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

62 INTRODUCTION TO MATHEMATICS
portant symbols should have an important
meaning. This is one of the merits of the
Arabic notation for numbers; by means of
ten symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and by
simple juxtaposition it symbolizes any number
whatever. Again in algebra, when we have
two variable numbers x and y, we have to
make a choice as to what shall be denoted by
their juxtaposition xy. Now the two most
important ideas on hand are those of addition
and multiplication. Mathematicians have
chosen to make their symbolism more concise
by defining xy to stand for xXy. Thus the
laws (3), (4), and (5) above are in general
written,
xy==yx, (xy)z=x(yz), x(y +z) =xy +xz,
thus securing a great gain in conciseness.
The same rule of symbolism is applied to the
juxtaposition of a definite number and a variable: we write Sx for 3 Xx, and 30x for 30 Xx.
It is evident that in substituting definite
numbers for the variables some care must be
taken to restore the X, so as not to conflict
with the Arabic notation. Thus when we
substitute 2 for x and 3 for y in xy, we must
write 2 X 3 for xy, and not 23 which means
20+3.
It is interesting to note how important for
the development of science a modest-looking
symbol may be. It may stand for the emphatic presentation for an idea, often a very