+28k
$q1 *$q2: -56+16i+24j+26k
$q2 *$q1: -56+18i+20j+28k


### Library

 function show([System.Numerics.Quaternion]$c) { function st([Double]$r) {             if(0 -le $r) {return "+$r"} else {return "$r"} } return "$($c.w)$(st $c.y)i$(st $c.y)j$(st $c.z)k"}$q  = [System.Numerics.Quaternion]::new(1, 2, 3, 4)$q1 = [System.Numerics.Quaternion]::new(2, 3, 4, 5)$q2 = [System.Numerics.Quaternion]::new(3, 4, 5, 6)$r = 7"$q: $(show$q)""$q1:$(show $q1)""$q2: $(show$q2)""$r:$r""norm $q:$($q.Length())""negate $q: $(show ([System.Numerics.Quaternion]::Negate($q)))""conjugate $q:$(show ([System.Numerics.Quaternion]::Conjugate($q)))""$q + $r:$(show ([System.Numerics.Quaternion]::new($q.w +$r, $q.x,$q.y, $q.z)))""$q1 + $q2:$(show ([System.Numerics.Quaternion]::Add($q1,$q2)))""$q * $r: $(show ([System.Numerics.Quaternion]::new($q.w * $r,$q.x * $r,$q.y * $r,$q.z * $r)))""$q1 * $q2:$(show ([System.Numerics.Quaternion]::Multiply($q1,$q2)))""$q2 * $q1: $(show ([System.Numerics.Quaternion]::Multiply($q2,$q1)))"  Output: norm$q: 5.47722557505166
negate $q: -1-2i-3j-4k conjugate$q: 1-2i-3j-4k
$q +$r: 8+2i+3j+4k
$q1 +$q2: 5+7i+9j+11k
$q *$r: 7+14i+21j+28k
$q1 *$q2: -56+16i+24j+26k
$q2 *$q1: -56+18i+20j+28k


## Prolog

% A quaternion is represented as a complex term qx/4add(qx(R0,I0,J0,K0), qx(R1,I1,J1,K1), qx(R,I,J,K)) :-	!, R is R0+R1, I is I0+I1, J is J0+J1, K is K0+K1.add(qx(R0,I,J,K), F, qx(R,I,J,K)) :-	number(F), !, R is R0 + F.add(F, qx(R0,I,J,K), Qx) :-	add(qx(R0,I,J,K), F, Qx).mul(qx(R0,I0,J0,K0), qx(R1,I1,J1,K1), qx(R,I,J,K)) :- !,	R is R0*R1 - I0*I1 - J0*J1 - K0*K1,	I is R0*I1 + I0*R1 + J0*K1 - K0*J1,	J is R0*J1 - I0*K1 + J0*R1 + K0*I1,	K is R0*K1 + I0*J1 - J0*I1 + K0*R1.mul(qx(R0,I0,J0,K0), F, qx(R,I,J,K)) :-	number(F), !, R is R0*F, I is I0*F, J is J0*F, K is K0*F.mul(F, qx(R0,I0,J0,K0), Qx) :-	mul(qx(R0,I0,J0,K0),F,Qx).abs(qx(R,I,J,K), Norm) :-	Norm is sqrt(R*R+I*I+J*J+K*K).negate(qx(Ri,Ii,Ji,Ki),qx(R,I,J,K)) :-	R is -Ri, I is -Ii, J is -Ji, K is -Ki.conjugate(qx(R,Ii,Ji,Ki),qx(R,I,J,K)) :-	I is -Ii, J is -Ji, K is -Ki.

Test:

data(q,  qx(1,2,3,4)).data(q1, qx(2,3,4,5)).data(q2, qx(3,4,5,6)).data(r, 7). test :-	data(Name, qx(A,B,C,D)), abs(qx(A,B,C,D), Norm),	writef('abs(%w) is %w\n', [Name, Norm]), fail.test :- data(q, Qx), negate(Qx, Nqx),	writef('negate(%w) is %w\n', [q, Nqx]), fail.test :- data(q, Qx), conjugate(Qx, Nqx),	writef('conjugate(%w) is %w\n', [q, Nqx]), fail.test :- data(q1, Q1), data(q2, Q2), add(Q1, Q2, Qx),	writef('q1+q2 is %w\n', [Qx]), fail.test :- data(q1, Q1), data(q2, Q2), add(Q2, Q1, Qx),	writef('q2+q1 is %w\n', [Qx]), fail.test :- data(q, Qx), data(r, R), mul(Qx, R, Nqx),	writef('q*r is %w\n', [Nqx]), fail.test :- data(q, Qx), data(r, R), mul(R, Qx, Nqx),	writef('r*q is %w\n', [Nqx]), fail.test :- data(q1, Q1), data(q2, Q2), mul(Q1, Q2, Qx),	writef('q1*q2 is %w\n', [Qx]), fail.test :- data(q1, Q1), data(q2, Q2), mul(Q2, Q1, Qx),	writef('q2*q1 is %w\n', [Qx]), fail.test.
Output:
 ?- test.
abs(q) is 5.477225575051661
abs(q1) is 7.3484692283495345
abs(q2) is 9.273618495495704
negate(q) is qx(-1,-2,-3,-4)
conjugate(q) is qx(1,-2,-3,-4)
q1+q2 is qx(5,7,9,11)
q2+q1 is qx(5,7,9,11)
q*r is qx(7,14,21,28)
r*q is qx(7,14,21,28)
q1*q2 is qx(-56,16,24,26)
q2*q1 is qx(-56,18,20,28)

## PureBasic

Structure Quaternion    a.f  b.f  c.f  d.fEndStructure Procedure.f QNorm(*x.Quaternion)  ProcedureReturn Sqr(Pow(*x\a, 2) + Pow(*x\b, 2) + Pow(*x\c, 2) + Pow(*x\d, 2))EndProcedure ;If supplied, the result is returned in the quaternion structure *res,;otherwise a new quaternion is created.  A pointer to the result is returned.Procedure QNeg(*x.Quaternion, *res.Quaternion = 0)  If *res = 0: *res.Quaternion = AllocateMemory(SizeOf(Quaternion)): EndIf  If *res    *res\a = -*x\a    *res\b = -*x\b    *res\c = -*x\c    *res\d = -*x\d  EndIf  ProcedureReturn *resEndProcedure Procedure QConj(*x.Quaternion, *res.Quaternion = 0)  If *res = 0: *res.Quaternion = AllocateMemory(SizeOf(Quaternion)): EndIf  If *res    *res\a =  *x\a    *res\b = -*x\b    *res\c = -*x\c    *res\d = -*x\d  EndIf  ProcedureReturn *resEndProcedure Procedure QAddReal(r.f, *x.Quaternion, *res.Quaternion = 0)  If *res = 0: *res.Quaternion = AllocateMemory(SizeOf(Quaternion)): EndIf  If *res    *res\a = *x\a + r    *res\b = *x\b    *res\c = *x\c    *res\d = *x\d  EndIf  ProcedureReturn *resEndProcedure Procedure QAddQuaternion(*x.Quaternion, *y.Quaternion, *res.Quaternion = 0)  If *res = 0: *res.Quaternion = AllocateMemory(SizeOf(Quaternion)): EndIf  If *res    *res\a = *x\a + *y\a    *res\b = *x\b + *y\b    *res\c = *x\c + *y\c    *res\d = *x\d + *y\d  EndIf  ProcedureReturn *resEndProcedure Procedure QMulReal_and_Quaternion(r.f, *x.Quaternion, *res.Quaternion = 0)  If *res = 0: *res.Quaternion = AllocateMemory(SizeOf(Quaternion)): EndIf  If *res    *res\a = *x\a * r    *res\b = *x\b * r    *res\c = *x\c * r    *res\d = *x\d * r  EndIf  ProcedureReturn *resEndProcedure Procedure QMulQuaternion(*x.Quaternion, *y.Quaternion, *res.Quaternion = 0)  If *res = 0: *res.Quaternion = AllocateMemory(SizeOf(Quaternion)): EndIf  If *res     *res\a = *x\a * *y\a - *x\b * *y\b - *x\c * *y\c - *x\d * *y\d     *res\b = *x\a * *y\b + *x\b * *y\a + *x\c * *y\d - *x\d * *y\c     *res\c = *x\a * *y\c - *x\b * *y\d + *x\c * *y\a + *x\d * *y\b     *res\d = *x\a * *y\d + *x\b * *y\c - *x\c * *y\b + *x\d * *y\a  EndIf  ProcedureReturn *resEndProcedure Procedure Q_areEqual(*x.Quaternion, *y.Quaternion)  If (*x\a <> *y\a) Or (*x\b <> *y\b) Or (*x\c <> *y\c) Or (*x\d <> *y\d)    ProcedureReturn 0 ;false  EndIf  ProcedureReturn 1 ;trueEndProcedure

