CHAPTER II

SCALES OF NOTATION

42. In this chapter and the next we will find it convenient
to make use of certain ideas of arithmetic with which the
reader may not be familiar. So we will take a few pages here
to give a brief outline of the corresponding theories. ()

In the following pages, unless otherwise stated, the letters
@, b, ¢, --- will be used to denote integers, positiv, zero, or
negativ. Thus ¢ may stand for 4, or 0, or — 5.

NUMERICAL VALUE

43. Def. |a| is called the numerical, arithmetical, or abso-
lute value of @, and stands for « itself, if @ is either positiv
or zero, but for — a, if @ is negativ.

Thus [6] = 6, [0] =0, |— 3| = 3.

44. Th. If a is negativ or zero, |a| = — a.
For — 0 = 0.

45. Th. The numerical value of a positiv or negativ integer
1s positiv; the numerical value of a zero integer is zero; the numeri-
cal value of an integer is never negativ.

This theorem follows easily from the definition.

46. Th. If a lies between — b and b, that is, if either
—b<a<borb<a< —bthen |a| < |b|; and conversely.

First suppose ~ <

Then — b <b.

Hence b is positiv. For, if b were zero, — 0 would be less
than 0; and, if b were negativ, — b would be positiv and we

would have a positiv number less than a negativ number.

(1) A fuller treatment of these subjects will be found in the author's
Elements of the Theory of Integers, rev. ed., Garden City, N. Y., 1931.

17