The
Quaternions denoted
H are a
division ring or
skew field. This means that you can
add,
subtract,
multiply and
divide. Notice though that
multiplication is non-
commutative.
They were invented by Hamilton in 1843 who was so pleased that he scratched the defining relations on Brougham Bridge on the Royal Canal in Dublin.
As a real vector space the Quaternions have basis 1,i,j,k
and the mutiplication can be deduced from the rules
i2=j2=k2=-1, ij=-ji=k, jk=-kj=i, ki=-ik=j.
For each quaternion
q=t+xi+yj+zk in
H we can define
b(q)=t-xi-yj-zk. It's easy to see that
qb(q)=t2+x2+y2+z2
Note that if q is nonzero, so that one of t,x,y,z is
nonzero, then qb(q) is a nonzero real number. It follows
that such a q has inverse
b(q)/(t2+x2+y2+z2)
The quaternions have a
concrete description as a subalgebra of
2x2
complex matrices. The quaternion
q=t+xi+yj+zk
corresponds to the matrix
-- --
| t+xi y+zi |
| -y+zi t-xi |
-- --
The subgroup of the group of units of H consisting
of {1,-1,i,-i,j,-j, k,-k} is called the Quaternion group (or
Pauli spin group). In this group of order 8 all of the elements except for 1 and -1 have order 4, with -1 having order 2.