Implementation & test

Procedure.s ShowQ(*x.Quaternion, NN = 0)  ProcedureReturn "{" + StrF(*x\a, NN) + "," + StrF(*x\b, NN) + "," + StrF(*x\c, NN) + "," + StrF(*x\d, NN) + "}"EndProcedure If OpenConsole()  Define.Quaternion Q0, Q1, Q2, res, res2  Define.f          r = 7   Q0\a = 1: Q0\b = 2: Q0\c = 3: Q0\d = 4  Q1\a = 2: Q1\b = 3: Q1\c = 4: Q1\d = 5  Q2\a = 3: Q2\b = 4: Q2\c = 5: Q2\d = 6   PrintN("Q0 = " + ShowQ(Q0, 0))  PrintN("Q1 = " + ShowQ(Q1, 0))  PrintN("Q2 = " + ShowQ(Q2, 0))   PrintN("Normal of Q0 = " + StrF(QNorm(Q0)))  PrintN("Neg(Q0)  = " + ShowQ(QNeg(Q0, res)))  PrintN("Conj(Q0) = " + ShowQ(QConj(Q0, res)))  PrintN("r + Q0   = " + ShowQ(QAddReal(r, Q0, res)))  PrintN("Q0 + Q1  = " + ShowQ(QAddQuaternion(Q0, Q1, res)))  PrintN("Q1 + Q2  = " + ShowQ(QAddQuaternion(Q1, Q2, res)))  PrintN("Q1 * Q2  = " + ShowQ(QMulQuaternion(Q1, Q2, res)))  PrintN("Q2 * Q1  = " + ShowQ(QMulQuaternion(Q2, Q1, res2)))  Print( "Q1 * Q2"): If Q_areEqual(res, res2): Print(" = "): Else: Print(" <> "): EndIf: Print( "Q2 * Q1")   Print(#CRLF$+ #CRLF$ + "Press ENTER to exit"): Input()  CloseConsole()EndIf

Result

Q0 = {1,2,3,4}
Q1 = {2,3,4,5}
Q2 = {3,4,5,6}
Normal of Q0 = 5.4772257805
Neg(Q0)  = {-1,-2,-3,-4}
Conj(Q0) = {1,-2,-3,-4}
r + Q0   = {8,2,3,4}
Q0 + Q1  = {3,5,7,9}
Q1 + Q2  = {5,7,9,11}
Q1 * Q2  = {-56,16,24,26}
Q2 * Q1  = {-56,18,20,28}
Q1 * Q2 <> Q2 * Q1

## Python

This example extends Pythons namedtuples to add extra functionality.

from collections import namedtupleimport math class Q(namedtuple('Quaternion', 'real, i, j, k')):    'Quaternion type: Q(real=0.0, i=0.0, j=0.0, k=0.0)'      __slots__ = ()      def __new__(_cls, real=0.0, i=0.0, j=0.0, k=0.0):        'Defaults all parts of quaternion to zero'        return super().__new__(_cls, float(real), float(i), float(j), float(k))     def conjugate(self):        return Q(self.real, -self.i, -self.j, -self.k)     def _norm2(self):        return sum( x*x for x in self)     def norm(self):        return math.sqrt(self._norm2())     def reciprocal(self):        n2 = self._norm2()        return Q(*(x / n2 for x in self.conjugate()))      def __str__(self):        'Shorter form of Quaternion as string'        return 'Q(%g, %g, %g, %g)' % self     def __neg__(self):        return Q(-self.real, -self.i, -self.j, -self.k)     def __add__(self, other):        if type(other) == Q:            return Q( *(s+o for s,o in zip(self, other)) )        try:            f = float(other)        except:            return NotImplemented        return Q(self.real + f, self.i, self.j, self.k)     def __radd__(self, other):        return Q.__add__(self, other)     def __mul__(self, other):        if type(other) == Q:            a1,b1,c1,d1 = self            a2,b2,c2,d2 = other            return Q(                 a1*a2 - b1*b2 - c1*c2 - d1*d2,                 a1*b2 + b1*a2 + c1*d2 - d1*c2,                 a1*c2 - b1*d2 + c1*a2 + d1*b2,                 a1*d2 + b1*c2 - c1*b2 + d1*a2 )        try:            f = float(other)        except:            return NotImplemented        return Q(self.real * f, self.i * f, self.j * f, self.k * f)     def __rmul__(self, other):        return Q.__mul__(self, other)     def __truediv__(self, other):        if type(other) == Q:            return self.__mul__(other.reciprocal())        try:            f = float(other)        except:            return NotImplemented        return Q(self.real / f, self.i / f, self.j / f, self.k / f)     def __rtruediv__(self, other):        return other * self.reciprocal()     __div__, __rdiv__ = __truediv__, __rtruediv__ Quaternion = Q        q  = Q(1, 2, 3, 4)q1 = Q(2, 3, 4, 5)q2 = Q(3, 4, 5, 6)r  = 7

Continued shell session Run the above with the -i flag to python on the command line, or run with idle then continue in the shell as follows:

