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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 the quantity produced by the division of two numbers.[1] The quotient has widespread use throughout mathematics, and is commonly referred to as a fraction or a ratio. For example, when dividing twenty (the dividend) by three (the divisor), the quotient is six and two thirds. In this sense, a quotient is the ratio of a dividend to its divisor.


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 without the remainder becoming negative. For example, the divisor three may be subtracted up to six times from the dividend twenty 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.[2]

Quotient of two integers[edit]

The definition of a rational number is the quotient of two integers (as long as the denominator is not a zero).

More formal definitions:[3]

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.

Even more formally:

if r is a real number, then r is rational ⇔ ∃ 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 of a square to the side.

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]


  1. ^ "Quotient". Dictionary.com.
  2. ^ Weisstein, Eric W. "Quotient". MathWorld.
  3. ^ S., Epp, Susanna (2011-01-01). Discrete mathematics with applications. Brooks/Cole. p. 163. ISBN 9780495391326. OCLC 970542319.