MATHEMATICAL RECREATIONS 175

445. Th. If m=a-+b<2m,
in particular, if n = a <mand n = b < m,
then 0 < c+d=mand 0= e + f < m.

APPLICATIONS TO NUMBERS IN THE DECIMAL SYSTEM

446. Now suppose that ¢ and b are positiv or zero integers
written decimally. Take m = 10.

Th. Then
#n=->3
c=10 — a d=10—1b
e=a—15 f=b+:35
c+e=25 d+f=25
c+d=20— (a +0)
e+ f=(a+0b) — 10
(c+d)+(+f) =1
c+d=10— (e +§)
e+ f=10 — (c + d)

447. Th. a4+ b =10+ (e + )
That is, the sum of two numbers equals 10 plus the sum of their
excesses over 5.

448. Th. ab = (e + )10 + cd.

That is, the product of two numbers equals 10 times the sum of
their excesses over 5 plus the product of their complements with
respect to 10.

449. Th. ab=[5+(+ )15+ ¢

- That is, the product of two numbers equals 5 times the sum
of 5 and their excesses over 5 plus the product of these excesses.

450. Th. a+b=ce-+ f (mod 10)

That is, the sum of two numbers is congruent to the sum of their
excesses over 5 with respect to the modulus 10.