>>> qQuaternion(real=1.0, i=2.0, j=3.0, k=4.0)>>> q1Quaternion(real=2.0, i=3.0, j=4.0, k=5.0)>>> q2Quaternion(real=3.0, i=4.0, j=5.0, k=6.0)>>> r7>>> q.norm()5.477225575051661>>> q1.norm()7.3484692283495345>>> q2.norm()9.273618495495704>>> -qQuaternion(real=-1.0, i=-2.0, j=-3.0, k=-4.0)>>> q.conjugate()Quaternion(real=1.0, i=-2.0, j=-3.0, k=-4.0)>>> r + qQuaternion(real=8.0, i=2.0, j=3.0, k=4.0)>>> q + rQuaternion(real=8.0, i=2.0, j=3.0, k=4.0)>>> q1 + q2Quaternion(real=5.0, i=7.0, j=9.0, k=11.0)>>> q2 + q1Quaternion(real=5.0, i=7.0, j=9.0, k=11.0)>>> q * rQuaternion(real=7.0, i=14.0, j=21.0, k=28.0)>>> r * qQuaternion(real=7.0, i=14.0, j=21.0, k=28.0)>>> q1 * q2Quaternion(real=-56.0, i=16.0, j=24.0, k=26.0)>>> q2 * q1Quaternion(real=-56.0, i=18.0, j=20.0, k=28.0)>>> assert q1 * q2 != q2 * q1>>> >>> i, j, k = Q(0,1,0,0), Q(0,0,1,0), Q(0,0,0,1)>>> i*iQuaternion(real=-1.0, i=0.0, j=0.0, k=0.0)>>> j*jQuaternion(real=-1.0, i=0.0, j=0.0, k=0.0)>>> k*kQuaternion(real=-1.0, i=0.0, j=0.0, k=0.0)>>> i*j*kQuaternion(real=-1.0, i=0.0, j=0.0, k=0.0)>>> q1 / q2Quaternion(real=0.7906976744186047, i=0.023255813953488358, j=-2.7755575615628914e-17, k=0.046511627906976744)>>> q1 / q2 * q2Quaternion(real=2.0000000000000004, i=3.0000000000000004, j=4.000000000000001, k=5.000000000000001)>>> q2 * q1 / q2Quaternion(real=2.0, i=3.465116279069768, j=3.906976744186047, k=4.767441860465116)>>> q1.reciprocal() * q1Quaternion(real=0.9999999999999999, i=0.0, j=0.0, k=0.0)>>> q1 * q1.reciprocal()Quaternion(real=0.9999999999999999, i=0.0, j=0.0, k=0.0)>>> 

## R

Using the quaternions package.

 library(quaternions) q <- Q(1, 2, 3, 4)q1 <- Q(2, 3, 4, 5)q2 <- Q(3, 4, 5, 6)r <- 7.0 display <- function(x){  e <- deparse(substitute(x))  res <- if(class(x) == "Q") paste(x$r, "+", x$i, "i+", x$j, "j+", x$k, "k", sep = "") else x  cat(noquote(paste(c(e, " = ", res, "\n"), collapse="")))  invisible(res)} display(norm(q))display(-q)display(Conj(q))display(r + q)display(q1 + q2)display(r*q)display(q*r)if(display(q1*q2) == display(q2*q1)) cat("q1*q2 == q2*q1\n") else cat("q1*q2 != q2*q1\n") ## norm(q) = 5.47722557505166## -q = -1+-2i+-3j+-4k## Conj(q) = 1+-2i+-3j+-4k## r + q = 8+2i+3j+4k## q1 + q2 = 5+7i+9j+11k## r * q = 7+14i+21j+28k## q * r = 7+14i+21j+28k## q1 * q2 = -56+16i+24j+26k## q2 * q1 = -56+18i+20j+28k## q1*q2 != q2*q1  

## Racket

#lang racket (struct quaternion (a b c d)  #:transparent) (define-match-expander quaternion:   (λ (stx)    (syntax-case stx ()      [(_ a b c d)        #'(or (quaternion a b c d)             (and a (app (λ(_) 0) b) (app (λ(_) 0) c) (app (λ(_) 0) d)))]))) (define (norm q)  (match q    [(quaternion: a b c d)     (sqrt (+ (sqr a) (sqr b) (sqr c) (sqr d)))])) (define (negate q)  (match q    [(quaternion: a b c d)     (quaternion (- a) (- b) (- c) (- d))])) (define (conjugate q)  (match q    [(quaternion: a b c d)     (quaternion a (- b) (- c) (- d))])) (define (add q1 q2 . q-rest)  (let ((ans (match* (q1 q2)               [((quaternion: a1 b1 c1 d1) (quaternion: a2 b2 c2 d2))                (quaternion (+ a1 a2) (+ b1 b2) (+ c1 c2) (+ d1 d2))])))    (if (empty? q-rest)        ans        (apply add (cons ans q-rest))))) (define (multiply q1 q2 . q-rest)  (let ((ans (match* (q1 q2)               [((quaternion: a1 b1 c1 d1) (quaternion: a2 b2 c2 d2))                (quaternion (- (* a1 a2) (* b1 b2) (* c1 c2) (* d1 d2))                            (+ (* a1 b2) (* b1 a2) (* c1 d2) (- (* d1 c2)))                            (+ (* a1 c2) (- (* b1 d2)) (* c1 a2) (* d1 b2))                            (+ (* a1 d2) (* b1 c2) (- (* c1 b2)) (* d1 a2)))])))    (if (empty? q-rest)        ans        (apply multiply (cons ans q-rest))))) ;; Tests(module+ main  (define i (quaternion 0 1 0 0))  (define j (quaternion 0 0 1 0))  (define k (quaternion 0 0 0 1))  (displayln (multiply i j k))  (newline)   (define q (quaternion 1 2 3 4))  (define q1 (quaternion 2 3 4 5))  (define q2 (quaternion 3 4 5 6))  (define r 7)   (for ([quat (list q q1 q2)])    (displayln quat)    (displayln (norm quat))    (displayln (negate quat))    (displayln (conjugate quat))    (newline))   (add r q)  (add q1 q2)  (multiply r q)   (newline)  (multiply q1 q2)  (multiply q2 q1)  (equal? (multiply q1 q2)          (multiply q2 q1)))
Output:
#(struct:quaternion -1 0 0 0)

#(struct:quaternion 1 2 3 4)
5.477225575051661
#(struct:quaternion -1 -2 -3 -4)
#(struct:quaternion 1 -2 -3 -4)

#(struct:quaternion 2 3 4 5)
7.3484692283495345
#(struct:quaternion -2 -3 -4 -5)
#(struct:quaternion 2 -3 -4 -5)

#(struct:quaternion 3 4 5 6)
9.273618495495704
#(struct:quaternion -3 -4 -5 -6)
#(struct:quaternion 3 -4 -5 -6)

(quaternion 8 2 3 4)
(quaternion 5 7 9 11)
(quaternion 7 14 21 28)

(quaternion -56 16 24 26)
(quaternion -56 18 20 28)
#f


## REXX

The REXX language has no native quaternion support, but subroutines can be easily written.

