1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 71

where i,j, k, m = 1, 2, ... , n, giving n^4 equations that are to be satisfied by

the na multiplication constants. If the equations (1.19-6) are satisfied, the

hypercomplex system forms what is known as a linear associative algebra ·
of order n.
In the case n = 4 the constants Cjkl can be chosen so as to satisfy (1.19-6).

It suffices to define the product of any two units ei and ek by means of

the multiplication table.

e1 e2 ea e4

ei ei e2 ea e4

e2 e2 -e1 e4 -ea

ea ea -e4 -e1 e2

e4 e4 ea -e2 -e1

From the table it is easy, but laborious, to find the values of the 64 constants
Cjkl~ and to check (1.19-6). For instance, c212 = 1, c211 = c21a = c214 = O,
and so on.
With this definition of multiplication, and with F the real number field,
the system of hypercomplex numbers becomes the system of real quater-
nions, as introduced by Hamilton. The subset of quaternions of the special
form (x 1 ,0,0,0) is readily seen to be isomorphic to the real number sys-
tem, so the real number system can be embedded in the quaternion system.
Using Hamilton's notations

we have

e1=(1,0,0,0)=1

ea = (0, 0, 1, 0) = j

e 2 = (0, 1, O, 0) = i

e4=(0,0,0,1) = k

(x1, X2, Xa, X4) = X1 + X2i + Xaj + X4k

Then the multiplication table can be summarized as follows:

12 =1, i^2 =j^2 =k^2 =-1

i. 1 = 1. i = i, j. 1 = 1. j = j, k. 1 = 1. k = k

ij = -ji = k, jk=-kj=i,

and the product of two quaternions is given by

(x1 + x2i + xaj + X4k)(Y1 + Y2i + Yai + y4k)

ki = -ik = j

= (x1Y1 - X2Y2 - XaYa - x4y4) + (x1Y2 + X2Y1

+ XaY4 - X4ya)i + (x1Ya - X2Y4 + XaY1 + X4Y2)j

+ (x1y4 + x2Ys - xsY2 + x4y1)k (1.19-7)