CHAPTER V
MATHEMATICAL RECREATIONS
COMPLEMENTS AND EXCESSES

420. Def. If m and a are two finite numbers, m — @ may
be called the complement of ¢ with respect to .

421. Th. Thus, if ¢ is the complement of a with respect to m,

c=m-—a
m=a +c¢
a=m—c¢

422. Th. If m = 0, the complement of a with respect to m
1s the same as the opposit of a.

423. Th. Also, the complement of the complement of a number
with respect to the same number is the given number.

Form — (m — a) = a

As a special case of this theorem the opposit of the opposit
of a number is the number itself.

424. Th. If c is the complement of a with respect to m, then
0<a<ml2) =ml2<c<m)
In this case, since 0 < m/2, m is positiv.

425. Def. We will suppose that m is a fixed number, which
we will call the base, and speak of m — a simply as the com-
plement of a.

426. Th. If a and b are two finite numbers, ¢ the complement
of a and d the complement of b, then
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