/*REXX program performs some operations on  quaternion type numbers and displays results*/                       q = 1 2 3 4    ;                     q1 = 2 3 4 5                       r = 7          ;                     q2 = 3 4 5 6call qShow  q                         ,     'q'call qShow  q1                        ,     'q1'call qShow  q2                        ,     'q2'call qShow  r                         ,     'r'call qShow  qNorm(q)                  ,     'norm q'                  ,        "task 1:"call qShow  qNeg(q)                   ,     'negative q'              ,        "task 2:"call qShow  qConj(q)                  ,     'conjugate q'             ,        "task 3:"call qShow  qAdd( r, q  )             ,     'addition r+q'            ,        "task 4:"call qShow  qAdd(q1, q2 )             ,     'addition q1+q2'          ,        "task 5:"call qShow  qMul( q, r  )             ,     'multiplication q*r'      ,        "task 6:"call qShow  qMul(q1, q2 )             ,     'multiplication q1*q2'    ,        "task 7:"call qShow  qMul(q2, q1 )             ,     'multiplication q2*q1'    ,        "task 8:"exit                                             /*stick a fork in it,  we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/qConj: procedure; parse arg x;    call qXY;    return    x.1    (-x.2)    (-x.3)    (-x.4)qNeg:  procedure; parse arg x;    call qXY;    return   -x.1    (-x.2)    (-x.3)    (-x.4)qNorm: procedure; parse arg x;    call qXY;    return sqrt(x.1**2 +x.2**2 +x.3**2 +x.4**2)qAdd:  procedure; parse arg x,y;  call qXY 2;  return x.1+y.1  x.2+y.2  x.3+y.3  x.4+y.4/*──────────────────────────────────────────────────────────────────────────────────────*/qMul:  procedure; parse arg x,y;  call qXY y       return x.1*y.1 -x.2*y.2 -x.3*y.3 -x.4*y.4     x.1*y.2 +x.2*y.1 +x.3*y.4 -x.4*y.3 ,              x.1*y.3 -x.2*y.4 +x.3*y.1 +x.4*y.2     x.1*y.4 +x.2*y.3 -x.3*y.2 +x.4*y.1/*──────────────────────────────────────────────────────────────────────────────────────*/qShow: procedure; parse arg x;          call qXY;   $= do m=1 for 4; _=x.m; if _==0 then iterate; if _>=0 then _="+"_ if m\==1 then _= _ || substr('~ijk', m, 1);$=strip($|| _,,"+") end /*m*/ say left(arg(3), 9) right(arg(2), 20) ' ──► '$       return $/*──────────────────────────────────────────────────────────────────────────────────────*/qXY: do n=1 for 4; x.n= word( word(x, n) 0, 1) / 1; end /*n*/ if arg()==1 then do m=1 for 4; y.m= word( word(y, m) 0, 1) / 1; end /*m*/ return/*──────────────────────────────────────────────────────────────────────────────────────*/sqrt: procedure; parse arg x; if x=0 then return 0; d= digits(); i=; m.=9; h=d+6 numeric digits; numeric form; if x<0 then do; x= -x; i= 'i'; end parse value format(x, 2, 1, , 0) 'E0' with g 'E' _ .; g= g *.5'e'_ % 2 do j=0 while h>9; m.j=h; h= h % 2 + 1; end /*j*/ do k=j+5 to 0 by -1; numeric digits m.k; g= (g + x/g)* .5; end /*k*/ numeric digits d; return (g/1)i /*make complex if X<0. */ output when using the default input:  q ──► 1+2i+3j+4k q1 ──► 2+3i+4j+5k q2 ──► 3+4i+5j+6k r ──► 7 task 1: norm q ──► 5.47722558 task 2: negative q ──► -1-2i-3j-4k task 3: conjugate q ──► 1-2i-3j-4k task 4: addition r+q ──► 8+2i+3j+4k task 5: addition q1+q2 ──► 5+7i+9j+11k task 6: multiplication q*r ──► 7+14i+21j+28k task 7: multiplication q1*q2 ──► -56+16i+24j+26k task 8: multiplication q2*q1 ──► -56+18i+20j+28k  ## Ruby Works with: Ruby version 1.9 class Quaternion def initialize(*parts) raise ArgumentError, "wrong number of arguments (#{parts.size} for 4)" unless parts.size == 4 raise ArgumentError, "invalid value of quaternion parts #{parts}" unless parts.all? {|x| x.is_a?(Numeric)} @parts = parts end def to_a; @parts; end def to_s; "Quaternion#{@parts.to_s}" end alias inspect to_s def complex_parts; [Complex(*to_a[0..1]), Complex(*to_a[2..3])]; end def real; @parts.first; end def imag; @parts[1..3]; end def conj; Quaternion.new(real, *imag.map(&:-@)); end def norm; Math.sqrt(to_a.reduce(0){|sum,e| sum + e**2}) end # In Rails: Math.sqrt(to_a.sum { e**2 }) def ==(other) case other when Quaternion; to_a == other.to_a when Numeric; to_a == [other, 0, 0, 0] else false end end def -@; Quaternion.new(*to_a.map(&:-@)); end def -(other); self + -other; end def +(other) case other when Numeric Quaternion.new(real + other, *imag) when Quaternion Quaternion.new(*to_a.zip(other.to_a).map { |x,y| x + y }) # In Rails: zip(other).map(&:sum) end end def *(other) case other when Numeric Quaternion.new(*to_a.map { |x| x * other }) when Quaternion # Multiplication of quaternions in C x C space. See "Cayley-Dickson construction". a, b, c, d = *complex_parts, *other.complex_parts x, y = a*c - d.conj*b, a*d + b*c.conj Quaternion.new(x.real, x.imag, y.real, y.imag) end end # Coerce is called by Ruby to return a compatible type/receiver when the called method/operation does not accept a Quaternion def coerce(other) case other when Numeric then [Scalar.new(other), self] else raise TypeError, "#{other.class} can't be coerced into #{self.class}" end end class Scalar def initialize(val); @val = val; end def +(other); other + @val; end def *(other); other * @val; end def -(other); Quaternion.new(@val, 0, 0, 0) - other; end endend if __FILE__ ==$0  q  = Quaternion.new(1,2,3,4)  q1 = Quaternion.new(2,3,4,5)  q2 = Quaternion.new(3,4,5,6)  r  = 7  expressions = ["q", "q1", "q2",                 "q.norm", "-q", "q.conj", "q + r", "r + q","q1 + q2", "q2 + q1",                 "q * r", "r * q", "q1 * q2", "q2 * q1", "(q1 * q2 != q2 * q1)",                 "q - r", "r - q"]  expressions.each do |exp|    puts "%20s = %s" % [exp, eval(exp)]  endend
Output:
                   q = Quaternion[1, 2, 3, 4]
q1 = Quaternion[2, 3, 4, 5]
q2 = Quaternion[3, 4, 5, 6]
q.norm = 5.477225575051661
-q = Quaternion[-1, -2, -3, -4]
q.conj = Quaternion[1, -2, -3, -4]
q + r = Quaternion[8, 2, 3, 4]
r + q = Quaternion[8, 2, 3, 4]
q1 + q2 = Quaternion[5, 7, 9, 11]
q2 + q1 = Quaternion[5, 7, 9, 11]
q * r = Quaternion[7, 14, 21, 28]
r * q = Quaternion[7, 14, 21, 28]
q1 * q2 = Quaternion[-56, 16, 24, 26]
q2 * q1 = Quaternion[-56, 18, 20, 28]
(q1 * q2 != q2 * q1) = true
q - r = Quaternion[-6, 2, 3, 4]
r - q = Quaternion[6, -2, -3, -4]


