64 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

130. Hence we can change in two steps from the octimal
to either the quaternary or sexidenal, and reversely, but there
is no such simple rule for changing directly, because [0 is not
an integral power of 8, and 8 is not an integral power of 4.

131. Of course we can derive similar rules for interchange
between the scales having as bases 7, 72, 73, -+ -, in particular
3:9, 27, 5=

RADIX NOTATIONS APPLIED TO FRACTIONS

132. The various notations, binary, ternary, etc., can be
applied to fractions as well as to whole numbers, just as the
decimal notation can be (%).

Thus in the binary notation 1/2 = .1, 1/4 = .01; in the
ternary 1/2 = .1111 ---, 1/3 = .1; in the septenary 2/7 = .2,
3/4 = .515151 --- = .51; all of which results may be obtaind
by applying the process of short division in the various
notations.

133. It is easy to see that the series of figures thus repre-
senting a fraction will not terminate, if the denominator of
the fraction, when the fraction is in its lowest terms, has any
prime factor which is not a factor of the radix of the notation.

134. The most useful fractions are 1/2, 1/4, 1/8, 1/I0 and
their proper multiples. The fractions 1/3, 1/6, 1/8, 1/5, 1/'¢
and their proper multiples are also much used. Let us see
how these fractions are represented in some of the systems we
have described. The results are given in the following tables.

(*) See Chrystal, l.c., p. 170.