SCALES OF NOTATION 27

Therfor |g| — |f| = 1, whether b is positiv or negativ.

Next, if a is negativ, f and g are both negativ or both positiv,
according as b is positiv or negativ. § 83.

Hence f = F |f| and g = F |g| according as b is positiv
or negativ.

Also g — f = = 1 according as b is positiv or negativ. § 76.

Hence F |g| & |f| = & 1 according as b is positiv or
negativ.

Therfor |g| — |f| = — 1 whether b is positiv or negativ.

Combining our two results, we have that [g| — |[f| = £ 1
according as a is positiv or negativ.

90. Th. If a->3b, 1 < |b| < |a|, and f and g are the
lower and upper quotients obtaind when a is divided by b, then
Ifl < la| and |g| < |al.

Since 1 < |b], |b] is positiv.

Hence b is positiv or negativ.

Also, since 1 < |b], 2= |b]

Since a —>% b, a is positiv or negativ. § 60.

Suppose a is positiv.

Then, if b is positiv, f is positiv and hence fb is positiv;

if b is negativ, f is negativ and hence fb is positiv. § 83.

Thus fb is positiv whether b is positiv or negativ.

Hence |fb| = fb

Since a is positiv, |a¢| = a

Since f is the lower quotient, fo < a

Hence /o] < |a|

Since f is positiv or negativ, |f| is positiv.

Hence, since 2 = |b], L% 2= 11 1D

Since |f| is positiv, 1=0f

Hence -+ [fE= TR AT
Then

1+ Ifl = Ifl + Ifl = Ifl X2= [fl13] = |fb] < |a
Therfor 14+ |f] < |al

Hence, since |f| < 1 4+ |f| and, asais positiv, |g] =1+ |f]
(§ 89), we have |f| < |a| and |g| < |a|