Arithmetic operations v t e

Addition (+) ${\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term))\,+\,{\text{term))\\\scriptstyle {\text{summand))\,+\,{\text{summand))\\\scriptstyle {\text{addend))\,+\,{\text{addend))\\\scriptstyle {\text{augend))\,+\,{\text{addend))\end{matrix))\right\}\,=\,}$ ${\displaystyle \scriptstyle {\text{sum))}$ Subtraction (−) ${\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term))\,-\,{\text{term))\\\scriptstyle {\text{minuend))\,-\,{\text{subtrahend))\end{matrix))\right\}\,=\,}$ ${\displaystyle \scriptstyle {\text{difference))}$ Multiplication (×) ${\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor))\,\times \,{\text{factor))\\\scriptstyle {\text{multiplier))\,\times \,{\text{multiplicand))\end{matrix))\right\}\,=\,}$ ${\displaystyle \scriptstyle {\text{product))}$ Division (÷) ${\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend))}{\scriptstyle {\text{divisor))))\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator))}{\scriptstyle {\text{denominator))))\end{matrix))\right\}\,=\,}$ ${\displaystyle \scriptstyle \left\((\begin{matrix}\scriptstyle {\text{fraction))\\\scriptstyle {\text{quotient))\\\scriptstyle {\text{ratio))\end{matrix))\right.}$ Exponentiation (^) ${\displaystyle \scriptstyle {\text{base))^{\text{exponent))\,=\,}$ ${\displaystyle \scriptstyle {\text{power))}$ nth root (√) ${\displaystyle \scriptstyle {\sqrt[{\text{degree))]{\scriptstyle {\text{radicand))))\,=\,}$ ${\displaystyle \scriptstyle {\text{root))}$ Logarithm (log) ${\displaystyle \scriptstyle \log _{\text{base))({\text{anti-logarithm)))\,=\,}$ ${\displaystyle \scriptstyle {\text{logarithm))}$

v t e

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. It has two definitions: either the integer part of a division (in the case of Euclidean division),^[2] or as a fraction or a ratio (in the case of a general division). For example, when dividing 20 (the dividend) by 3 (the divisor), the quotient is 6 (with a remainder of 2) in the first sense, and ${\displaystyle 6{\tfrac {2}{3))=6.66...}$ (a repeating decimal) in the second sense. Ratios can be defined as dimensionless quotients;^[3] non-dimensionless quotients are also known as rates.^[4]

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.

$${\displaystyle {\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.^[5]

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:^[6]

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

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.^[7]

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.