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# Quotient  ( Real Numbers )

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In , a quotient (from 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 (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 $6\tfrac\left\{2\right\}\left\{3\right\}$ 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{align} & \leftarrow \text{dividend or numerator} \\ & \leftarrow \text{divisor or denominator} \end{align} \Biggr \} \leftarrow \text{quotient}

Integer part definition
The quotient is also less commonly defined as the greatest of times a divisor may be subtracted from a dividend—before making the 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 of the ratio of two numbers.

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

A more detailed definition goes as follows:

(2011). 9780495391326, Brooks/Cole.

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 may be formed by breaking a group into a number of similar , while a quotient space may be formed in a similar process by breaking a into a number of similar .

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