Complex Numbers 69
1.19 HYPERCOMPLEX NUMBERS. QUATERNIONSOrdinary complex numbers have been defined as ordered pairs (x 1 , x 2 ) of
real numbers subject to certain rules of equality, addition, and multipli-
cation. A possible generalization suggests itself at once, namely, that of
considering ordered n-tuples (x 1 , x 2 , ••• , Xn) of real numbers subject also
to certain rules of equality, addition, and multiplication. This idea was
first developed by Hamilton [13] and by Grassman [11].
If equality and addition are defined componentwise, as for ordinary
complex numbers, it can be shown that no matter what definition of mul-
tiplication is chosen, the set of those n-tuples (n > 2), called hypercomplex
numbers, does not constitute a (commutative) field, and that only in the
case n = 4, with a suitable definition of multiplication, one obtains a non-
commutative field. In all other cases the construction leads to some other
algebraic system.
Let x = (x1, x2, ... , Xn) and y = (yi, y2, ... , Yn) be two n-tuples with
components Xk,Yk (k = 1, ... ,n) belonging to a certain given field F (in
particular, the real or the complex field), and suppose that equality, addi-
tion, and multiplication by the elements of F (sometimes called the scalars)
are defined as follows:
- Equality: x = y iff Xk = Yk (k = 1, ... , n)
- Addition: x + y = (x1 + Y1, ... , Xn + Yn)
3. Multiplication by scalars: ex = ( cx 1 , ..• , cxn) for any c E F
Then it is easily seen, as in Section 1.5, that the set of elements X =
{x, y, ... } constitutes a linear or vector system of dimension n over F. The
n-tuples are then called n-dimensional vectors. They may also be thought
of as points in an n-dimensional space with coordinates Xk ( k = 1, ... , n ).
From (2) and (3) we have
x = (x1, X2, ... , Xn) = x1(l, O, ... , 0) + x2(0, 1, ... , 0) + · · · + Xn(O, 0, ... , 1)
and if we let
e 1 = (1, O, ... , 0), e2 = (0, 1, ... , 0), ... , en= (0,0, ... ,1)
we get the representation
(1.19-1)which is similar to the binomial form of the complex number.
The vectors ek are called unit vectors in X. They are linearly inde-
pendent in X and thus constitute a basis in X; i.e., any x E X can be
expressed as a linear combination of the ek with coefficients in F.