SCALES OF NOTATION 19

48. Th. If either a or b is positiv and the other negativ,
la + 8| < la| + [8].()
Forexample, |7+ — 3| = |4| =4<74+3=|7| + | — 3|
49. Th. |a +b| Z |a| + |b]
This is got by combining §§ 47, 48.

50. Th. |la+b+4+c+ ---| Z |le| + [b] + |c| + -

In words: The numerical value of the sum of any number of
integers is equal to or less than (not greater than) the sum of their
numerical values.

51. Th. If none of the integers a, b, c, - -+ is negativ or none
of them s positiv,
6.+ b+ aden oo fnl o WS ohff o

52. Th. If at least one of the integers a, b, c, is positiv and
at least one negativ,

o + b a4 oo <o Sl Rigl e

53. Th. The numerical value of the product of any number
of integers is equal to the product of their numerical values.(*)

For example, |3 X (—5) X2| =|—30] =30=3 X5 X 2
= 3] X [=5] X [2]

DIVISIBILITY

54. Def. One integer @ is said to be divisible by, or a
multiple of, a second integer b, if b is not zero and there exists
a third integer ¢, such that @ = ¢gb. The third integer ¢ is
called the quotient of @ and b.

Thus 6 is divisible by 2, the quotient being 3; 0 is divisible
by 4, the quotient being 0; the integer 5 is not divisible by
the integer 2, since there is no third integer g, such that

(*) For proof, see case 4, § 400, Theory of Integers.
(2) Theory of Integers, § 405.