The integer quotient of and may be defined as
. This formulation leads to the following characterization of
the modulus function:^{1.2}

Notation: We shall generally use the infix notation for . For the purpose of resolving ambiguous expressions, the precedence of this operator is highr than that of addition and lower than that of multiplication.

Although is of interest mainly when , , and , the definition is less restrictive and arbitrary real arguments must be considered. The following closure properties are worth noting.

PROOF: By Definitions 1.2.1 and 1.1.1,

In the usual case , we also have an upper bound on .

PROOF: By Definitions 1.2.1 and 1.1.1,

PROOF: By Lemmas 1.2.2 and 1.2.3, and .

When , we have another upper bound on .

PROOF: By Lemma 1.2.1, . If , then , by Lemma 1.1.3, and . If , then , and again, .

PROOF: Since , by Definition 1.1.1. Now by Lemma 1.2.1,

We have two equivalent conditions for .

PROOF: This follows immediately from Lemma 1.2.1.

PROOF: By Lemma 1.2.1,

PROOF: If , then . Otherwise, Lemma 1.2.8 applies.

In the event that is a multiple of , the following lemma may be used to identify the muliplier.

PROOF: Since , we have . By Lemma 1.2.8, , which implies .

PROOF: This is Lemma 1.2.10 with .

PROOF: By Lemma 1.2.8,

and | |||

and | |||

and |

We have the following generalization of Lemma 1.2.8.

PROOF: By Lemma 1.2.1,

PROOF: Since , . The result follows from Lemma 1.2.13.

PROOF: By Lemma 1.1.4,

PROOF: By Lemmas 1.2.15 and 1.2.6,

PROOF: By Lemmas 1.2.15 and 1.2.5,

PROOF: By Lemma 1.2.1,
, and hence

PROOF: By Lemma 1.2.1,
, and hence

PROOF: By Lemma 1.2.1,

Lemma 1.2.20 is used most frequently with power-of-two moduli.

PROOF: This is the case of Lemma 1.2.20 with and .

PROOF: By Lemma 1.2.1,

PROOF: By Definition 1.2.1 and Lemma 1.1.4,

PROOF: By Lemma 1.2.18,

The next four lemmas are trivial consequences of preceding results, but are worth noting because of the frequency with which the modulus 2 occurs in our applications.

PROOF: By Lemmas 1.2.2 and 1.2.3, and .

PROOF: By Lemma 1.2.13,

PROOF: This follows from Lemma 1.2.13 and the observation that .

PROOF: By Lemma 1.2.1,

David Russinoff 2017-08-01