172 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

a=m-—c¢c
b=m—d
a+b=2m— (c+d)
a—b=—(c—d)
ab = (m — c)(m — d)

Il

m? — (c + d)ym + cd
[m — (c + d)Jm + cd

427. In considering the complements of various numbers
with respect to the same base we consider differences whose
minuends are the same.

Now let us consider differences whose subtrahends are the
same.

428. Def. If a and » are two finite numbers, ¢ — #» may
be called the excess of a over 7.

429. Th. Thus, if e is the excess of a over n,

e=a —n
a=mn-+e
n=a—e

430. Th. If n = 0, the excess of a over n is a itself.

431. Def. We will suppose that # is a fixed number, which
we will call the base, and speak of ¢ — 7 simply as the excess
of a.

432. Th. If a and b are two finite numbers, e the excess of
a and f the excess of b, then

a=mn-+e

b=n+4f
a+b=2n+ (e + )
a—b=e¢—f

ab = (n + e)(n + f)

Il

n? + (e + f)n + of
[n+ (e+f)ln + ¢f