28 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

Now suppose a is negativ.
Then, if bis positiv, g is negativ and hence gb is negativ;

if b is negativ, g is positiv and hence gb is negativ. § 83.
Thus gb is negativ whether b is positiv or negativ.

Hence |gb| = — gb

Since a is negativ, |¢| = — a

Since g is the upper quotient, gb > a

Hence —gbh< —a

And lgb| < |a]

Since g is positiv or negativ, |g| is positiv.
Hence, since 2 = |b], lg] X 2= |g]|]|b]
Since |g| is positiv, 1= |g]
Hence 14+ [g] = lg] + g
Then

1+ [gl = lel + lgl = gl X 2= [g][p] = |gb] < |a]
Therfor 14 |g| < |a]

Hence, since [g| <1+ [g| and, asaisnegativ, |f| = 1 + |g]
(§ 89), we have [f| < |a| and |g| < |a|

Exs.

 

 

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GENERALIZED DIVISION

91. Def. We are going to find it convenient now to gen-
eralize our idea of division. Having given any three integers
a, b, and g, if we subtract ¢b from a, we will say that we have