26 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

Exs.

 

 

 

88. Th. Ifa>>band 1< |b| < |a|, then
1+ |a:b| < |a| and |a :b| < |a]

Since 1 < |b], 2= |b|

Let ¢ = a : b, whence a = ¢b

Since a = gb and 1 < |b| < |al, I<dal 1)
Hence |g| is positiv.

Hence, since 2 = |b|, lg] X 2= |q]||b]
Since 1 < |¢], P lel < gl el
Then

1+ la:b] =14 |q] < |g| + |q
= |q| X 2= [q]|[b] = |gb]| = |a]

Therfor 14+ |a:b] < |al

Hence, since |a : b] <1+ |a :b], la : 8| < |a|
Ex.1. a=6 b=—-3¢q¢g=—21+]|q =3<6=]a]
Ex.2. a=—4b=—-2¢=2 1+ |¢ =3<4=]a|

8. Th. If a->3b, 1 < |b| < |a|, and f and g are the
lower and upper quotients obtaind by dividing a by b, then
lg| — |f] = == 1, according as a is positiv or negativ.

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

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

Alsog — f = =+ 1, according as b is positiv or negativ. § 76.

Hence =+ |g| F |f| = % 1, according as b is positiv or
negativ.

(*) By § 501, Theory of Integers.