Quotient

Short description: Mathematical result of division

Arithmetic operationsv * d * e

| Addition (+)

|  term+termsummand+summandaddend (broad sense)+addend (broad sense)augend+addend (strict sense)}=|  sum
Subtraction (−)
|  term−termminuend−subtrahend}=|  difference
Multiplication (×)
|  factor×factormultiplier×multiplicand}=|  product
Division (÷)
|  dividenddivisor numeratordenominator}=|  fractionquotientratio
Exponentiation
|  baseexponent=|  power
nth root (√)
|  radicanddegree=|  root
Logarithm (log)
|  logbase(anti-logarithm)=|  logarithm
The quotient of 12 apples by 3 apples is 4.

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 623 in the proper division sense. In the second sense, a quotient is simply the ratio of a dividend to its divisor.

## Contents

* 1 Notation
* 2 Integer part definition
* 3 Quotient of two integers
* 4 More general quotients
* 5 See also
* 6 References

## 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.

12<-dividend or numerator<-divisor or denominator}<-quotient

## 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,

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

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:[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 r=ab and b≠0.

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

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

* 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

1. ↑ "Quotient". http://dictionary.reference.com/browse/quotient.
2. ↑ Weisstein, Eric W.. "Integer Division" (in en). https://mathworld.wolfram.com/IntegerDivision.html#:~:text=Integer division is division in,and is the floor function..
3. ↑ Weisstein, Eric W.. "Quotient". http://mathworld.wolfram.com/Quotient.html.
4. ↑ Epp, Susanna S. (2011-01-01). Discrete mathematics with applications. Brooks/Cole. pp. 163. ISBN 9780495391326. OCLC 970542319.
5. ↑ "Irrationality of the square root of 2.". https://www.math.utah.edu/~pa/math/q1.html.

* v
* t
* e

Fractions and ratios

| Division and ratio|

* Dividend : Divisor = Quotient

Fraction|

* Numerator / Denominator = Quotient

* Algebraic
* Aspect
* Binary
* Continued
* Decimal
* Dyadic
* Egyptian
* Golden
* Silver
* Integer
* Irreducible
* Reduction
* LCD
* Musical interval
* Paper size
* Percentage
* Unit

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