Quotient

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 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. ==See also==