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In
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
, 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 o ...
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 division (mathematics), dividing one integer (the dividend) by another (the divisor), in a way that produces a quotient and a remainder smaller than the divisor ...
), or as a
fraction A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths ...
or a
ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

ratio
(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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
can be defined as the quotient of two
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
defined on it, a "
quotient set Set, The Set, or SET may refer to: Science, technology, and mathematics Mathematics * Set (mathematics), a collection of distinct elements or members * Category of sets, the category whose objects and morphisms are sets and total functions, respe ...
" may be created which contains those equivalence classes as elements. A
quotient group A quotient group or factor group is a mathematical group (mathematics), 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 "factore ...
may be formed by breaking a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
into a number of similar
cosets In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, while a quotient space may be formed in a similar process by breaking a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
into a number of similar
linear subspace In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s. For quotient objects, the horizontal fraction bar is not used, only the diagonal
fraction slash The slash is an oblique slanting line #Conjunction, punctuation mark . Once used to mark full stop, periods and commas, the slash is now most often used to represent #XOR, exclusive or #And, inclusive or, #Division, division and #Fractions, fra ...
, for example: .


See also

*
Product (mathematics) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
*
Quotient category In mathematics, a quotient category is a category (mathematics), category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of small categories, category of (locally small) categories, anal ...
*
Quotient graph In graph theory, a quotient graph ''Q'' of a graph ''G'' is a graph whose vertices are blocks of a partition of a set, partition of the vertices of ''G'' and where block ''B'' is adjacent to block ''C'' if some vertex in ''B'' is adjacent to some ve ...
* Quotient in
integer division Division is one of the four basic operations of arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη# ...
*
Quotient moduleIn algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In i ...
*
Quotient objectIn category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...
*
Quotient of a formal languageIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, also left and right quotient *
Quotient ring In ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...
*
Quotient set Set, The Set, or SET may refer to: Science, technology, and mathematics Mathematics * Set (mathematics), a collection of distinct elements or members * Category of sets, the category whose objects and morphisms are sets and total functions, respe ...
*
Quotient space (topology) as the quotient space of a disk, by ''gluing'' together to a single point the points (in blue) of the boundary of the disk. In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, to ...
* Quotient type *
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 expression ...


References

{{Authority control Real numbers Division (mathematics)