12 apples divided into 4 groups of 3 each.
The quotient of 12 apples by 3 apples is 4.

In arithmetic, a quotient (from Latin: quotiens 'how many times', pronounced /ˈkwʃənt/) is a quantity produced by the division of two numbers.[1] The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in the case of Euclidean division),[2] or as a fraction or a ratio (in the case of a general division). For example, when dividing 20 (the dividend) by 3 (the divisor), the quotient is 6 (with a remainder of 2) in the first sense, and (a repeating decimal) in the second sense. Ratios can be defined as dimensionless quotients;[3] non-dimensionless quotients are also known as rates.[4]


Main article: Division (mathematics) § 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.

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,


20 − 3 − 3 − 3 − 3 − 3 − 3 − 3 < 0.

In this sense, a quotient is the integer part of the ratio of two numbers.[5]

Quotient of two integers

Main article: Rational number

A rational number can be defined as the quotient of two integers (as long as the denominator is non-zero).

A more detailed definition goes as follows:[6]

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.[7]

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 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


  1. ^ "Quotient". Dictionary.com.
  2. ^ Weisstein, Eric W. "Integer Division". mathworld.wolfram.com. Retrieved 2020-08-27.
  3. ^ "ISO 80000-1:2022(en) Quantities and units — Part 1: General". iso.org. Retrieved 2023-07-23.
  4. ^ "The quotient of two numbers (or quantities); the relative sizes of two numbers (or quantities)", "The Mathematics Dictionary" [1]
  5. ^ Weisstein, Eric W. "Quotient". MathWorld.
  6. ^ Epp, Susanna S. (2011-01-01). Discrete mathematics with applications. Brooks/Cole. p. 163. ISBN 9780495391326. OCLC 970542319.
  7. ^ "Irrationality of the square root of 2". www.math.utah.edu. Retrieved 2020-08-27.