THE MAKING OF NUMBERS I3

10203. Here the nought symbol (or cipher) is used to fill the
gaps where no other digits are required and so to prevent the
other digits from running into each other.

If, however, we accept nought as a number in its own right,
we have to make allowances for certain peculiarities observ-
able in its relation to other numbers.

The division of one number by another can be resolved
giving a distinct answer which, for each particular division, is
invariable. It is impossible, however, to divide any other
number by nought and to express the result in any definite
form. The result of such a division is said to be infinity and,
in this connexion, it really means that the answer is as large
as we like to make it.

The remarkable point about this is that we can use the
smallest number imaginable as a divisor and still obtain a
definite result, no matter how small the number.

Thus, in the equationgzz, where  is the result of divid-

ing x by », £ is infinity if y equals nought, and it is clear that
we can never express the meaning of infinity by the use of
digits. Yet, if 2" is not equal to nought, then < has a definite
meaning in terms of digits, and this is so no matter whether
the value of ¥ be positive or negative. As we make 1 smaller
and smaller (that is, as we approach nought), so { becomes
larger and larger (as we approach infinity).

But we can make 1" progressively smaller and yet never
reach nought at all. Whatever the original value of 1, we can
obviously divide it by 2 and so obtain a smaller number
which can never be equal to nought, even though we may
approach it ever more closely. Like a desert mirage, the
nearer we get to it, so the further off it seems.

On the other hand, the division of nought by a non-zero
number has the peculiarity that the answer is always nought.

Similarly, multiplication by nought always gives the an-
swer nought irrespective of what numbers are multiplied.
There is nothing really remarkable about this; for if we are
short of water, it makes no difference how many buckets we

TR e

B

o
N S SR

G R BB R

f
s
3
&
&
-2
&
}"_

REERENRSN

SIS AR it Bt