Quotient
Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a quotient (from ) is the result of division
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...

. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...

, and 3 the divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.
A quotient can also mean just the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

part of the result of dividing two integers. For example, the quotient of 13 and 5 would be 2 while the remainder
Remainder
In arithmetic, the remainder is the amount "left over" after the division of two integers which cannot be expressed with an integer quotient....

would be 3. For more, see the division algorithm
Division algorithm
In mathematics, and more particularly in arithmetic, the usual process of division of integers producing a quotient and a remainder can be specified precisely by a theorem stating that these exist uniquely with given properties. An integer division algorithm is any effective method for producing...

.

In more abstract branches of mathematics, the word quotient is often used to describe sets, spaces, or algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...

s whose elements are the equivalence classes of some equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

on another set, space, or algebraic structure. See:
• quotient set
• quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

• quotient ring
Quotient ring
In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...

• quotient module
Quotient module
In abstract algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is analogous to how one obtains the ring of integers modulo an integer n, see modular arithmetic...

• quotient space (linear algebra)
Quotient space (linear algebra)
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N ....

• quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...

of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

• quotient object
• quotient category
Quotient category
In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. The notion is similar to that of a quotient group or quotient space, but in the categorical setting.-Definition:Let C be a category...

• right quotient
Right quotient
The right quotient of a formal language L_1 with a formal language L_2 is the language consisting of strings w such that wx is in L_1 for some string x in L_2...

and left quotient
Left quotient
If L_1 and L_2 are formal languages, then the left quotient of L_1 with L_2 is the language consisting of strings w such that xw is in L_2 for some string x in L_1...

(operations on formal language
Formal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...

s)

The quotient rule is a method for finding derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

s in calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

.

## Other meanings

Quotients also come up in certain tests, like the IQ test, which stands for intelligence quotient
Intelligence quotient
An intelligence quotient, or IQ, is a score derived from one of several different standardized tests designed to assess intelligence. When modern IQ tests are constructed, the mean score within an age group is set to 100 and the standard deviation to 15...

. In this case, one's quotient is basically one's score. In recent decades, as more emphasis has been placed on full personal development, other similar quotients have appeared. These include moral
Morality
Morality is the differentiation among intentions, decisions, and actions between those that are good and bad . A moral code is a system of morality and a moral is any one practice or teaching within a moral code...

quotient, emotional quotient
Emotional intelligence
Emotional intelligence is a skill or ability in the case of the trait EI model, a self-perceived ability to identify, assess, and control the emotions of oneself, of others, and of groups. Various models and definitions have been proposed of which the ability and trait EI models are the most...

, adversity quotient, social quotient, creativity
Creativity
Creativity refers to the phenomenon whereby a person creates something new that has some kind of value. What counts as "new" may be in reference to the individual creator, or to the society or domain within which the novelty occurs...

quotient, etc.