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In mathematics, the term ''modulo'' ("with respect to a modulus of", the
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
ablative In grammar, the ablative case (pronounced ; sometimes abbreviated ) is a grammatical case for nouns, pronouns, and adjectives in the grammars of various languages; it is sometimes used to express motion away from something, among other uses. T ...
of '' modulus'' which itself means "a small measure") is often used to assert that two distinct mathematical objects can be regarded as equivalent—if their difference is accounted for by an additional factor. It was initially introduced into mathematics in the context of
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his bo ...
by
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
in 1801. Since then, the term has gained many meanings—some exact and some imprecise (such as equating "modulo" with "except for"). For the most part, the term often occurs in statements of the form: :''A'' is the same as ''B'' modulo ''C'' which means :''A'' and ''B'' are the same—except for differences accounted for or explained by ''C''.


History

''Modulo'' is a
mathematical jargon The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in ...
that was introduced into mathematics in the book ''
Disquisitiones Arithmeticae The (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on th ...
'' by
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
in 1801. Given the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s ''a'', ''b'' and ''n'', the expression "''a'' ≡ ''b'' (mod ''n'')", pronounced "''a'' is congruent to ''b'' modulo ''n''", means that ''a'' − ''b'' is an integer multiple of ''n'', or equivalently, ''a'' and ''b'' both share the same remainder when divided by ''n''. It is the
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
ablative In grammar, the ablative case (pronounced ; sometimes abbreviated ) is a grammatical case for nouns, pronouns, and adjectives in the grammars of various languages; it is sometimes used to express motion away from something, among other uses. T ...
of '' modulus'', which itself means "a small measure." The term has gained many meanings over the years—some exact and some imprecise. The most general precise definition is simply in terms of an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
''R'', where ''a'' is ''equivalent'' (or ''congruent)'' to ''b'' modulo ''R'' if ''aRb''. More informally, the term is found in statements of the form: :''A'' is the same as ''B'' modulo ''C'' which means :''A'' and ''B'' are the same—except for differences accounted for or explained by ''C''.


Usage


Original use

Gauss originally intended to use "modulo" as follows: given the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s ''a'', ''b'' and ''n'', the expression ''a'' ≡ ''b'' (mod ''n'') (pronounced "''a'' is congruent to ''b'' modulo ''n''") means that ''a'' − ''b'' is an integer multiple of ''n'', or equivalently, ''a'' and ''b'' both leave the same remainder when divided by ''n''. For example: : 13 is congruent to 63 modulo 10 means that : 13 − 63 is a multiple of 10 (equiv., 13 and 63 differ by a multiple of 10).


Computing

In
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, ...
and
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...
, the term can be used in several ways: * In
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, ...
, it is typically the
modulo operation In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is th ...
: given two numbers (either integer or real), ''a'' and ''n'', ''a'' modulo ''n'' is the
remainder In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient ( integer division). In alge ...
of the numerical
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
of ''a'' by ''n'', under certain constraints. * In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...
as applied to functional programming, "operating modulo" is special jargon which refers to mapping a functor to a category by highlighting or defining remainders.


Structures

The term "modulo" can be used differently—when referring to different mathematical structures. For example: * Two members ''a'' and ''b'' of a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
are congruent modulo a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
,
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
''ab''−1 is a member of the normal subgroup (see
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
and
isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist fo ...
for more). * Two members of a ring or an algebra are congruent modulo an ideal, if the difference between them is in the ideal. ** Used as a verb, the act of factoring out a normal subgroup (or an ideal) from a group (or ring) is often called "''modding out'' the..." or "we now ''mod out'' the...". * Two subsets of an infinite set are equal modulo finite sets precisely if their
symmetric difference In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. Th ...
is finite, that is, you can remove a finite piece from the first subset, then add a finite piece to it, and get the second subset as a result. * A
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the contex ...
of maps leads to the definition of a quotient space as being one space modulo another; thus, for example, that a
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
is the space of closed forms modulo exact forms.


Modding out

In general, ''modding out'' is a somewhat informal term that means declaring things equivalent that otherwise would be considered distinct. For example, suppose the sequence 1 4 2 8 5 7 is to be regarded as the same as the sequence 7 1 4 2 8 5, because each is a cyclicly-shifted version of the other: :: \begin & 1 & & 4 & & 2 & & 8 & & 5 & & 7 \\ \searrow & & \searrow & & \searrow & & \searrow & & \searrow & & \searrow & & \searrow \\ & 7 & & 1 & & 4 & & 2 & & 8 & & 5 \end In that case, one is ''"modding out by cyclic shifts''".


See also

*
Essentially unique In mathematics, the term essentially unique is used to describe a weaker form of uniqueness, where an object satisfying a property is "unique" only in the sense that all objects satisfying the property are equivalent to each other. The notion of ess ...
*
List of mathematical jargon The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in ...
*
Up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...


References


External links


Modulo
in the
Jargon File The Jargon File is a glossary and usage dictionary of slang used by computer programmers. The original Jargon File was a collection of terms from technical cultures such as the MIT AI Lab, the Stanford AI Lab (SAIL) and others of the old ARPANET ...
{{DEFAULTSORT:Modulo (Jargon) Mathematical terminology