# Quotient Information

https://en.wikipedia.org/wiki/Quotient

In arithmetic, a quotient (from Latin: quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers.  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),  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 $6{\tfrac {2}{3}}$ in the proper division sense. In the second sense, a quotient is simply the ratio of a dividend to its divisor.

## 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.

{\dfrac {1}{2}}\quad {\begin{aligned}&\leftarrow {\text{dividend or numerator}}\\&\leftarrow {\text{divisor or denominator}}\end{aligned}}{\Biggr \}}\leftarrow {\text{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. 

## Quotient of two integers

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: 

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 $r={\tfrac {a}{b}}$ and $b\neq 0$ .

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. 

## 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.