Quotient
Arithmetic operations  


In arithmetic, a quotient (from Latin: quotiens 'how many times', pronounced /ˈkwoʊʃənt/) is a quantity produced by the division of two numbers.^{[1]} 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),^{[2]} or as a fraction or a 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 in the proper division sense. In the second sense, a quotient is simply the ratio of a dividend to its divisor.
Notation[edit]
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.
Integer part definition[edit]
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 of the ratio of two numbers.^{[3]}
Quotient of two integers[edit]
A rational number can be defined as the quotient of two integers (as long as the denominator is nonzero).
A more detailed definition goes as follows:^{[4]}
 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 and .
The existence of irrational numbers—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.^{[5]}
More general quotients[edit]
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 defined on it, a "quotient set" may be created which contains those equivalence classes as elements. A quotient group 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 into a number of similar linear subspaces.
See also[edit]
 Product (mathematics)
 Quotient category
 Quotient graph
 Integer division
 Quotient module
 Quotient object
 Quotient of a formal language, also left and right quotient
 Quotient ring
 Quotient set
 Quotient space (topology)
 Quotient type
 Quotition and partition
References[edit]
 ^ "Quotient". Dictionary.com.
 ^ Weisstein, Eric W. "Integer Division". mathworld.wolfram.com. Retrieved 20200827.
 ^ Weisstein, Eric W. "Quotient". MathWorld.
 ^ Epp, Susanna S. (20110101). Discrete mathematics with applications. Brooks/Cole. p. 163. ISBN 9780495391326. OCLC 970542319.
 ^ "Irrationality of the square root of 2". www.math.utah.edu. Retrieved 20200827.