An introduction to mathematics, by A. N. Whitehead.

96 INTRODUCTION TO MATHEMATICS
length and in a certain direction. The line
OP may be called the vector line from 0 to
P, or the step from O to P. We see, therefore,
that we have in this chapter only extended
the interpretation which we gave formerly of
the positive and negative numbers. This
method of representation by vectors is very
useful when we consider the meaning to be
assigned to the operations of the addition and
multiplication of ordered couples.
We will now go on to this question, and
ask what meaning we shall find it convenient
to assign to the addition of the two ordered
couples (x, y) and (x', y'). The interpretation must, (a) make the result of addition
to be another ordered couple, (b) make the
operation commutative so that (x, y) +
(x', y') = (x, y') +(x, y), (c) make the operation associative so that
{(x, y)+(x', y') + (u, )
= (x, y) + {(x', y') + (u, v)},
(d) make the result of subtraction unique,
so that when we seek to determine the unknown ordered couple (x, y) so as to satisfy
the equation
(x, y)+(a, b)=(c, d),
there is one and only one answer which we
can represent by
(x, y)= (c, d)-(a, b).