tasks a-s

Task/Quaternion-type/GAP/quaternion-type.gap

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+# 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
+
+# Computing norm may be difficult, since the result would be in a quadratic field (Sqrt exists in GAP, is quite unusual, see ?E in GAP, and the following example
+Sqrt(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
+
+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 it
+r:=5/3*b[1];
+# (5/3)*e
+r + 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)*k
+5/3 * q;
+# (5/3)*e+(10/3)*i+(5)*j+(20/3)*k
+
+# Negative
+-q;
+(-1)*e+(-2)*i+(-3)*j+(-4)*k