# Quotient

Calculation results | |
---|---|

Addition (+) | |

Subtraction (−) | |

Multiplication (×) | |

Division (÷) | |

Modulo (mod) | |

Exponentiation | |

nth root (√) | |

Logarithm (log) | |

In arithmetic, a **quotient** (from Latin: *quotiens* "how many times", pronounced *ˈkwoʊʃənt*) is the result of division.^{[1]} For example, when dividing twenty-one (the *dividend*) by three (the *divisor*), the *quotient* is seven.

The quotient may sometimes be defined as the number of times the divisor may be subtracted from the dividend without the remainder becoming negative. For example, the divisor 3 may be subtracted up to 7 times from the dividend 21 before the remainder becomes negative: 21-3-3-3-3-3-3-3 = 0.

Outside of arithmetic, many branches of math have co-opted 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

- Ratio
- Quotient set
- Quotient group
- Quotient ring
- Quotient module
- Quotient space (linear algebra)
- Quotient space (topology)
- Quotient object
- Quotient category
- Right quotient
- Left quotient
- Quotient type