## Rust

use std::fmt::{Display, Error, Formatter};use std::ops::{Add, Mul, Neg}; #[derive(Clone,Copy,Debug)]struct Quaternion {    a: f64,    b: f64,    c: f64,    d: f64} impl Quaternion {    pub fn new(a: f64, b: f64, c: f64, d: f64) -> Quaternion {        Quaternion {            a: a,            b: b,            c: c,            d: d        }    }     pub fn norm(&self) -> f64 {        (self.a.powi(2) + self.b.powi(2) + self.c.powi(2) + self.d.powi(2)).sqrt()    }     pub fn conjugate(&self) -> Quaternion {        Quaternion {            a: self.a,            b: -self.b,            c: -self.c,            d: -self.d        }    }} impl Add for Quaternion {    type Output = Quaternion;     #[inline]    fn add(self, other: Quaternion) -> Self::Output {        Quaternion {            a: self.a + other.a,            b: self.b + other.b,            c: self.c + other.c,            d: self.d + other.d        }    }} impl Add<f64> for Quaternion {    type Output = Quaternion;     #[inline]    fn add(self, other: f64) -> Self::Output {        Quaternion {            a: self.a + other,            b: self.b,            c: self.c,            d: self.d        }    }} impl Add<Quaternion> for f64 {    type Output = Quaternion;     #[inline]    fn add(self, other: Quaternion) -> Self::Output {        Quaternion {            a: other.a + self,            b: other.b,            c: other.c,            d: other.d        }    }} impl Display for Quaternion {    fn fmt(&self, f: &mut Formatter) -> Result<(), Error> {        write!(f, "({} + {}i + {}j + {}k)", self.a, self.b, self.c, self.d)    }} impl Mul for Quaternion {    type Output = Quaternion;     #[inline]    fn mul(self, rhs: Quaternion) -> Self::Output {        Quaternion {            a: self.a * rhs.a - self.b * rhs.b - self.c * rhs.c - self.d * rhs.d,            b: self.a * rhs.b + self.b * rhs.a + self.c * rhs.d - self.d * rhs.c,            c: self.a * rhs.c - self.b * rhs.d + self.c * rhs.a + self.d * rhs.b,            d: self.a * rhs.d + self.b * rhs.c - self.c * rhs.b + self.d * rhs.a,        }    }} impl Mul<f64> for Quaternion {    type Output = Quaternion;     #[inline]    fn mul(self, other: f64) -> Self::Output {        Quaternion {            a: self.a * other,            b: self.b * other,            c: self.c * other,            d: self.d * other        }    }} impl Mul<Quaternion> for f64 {    type Output = Quaternion;     #[inline]    fn mul(self, other: Quaternion) -> Self::Output {        Quaternion {            a: other.a * self,            b: other.b * self,            c: other.c * self,            d: other.d * self        }    }} impl Neg for Quaternion {    type Output = Quaternion;     #[inline]    fn neg(self) -> Self::Output {        Quaternion {            a: -self.a,            b: -self.b,            c: -self.c,            d: -self.d        }    }} fn main() {    let q0 = Quaternion { a: 1., b: 2., c: 3., d: 4. };    let q1 = Quaternion::new(2., 3., 4., 5.);    let q2 = Quaternion::new(3., 4., 5., 6.);    let r: f64 = 7.;     println!("q0 = {}", q0);    println!("q1 = {}", q1);    println!("q2 = {}", q2);    println!("r  = {}", r);    println!();    println!("-q0 = {}", -q0);    println!("conjugate of q0 = {}", q0.conjugate());    println!();    println!("r + q0 = {}", r + q0);    println!("q0 + r = {}", q0 + r);    println!();    println!("r * q0 = {}", r * q0);    println!("q0 * r = {}", q0 * r);    println!();    println!("q0 + q1 = {}", q0 + q1);    println!("q0 * q1 = {}", q0 * q1);    println!();    println!("q0 * (conjugate of q0) = {}", q0 * q0.conjugate());    println!();    println!(" q0 + q1  * q2 = {}", q0 + q1 * q2);    println!("(q0 + q1) * q2 = {}", (q0 + q1) * q2);    println!();    println!(" q0 *  q1  * q2  = {}", q0 *q1 * q2);    println!("(q0 *  q1) * q2  = {}", (q0 * q1) * q2);    println!(" q0 * (q1  * q2) = {}", q0 * (q1 * q2));    println!();    println!("normal of q0 = {}", q0.norm());}
Output:
q0 = (1 + 2i + 3j + 4k)
q1 = (2 + 3i + 4j + 5k)
q2 = (3 + 4i + 5j + 6k)
r  = 7

-q0 = (-1 + -2i + -3j + -4k)
conjugate of q0 = (1 + -2i + -3j + -4k)

r + q0 = (8 + 2i + 3j + 4k)
q0 + r = (8 + 2i + 3j + 4k)

r * q0 = (7 + 14i + 21j + 28k)
q0 * r = (7 + 14i + 21j + 28k)

q0 + q1 = (3 + 5i + 7j + 9k)
q0 * q1 = (-36 + 6i + 12j + 12k)

q0 * (conjugate of q0) = (30 + 0i + 0j + 0k)

q0 + q1  * q2 = (-55 + 18i + 27j + 30k)
(q0 + q1) * q2 = (-100 + 24i + 42j + 42k)

q0 *  q1  * q2  = (-264 + -114i + -132j + -198k)
(q0 *  q1) * q2  = (-264 + -114i + -132j + -198k)
q0 * (q1  * q2) = (-264 + -114i + -132j + -198k)

normal of q0 = 5.477225575051661


## Scala

case class Quaternion(re: Double = 0.0, i: Double = 0.0, j: Double = 0.0, k: Double = 0.0) {  lazy val im = (i, j, k)  private lazy val norm2 = re*re + i*i + j*j + k*k  lazy val norm = math.sqrt(norm2)   def negative = Quaternion(-re, -i, -j, -k)  def conjugate = Quaternion(re, -i, -j, -k)  def reciprocal = Quaternion(re/norm2, -i/norm2, -j/norm2, -k/norm2)   def +(q: Quaternion) = Quaternion(re+q.re, i+q.i, j+q.j, k+q.k)  def -(q: Quaternion) = Quaternion(re-q.re, i-q.i, j-q.j, k-q.k)  def *(q: Quaternion) = Quaternion(	 re*q.re - i*q.i - j*q.j - k*q.k,	 re*q.i + i*q.re + j*q.k - k*q.j,	 re*q.j - i*q.k + j*q.re + k*q.i,	 re*q.k + i*q.j - j*q.i + k*q.re	   )  def /(q: Quaternion) = this * q.reciprocal   def unary_- = negative  def unary_~ = conjugate   override def toString = "Q(%.2f, %.2fi, %.2fj, %.2fk)".formatLocal(java.util.Locale.ENGLISH, re, i, j, k)} object Quaternion {  import scala.language.implicitConversions  import Numeric.Implicits._   implicit def number2Quaternion[T:Numeric](n: T) = Quaternion(n.toDouble)}

