SCALES OF NOTATION 23

82. In some of the following theorems we will have as part
of the hypothesis the statement
b—-=0~a-3>b~ |a| > |b]
It is interesting to note that this statement can be written in
another form as indicated in the following theorem.

Th. [b-=0~a-3b~ |a| > |b]|]
=[a-3>b~1<|b| < |e|]

83. Th. Ifb—=0,a->>b,and |a| > |b|; then if a and b
are both positiv or both negativ, both approximate quotients ob-
taind by dividing a by b are positiv; if either a or b is positiv
and the other negativ, these approximate quotients are both negativ;
and conversely, if either of these quotients is positiv, a and b are
either both positiv or both megativ; if either quotient is negativ,
either a or b is positiv and the other negativ.

Since b —= 0, |b| is positiv and b is positiv or negativ.
Then, since |a| > |b|, |a| is positiv and hence a is positiv
or negativ.

Let f be the lower approximate quotient and 7 the corre-
sponding positiv remainder.

Then a = fb + r, where |b] > 7. § 80.

Then, since |a| > |b], |a| > 7.

Hence, if a is positiv, @ > 7 and therfor @ — 7 is positiv.

If a is negativ, since 7 is positiv, @ — r is negativ.

Thus ¢ — 7 is positiv or negativ according as a is positiv
or negativ.

Nowf=(a—7r):b

Therfor, if @ and b are both positiv or both negativ, f is
positiv; if either a or b is positiv and the other negativ, f is
negativ.

Let g be the upper approximate quotient and s the corre-
sponding negativ remainder.

Then a = gb + s, where |b]| > |s]

Then, since |a| > |b], || > |s|

Now, if @ is positiv, since s is negativ, @ — s is positiv.

If ¢ is negativ, since also s is negativ and |a| > [s], we
have — @ > — s, whence a < s and @ — s is negativ.