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In
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ce ...
, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the
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 two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a division (in the case of
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
), or as a fraction or a
ratio In mathematics, a ratio shows how many times one number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in lan ...
(in the case of proper division). For example, when dividing 20 (the ''dividend'') by 3 (the ''divisor''), the ''quotient'' is "6 with a remainder of 2" in the Euclidean division sense, and 6\tfrac in the proper division sense. In the second sense, a quotient is simply the ratio of a dividend to its divisor.


Notation

The quotient is most frequently encountered as two numbers, or two variables, divided by a horizontal line. The words "dividend" and "divisor" refer to each individual part, while the word "quotient" refers to the whole. \dfrac \quad \begin & \leftarrow \text \\ & \leftarrow \text \end \Biggr \} \leftarrow \text


Integer part definition

The quotient is also less commonly defined as the greatest whole number of times a divisor may be subtracted from a dividend—before making the remainder negative. For example, the divisor 3 may be subtracted up to 6 times from the dividend 20, before the remainder becomes negative: : 20 − 3 − 3 − 3 − 3 − 3 − 3 ≥ 0, while : 20 − 3 − 3 − 3 − 3 − 3 − 3 − 3 < 0. In this sense, a quotient is the
integer part In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least inte ...
of the ratio of two numbers.


Quotient of two integers

A
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
can be defined as the quotient of two
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 (as long as the denominator is non-zero). A more detailed definition goes as follows: : A real number ''r'' is rational, if and only if it can be expressed as a quotient of two integers with a nonzero denominator. A real number that is not rational is irrational. Or more formally: : Given a real number ''r'', ''r'' is rational if and only if there exists integers ''a'' and ''b'' such that r = \tfrac a b and b \neq 0. The existence of
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s—numbers that are not a quotient of two integers—was first discovered in geometry, in such things as the ratio of the diagonal to the side in a square.


More general quotients

Outside of arithmetic, many branches of mathematics have borrowed the word "quotient" to describe structures built by breaking larger structures into pieces. Given a set with 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 ...
defined on it, a " quotient set" may be created which contains those equivalence classes as elements. A
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 ...
may be formed by breaking a group into a number of similar cosets, while a quotient space may be formed in a similar process by breaking a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
into a number of similar
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, ...
s.


See also

* Product (mathematics) *
Quotient category In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but in ...
*
Quotient graph In graph theory, a quotient graph ''Q'' of a graph ''G'' is a graph whose vertices are blocks of a partition of the vertices of ''G'' and where block ''B'' is adjacent to block ''C'' if some vertex in ''B'' is adjacent to some vertex in ''C'' with ...
* Integer division *
Quotient module In algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups by ...
*
Quotient object In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theor ...
* Quotient of a formal language, also left and right quotient *
Quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
* Quotient set *
Quotient space (topology) In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
*
Quotient type In type theory, a kind of foundation of mathematics, a quotient type is an algebraic data type that represents a type whose equality relation has been redefined by a given equivalence relation such that the elements of the type are partitioned i ...
*
Quotition and partition In arithmetic, quotition and partition are two ways of viewing fractions and division. In quotition division one asks, "how many parts are there?"; While in partition division one asks, "what is the size of each part?". For example, the expressio ...


References

{{Authority control Real numbers Division (mathematics)