Demonstration:

val q0=Quaternion(1.0, 2.0, 3.0, 4.0);val q1=Quaternion(2.0, 3.0, 4.0, 5.0);val q2=Quaternion(3.0, 4.0, 5.0, 6.0);val r=7; println("q0 = "+ q0)println("q1 = "+ q1)println("q2 = "+ q2)println("r  = "+ r)println() println("q0.re            = "+ q0.re)println("q0.im            = "+ q0.im)println("q0.norm          = "+ q0.norm)println("q0.negative      = "+ q0.negative)println("-q0              = "+ -q0)println("q0.conjugate     = "+ q0.conjugate)println("~q0              = "+ ~q0)println("q1+q2            = "+ (q1+q2))println("q2+q1            = "+ (q2+q1))println("q1+r             = "+ (q1+r))println("r+q1             = "+ (r+q1))println("q1-q2            = "+ (q1-q2))println("q2-q1            = "+ (q2-q1))println("q1-r             = "+ (q1-r))println("r-q1             = "+ (r-q1))println("q1*q2            = "+ q1*q2)println("q2*q1            = "+ q2*q1)println("q1*r             = "+ q1*r)println("r*q1             = "+ r*q1)println("(q1*q2)!=(q2*q1) = "+ ((q1*q2)!=(q2*q1)))println("q1/q2            = "+ q1/q2)println("q2/q1            = "+ q2/q1)println("q1/r             = "+ q1/r)println("r/q1             = "+ r/q1)
Output:
q0 = Q(1.00, 2.00i, 3.00j, 4.00k)
q1 = Q(2.00, 3.00i, 4.00j, 5.00k)
q2 = Q(3.00, 4.00i, 5.00j, 6.00k)
r  = 7

q0.re            = 1.0
q0.im            = (2.0,3.0,4.0)
q0.norm          = 5.477225575051661
q0.negative      = Q(-1.00, -2.00i, -3.00j, -4.00k)
-q0              = Q(-1.00, -2.00i, -3.00j, -4.00k)
q0.conjugate     = Q(1.00, -2.00i, -3.00j, -4.00k)
~q0              = Q(1.00, -2.00i, -3.00j, -4.00k)
q1+q2            = Q(5.00, 7.00i, 9.00j, 11.00k)
q2+q1            = Q(5.00, 7.00i, 9.00j, 11.00k)
q1+r             = Q(9.00, 3.00i, 4.00j, 5.00k)
r+q1             = Q(9.00, 3.00i, 4.00j, 5.00k)
q1-q2            = Q(-1.00, -1.00i, -1.00j, -1.00k)
q2-q1            = Q(1.00, 1.00i, 1.00j, 1.00k)
q1-r             = Q(-5.00, 3.00i, 4.00j, 5.00k)
r-q1             = Q(5.00, -3.00i, -4.00j, -5.00k)
q1*q2            = Q(-56.00, 16.00i, 24.00j, 26.00k)
q2*q1            = Q(-56.00, 18.00i, 20.00j, 28.00k)
q1*r             = Q(14.00, 21.00i, 28.00j, 35.00k)
r*q1             = Q(14.00, 21.00i, 28.00j, 35.00k)
(q1*q2)!=(q2*q1) = true
q1/q2            = Q(0.79, 0.02i, -0.00j, 0.05k)
q2/q1            = Q(1.26, -0.04i, 0.00j, -0.07k)
q1/r             = Q(0.29, 0.43i, 0.57j, 0.71k)
r/q1             = Q(0.26, -0.39i, -0.52j, -0.65k)

## Sidef

Translation of: Perl 6
class Quaternion(r, i, j, k) {     func qu(*r) { Quaternion(r...) }     method to_s  { "#{r} + #{i}i + #{j}j + #{k}k" }    method reals { [r, i, j, k] }    method conj  { qu(r, -i, -j, -k) }    method norm  { self.reals.map { _*_ }.sum.sqrt }     method ==(Quaternion b) { self.reals == b.reals }     method +(Number     b) { qu(b+r, i, j, k) }    method +(Quaternion b) { qu((self.reals ~Z+ b.reals)...) }     method neg { qu(self.reals.map{ .neg }...) }     method *(Number     b) { qu((self.reals»*»b)...) }    method *(Quaternion b) {        var (r,i,j,k) = b.reals...        qu(sum(self.reals ~Z* [r, -i, -j, -k]),           sum(self.reals ~Z* [i,  r,  k, -j]),           sum(self.reals ~Z* [j, -k,  r,  i]),           sum(self.reals ~Z* [k,  j, -i,  r]))    }} var q  = Quaternion(1, 2, 3, 4)var q1 = Quaternion(2, 3, 4, 5)var q2 = Quaternion(3, 4, 5, 6)var r  = 7 say "1) q norm  = #{q.norm}"say "2) -q      = #{-q}"say "3) q conj  = #{q.conj}"say "4) q  + r  = #{q + r}"say "5) q1 + q2 = #{q1 + q2}"say "6) q  * r  = #{q  * r}"say "7) q1 * q2 = #{q1 * q2}"say "8) q1q2 #{ q1*q2 == q2*q1 ? '==' : '!=' } q2q1"
Output:
1) q norm  = 5.47722557505166113456969782800802133952744694997983
2) -q      = -1 + -2i + -3j + -4k
3) q conj  = 1 + -2i + -3j + -4k
4) q  + r  = 8 + 2i + 3j + 4k
5) q1 + q2 = 5 + 7i + 9j + 11k
6) q  * r  = 7 + 14i + 21j + 28k
7) q1 * q2 = -56 + 16i + 24j + 26k
8) q1q2 != q2q1


