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D-2
The operator mod is used in this book, and in the literature, in two different ways, as a binary
operator and as a congruence relation. This appendix explains the distinction and precisely
defines the notation used in this book regarding parentheses. This notation is common but,
unfortunately, not universal.
D.1 THE BINARY OPERATOR mod
If a is an integer and n is a nonzero integer, we define a mod n to be the remainder when a is
divided by n. The integer n is called the modulus, and the remainder is called the residue. Thus,
for any integer a, we can always write:
a = a/n × n + (a mod n)
Formally, we define the operator mod as follows:
a mod n = a – a/n × nfor n ≠ 0
As a binary operation, mod takes two integer arguments and returns the remainder. For
example, 7 mod 3 = 1. The arguments may be integers, integer variables, or integer variable
expressions. For example, all of the following are valid, with the obvious meanings:
7 mod 3
7 mod m
x mod 3
x mod m
(x2 + y + 1) mod (2m + n)
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