22 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

75. Def. The number fis called the lower approximate quo-
tient, ¢ the upper approximate quotient when « is divided by b.

These approximate quotients are found from @ and b by an
operation that may be called approximate division to dis-
tinguish it from the division mentioned in § 56, which may be
called by contrast exact division, the quotient obtaind in exact
division also being called exact.

Thus when 25 is divided by 6, we find the lower quotient 4
and upper quotient 5; when — 25 is divided by — 6, we find
the lower quotient 5 and upper quotient 4.

76. Th. In every case g — f = =% 1, according as b is positiv
or negativ.

77. Def. 1If g is either an exact or lower or upper approxi-
mate quotient, obtaind by dividing a by b, a — gb is called
the remainder.

78. Th. The remainder corresponding to an exact quotient
is zero, the remainder corresponding to a lower quotient is positiv,
that corresponding to an upper quotient is negativ.

Thus, when 24 is divided by 6, the quotient is 4 and the
remainder is 24 — 4 X 6, which equals 0; when 25 is divided
by 6, the remainder corresponding to the lower quotient 4 is
25 — 4 X 6, which is positiv, the remainder corresponding to
the upper quotient 5 is 25 — 5 X 6, which is negativ.

79. Th. When a is approximately divided by b, the sum of
the numerical values of the positiv and negativ remainders is
equal to the numerical value of b.

80. Th. When a is approximately divided by b, both re-
mainders are numerically less than b.

81. Th. Ifa=qb+ rand |r| < |b|; then, if r = 0, q is
the exact quotient of a divided by b; if r is positiv, q is the lower
approximate quotient; if v is negativ, q is the upper approximate
quotient; in each case r is the corresponding remainder.