import Foundation struct Quaternion {  var a, b, c, d: Double   static let i = Quaternion(a: 0, b: 1, c: 0, d: 0)  static let j = Quaternion(a: 0, b: 0, c: 1, d: 0)  static let k = Quaternion(a: 0, b: 0, c: 0, d: 1)}extension Quaternion: Equatable {  static func ==(lhs: Quaternion, rhs: Quaternion) -> Bool {    return (lhs.a, lhs.b, lhs.c, lhs.d) == (rhs.a, rhs.b, rhs.c, rhs.d)  }}extension Quaternion: ExpressibleByIntegerLiteral {  init(integerLiteral: Double) {    a = integerLiteral    b = 0    c = 0    d = 0  }}extension Quaternion: Numeric {  var magnitude: Double {    return norm  }  init?<T>(exactly: T) { // stub to satisfy protocol requirements    return nil  }  public static func + (lhs: Quaternion, rhs: Quaternion) -> Quaternion {    return Quaternion(      a: lhs.a + rhs.a,      b: lhs.b + rhs.b,      c: lhs.c + rhs.c,      d: lhs.d + rhs.d    )  }  public static func - (lhs: Quaternion, rhs: Quaternion) -> Quaternion {    return Quaternion(      a: lhs.a - rhs.a,      b: lhs.b - rhs.b,      c: lhs.c - rhs.c,      d: lhs.d - rhs.d    )  }  public static func * (lhs: Quaternion, rhs: Quaternion) -> Quaternion {    return Quaternion(      a: lhs.a*rhs.a - lhs.b*rhs.b - lhs.c*rhs.c - lhs.d*rhs.d,      b: lhs.a*rhs.b + lhs.b*rhs.a + lhs.c*rhs.d - lhs.d*rhs.c,      c: lhs.a*rhs.c - lhs.b*rhs.d + lhs.c*rhs.a + lhs.d*rhs.b,      d: lhs.a*rhs.d + lhs.b*rhs.c - lhs.c*rhs.b + lhs.d*rhs.a    )  }  public static func += (lhs: inout Quaternion, rhs: Quaternion) {    lhs = Quaternion(      a: lhs.a + rhs.a,      b: lhs.b + rhs.b,      c: lhs.c + rhs.c,      d: lhs.d + rhs.d    )  }  public static func -= (lhs: inout Quaternion, rhs: Quaternion) {    lhs = Quaternion(      a: lhs.a - rhs.a,      b: lhs.b - rhs.b,      c: lhs.c - rhs.c,      d: lhs.d - rhs.d    )  }  public static func *= (lhs: inout Quaternion, rhs: Quaternion) {    lhs = Quaternion(      a: lhs.a*rhs.a - lhs.b*rhs.b - lhs.c*rhs.c - lhs.d*rhs.d,      b: lhs.a*rhs.b + lhs.b*rhs.a + lhs.c*rhs.d - lhs.d*rhs.c,      c: lhs.a*rhs.c - lhs.b*rhs.d + lhs.c*rhs.a + lhs.d*rhs.b,      d: lhs.a*rhs.d + lhs.b*rhs.c - lhs.c*rhs.b + lhs.d*rhs.a    )  }}extension Quaternion: CustomStringConvertible {  var description: String {    let formatter = NumberFormatter()    formatter.positivePrefix = "+"    let f: (Double) -> String = { formatter.string(from: $0 as NSNumber)! } return [f(a), f(b), "i", f(c), "j", f(d), "k"].joined() }}extension Quaternion { var norm: Double { return sqrt(a*a + b*b + c*c + d*d) } var conjugate: Quaternion { return Quaternion(a: a, b: -b, c: -c, d: -d) } public static func + (lhs: Double, rhs: Quaternion) -> Quaternion { var result = rhs result.a += lhs return result } public static func + (lhs: Quaternion, rhs: Double) -> Quaternion { var result = lhs result.a += rhs return result } public static func * (lhs: Double, rhs: Quaternion) -> Quaternion { return Quaternion(a: lhs*rhs.a, b: lhs*rhs.b, c: lhs*rhs.c, d: lhs*rhs.d) } public static func * (lhs: Quaternion, rhs: Double) -> Quaternion { return Quaternion(a: lhs.a*rhs, b: lhs.b*rhs, c: lhs.c*rhs, d: lhs.d*rhs) } public static prefix func - (x: Quaternion) -> Quaternion { return Quaternion(a: -x.a, b: -x.b, c: -x.c, d: -x.d) }} let q: Quaternion = 1 + 2 * .i + 3 * .j + 4 * .k // 1+2i+3j+4klet q1: Quaternion = 2 + 3 * .i + 4 * .j + 5 * .k // 2+3i+4j+5klet q2: Quaternion = 3 + 4 * .i + 5 * .j + 6 * .k // 3+4i+5j+6klet r: Double = 7 print(""" q = \(q) q1 = \(q1) q2 = \(q2) r = \(r) -q = \(-q) ‖q‖ = \(q.norm) conjugate of q = \(q.conjugate) r + q = q + r = \(r+q) = \(q+r) q₁ + q₂ = \(q1 + q2) = \(q2 + q1) qr = rq = \(q*r) = \(r*q) q₁q₂ = \(q1 * q2) q₂q₁ = \(q2 * q1) q₁q₂ ≠ q₂q₁ is \(q1*q2 != q2*q1)""") Output:  q = +1+2i+3j+4k q1 = +2+3i+4j+5k q2 = +3+4i+5j+6k r = 7.0 -q = -1-2i-3j-4k ‖q‖ = 5.477225575051661 conjugate of q = +1-2i-3j-4k r + q = q + r = +8+2i+3j+4k = +8+2i+3j+4k q₁ + q₂ = +5+7i+9j+11k = +5+7i+9j+11k qr = rq = +7+14i+21j+28k = +7+14i+21j+28k q₁q₂ = -56+16i+24j+26k q₂q₁ = -56+18i+20j+28k q₁q₂ ≠ q₂q₁ is true  ## Tcl Works with: Tcl version 8.6 or Library: TclOO package require TclOO # Support class that provides C++-like RAII lifetimesoo::class create RAII-support { constructor {} { upvar 1 { end } end lappend end [self] trace add variable end unset [namespace code {my destroy}] } destructor { catch { upvar 1 { end } end trace remove variable end unset [namespace code {my destroy}] } } method return {{level 1}} { incr level upvar 1 { end } end upvar$level { end } parent	trace remove variable end unset [namespace code {my destroy}]	lappend parent [self]	trace add variable parent unset [namespace code {my destroy}]	return -level $level [self] }} # Class of quaternionsoo::class create Q { superclass RAII-support variable R I J K constructor {{real 0} {i 0} {j 0} {k 0}} { next namespace import ::tcl::mathfunc::* ::tcl::mathop::* variable R [double$real] I [double $i] J [double$j] K [double $k] } self method return args { [my new {*}$args] return 2    }     method p {} {	return "Q($R,$I,$J,$K)"    }    method values {} {	list $R$I $J$K    }     method Norm {} {	+ [* $R$R] [* $I$I] [* $J$J] [* $K$K]    }     method conjugate {} {	Q return $R [-$I] [- $J] [-$K]    }    method norm {} {	sqrt [my Norm]    }    method unit {} {	set n [my norm]	Q return [/ $R$n] [/ $I$n] [/ $J$n] [/ $K$n]    }    method reciprocal {} {	set n2 [my Norm]	Q return [/ $R$n2] [/ $I$n2] [/ $J$n2] [/ $K$n2]    }    method - {{q ""}} {	if {[llength [info level 0]] == 2} {	    Q return [- $R] [-$I] [- $J] [-$K]	}	[my + [$q -]] return } method + q { if {[info object isa object$q]} {	    lassign [$q values] real i j k Q return [+$R $real] [+$I $i] [+$J $j] [+$K $k] } Q return [+$R [double $q]]$I $J$K    }    method * q {	if {[info object isa object $q]} { lassign [my values] a1 b1 c1 d1 lassign [$q values] a2 b2 c2 d2	    Q return [expr {$a1*$a2 - $b1*$b2 - $c1*$c2 - $d1*$d2}] \		[expr {$a1*$b2 + $b1*$a2 + $c1*$d2 - $d1*$c2}] \		[expr {$a1*$c2 - $b1*$d2 + $c1*$a2 + $d1*$b2}] \		[expr {$a1*$d2 + $b1*$c2 - $c1*$b2 + $d1*$a2}]	}	set f [double $q] Q return [*$R $f] [*$I $f] [*$J $f] [*$K $f] } method == q { expr { [info object isa object$q]	    && [info object isa typeof $q [self class]] && [my values] eq [$q values]	}    }     export - + * ==}

