70 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

a small base the disadvantage of requiring the writing of many
figures.

Where small numbers are concernd 4 would seem to be an
ideal base for computation, the tables being very easy to
memorize and the speed of computation very rapid.

For computation in general a base that is neither very large
nor very small would seem best. This requirement seems to
be met by the numbers 8,9, ¢, 0, ©, 2, ¥, &, 1.

Some may think that the addition and multiplication tables
required for the base [) would involve an impracticable amount
of memorizing. But we now generally know these tables in
the decimal system up to twelv plus twelv and twelv times.
twelv. And many business men say that the multiplication
table should be taught up to twenty times twenty(!). For
these reasons the larger bases of this group would seem prefer-
able to the smaller.

140. For representing whole numbers an even base has
apparently no advantage over an odd one. But we have to
deal with fractions. So the best of the above mentiond bases
would seem to be that with which the most useful fractions
are most simply represented.

The most useful fraction is 1/2. To represent this simply
the base must be an even number. So the choice is narrowd
down to 8§, ¥, B, £, 0.

In the system whose base is fourteen, sevenths and four-
teenths would be very simply represented, but no one would
claim that these are very important fractions.

In the decimal system fifths and tenths are very simply
represented, but except for the fact that ten is the base with
which we have been traind to work and that scientific litera-
ture and other literature is now based on the decimal system,
we would not have much more reason for dealing with fifths
and tenths than with sevenths and fourteenths.

(1) See J. W. A. Young, The Teaching of Mathematics, p. 218, New
York, Longmans, Green, and Co., 1907.