Demonstration code:

set q [Q new 1 2 3 4]set q1 [Q new 2 3 4 5]set q2 [Q new 3 4 5 6]set r 7 puts "q = [$q p]"puts "q1 = [$q1 p]"puts "q2 = [$q2 p]"puts "r =$r"puts "q norm = [$q norm]"puts "q1 norm = [$q1 norm]"puts "q2 norm = [$q2 norm]"puts "-q = [[$q -] p]"puts "q conj = [[$q conjugate] p]"puts "q + r = [[$q + $r] p]"# Real numbers are not objects, so no extending operations for themputs "q1 + q2 = [[$q1 + $q2] p]"puts "q2 + q1 = [[$q2 + $q1] p]"puts "q * r = [[$q * $r] p]"puts "q1 * q2 = [[$q1 * $q2] p]"puts "q2 * q1 = [[$q2 * $q1] p]"puts "equal(q1*q2, q2*q1) = [[$q1 * $q2] == [$q2 * \$q1]]"
Output:
q = Q(1.0,2.0,3.0,4.0)
q1 = Q(2.0,3.0,4.0,5.0)
q2 = Q(3.0,4.0,5.0,6.0)
r = 7
q norm = 5.477225575051661
q1 norm = 7.3484692283495345
q2 norm = 9.273618495495704
-q = Q(-1.0,-2.0,-3.0,-4.0)
q conj = Q(1.0,-2.0,-3.0,-4.0)
q + r = Q(8.0,2.0,3.0,4.0)
q1 + q2 = Q(5.0,7.0,9.0,11.0)
q2 + q1 = Q(5.0,7.0,9.0,11.0)
q * r = Q(7.0,14.0,21.0,28.0)
q1 * q2 = Q(-56.0,16.0,24.0,26.0)
q2 * q1 = Q(-56.0,18.0,20.0,28.0)
equal(q1*q2, q2*q1) = 0


## zkl

Translation of: D
class Quat{   fcn init(real=0,i1=0,i2=0,i3=0){      var [const] vector= // Quat(r,i,j,k) or Quat( (r,i,j,k) )              (if(List.isType(real)) real else vm.arglist).apply("toFloat");      var r,i,j,k; r,i,j,k=vector; // duplicate data for ease of coding      var [const]	// properties: This is one way to do it         norm2=vector.apply("pow",2).sum(0.0), // Norm squared	 abs=norm2.sqrt(),		       // Norm	 arg=(r/abs()).acos(),		// Theta !!!this may be incorrect...      ;   }   fcn toString { String("[",vector.concat(","),"]") }   var [const proxy]	// properties that need calculation (or are recursive)      conj   =fcn{ Quat(r,-i,-j,-k) },		// Conjugate      recip  =fcn{ n2:=norm2; Quat(r/n2,-i/n2,-j/n2,-k/n2) },// Reciprocal      pureim =fcn{ Quat(0, i, j, k) },   	// Pure imagery      versor =fcn{ self / abs; }, 		// Unit versor      iversor=fcn{ pureim / pureim.abs; },	// Unit versor of imagery part   ;    fcn __opEQ(z) { r == z.r and i == z.i and j == z.j and k == z.k }   fcn __opNEQ(z){ (not (self==z)) }    fcn __opNegate{ Quat(-r, -i, -j, -k) }   fcn __opAdd(z){      if (Quat.isInstanceOf(z)) Quat(vector.zipWith('+,z.vector));      else			Quat(r+z,i,j,k);   }   fcn __opSub(z){      if (Quat.isInstanceOf(z)) Quat(vector.zipWith('-,z.vector));      else			Quat(r-z,vector.xplode(1)); // same as above   }   fcn __opMul(z){      if (Quat.isInstanceOf(z)){	 Quat(r*z.r - i*z.i - j*z.j - k*z.k,	      r*z.i + i*z.r + j*z.k - k*z.j,	      r*z.j - i*z.k + j*z.r + k*z.i,	      r*z.k + i*z.j - j*z.i + k*z.r);      }      else Quat(vector.apply('*(z)));   }   fcn __opDiv(z){      if (Quat.isInstanceOf(z)) self*z.recip;      else			Quat(r/z,i/z,j/z,k/z);   }    fcn pow(r){ exp(r*iversor*arg)*abs.pow(r) }	// Power function   fcn log{ iversor*(r / abs).acos() + abs.log() }   fcn exp{					// e^q      inorm:=pureim.abs;      (iversor*inorm.sin() + inorm.cos()) * r.exp();   }}
    // Demo coder:=7;q:=Quat(2,3,4,5); q1:=Quat(2,3,4,5); q2:=Quat(3,4,5,6); println("1.          norm: q.abs: ", q.abs);println("2.                   -q: ", -q);println("3.    conjugate: q.conj: ", q.conj);println("4.          Quat(r) + q: ", Quat(r) + q);println("                  q + r: ", q + r);println("5.              q1 + q2: ", q1 + q2);println("6.          Quat(r) * q: ", Quat(r) * q);println("                  q * r: ", q * r);println("7.              q1 * q2: ", q1 * q2);println("                q2 * q1: ", q2 * q1);println("8.  q1 * q2 == q2 * q1 ? ", q1 * q2 == q2 * q1); i:=Quat(0,1); j:=Quat(0,0,1); k:=Quat(0,0,0,1);println("9.1               i * i: ", i * i);println("                  J * j: ", j * j);println("                  k * k: ", k * k);println("              i * j * k: ", i * j * k); println("9.2             q1 / q2: ", q1 / q2);println("9.3        q1 / q2 * q2: ", q1 / q2 * q2);println("           q2 * q1 / q2: ", q2 * q1 / q2);println("9.4      (i * pi).exp(): ", (i * (0.0).pi).exp());println("            exp(j * pi): ", (j * (0.0).pi).exp());println("            exp(k * pi): ", (k * (0.0).pi).exp());println("                q.exp(): ", q.exp());println("                q.log(): ", q.log());println("          q.log().exp(): ", q.log().exp());println("          q.exp().log(): ", q.exp().log()); s:=q.exp().log();println("9.5 let s=q.exp().log(): ", s);println("                s.exp(): ", s.exp());println("                s.log(): ", s.log());println("          s.log().exp(): ", s.log().exp());println("          s.exp().log(): ", s.exp().log());
Output:
1.          norm: q.abs: 7.34847
2.                   -q: [-2,-3,-4,-5]
3.    conjugate: q.conj: [2,-3,-4,-5]
4.          Quat(r) + q: [9,3,4,5]
q + r: [9,3,4,5]
5.              q1 + q2: [5,7,9,11]
6.          Quat(r) * q: [14,21,28,35]
q * r: [14,21,28,35]
7.              q1 * q2: [-56,16,24,26]
q2 * q1: [-56,18,20,28]
8.  q1 * q2 == q2 * q1 ? False
9.1               i * i: [-1,0,0,0]
J * j: [-1,0,0,0]
k * k: [-1,0,0,0]
i * j * k: [-1,0,0,0]
9.2             q1 / q2: [0.790698,0.0232558,-2.77556e-17,0.0465116]
9.3        q1 / q2 * q2: [2,3,4,5]
q2 * q1 / q2: [2,3.46512,3.90698,4.76744]
9.4      (i * pi).exp(): [-1,1.22465e-16,0,0]
exp(j * pi): [-1,0,1.22465e-16,0]
exp(k * pi): [-1,0,0,1.22465e-16]
q.exp(): [5.21186,2.22222,2.96296,3.7037]
q.log(): [1.99449,0.549487,0.732649,0.915812]
q.log().exp(): [2,3,4,5]
q.exp().log(): [2,0.33427,0.445694,0.557117]
9.5 let s=q.exp().log(): [2,0.33427,0.445694,0.557117]
s.exp(): [5.21186,2.22222,2.96296,3.7037]
s.log(): [0.765279,0.159215,0.212286,0.265358]
s.log().exp(): [2,0.33427,0.445694,0.557117]
s.exp().log(): [2,0.33427,0.445694,0.557117]
`