Quotient

The quotient of 12 apples by 3 apples is 4.

In arithmetic, a quotient (from Template:Langx 'how many times', pronounced Template:IPAc-en) is a quantity produced by the division of two numbers.<ref>Template:Cite web</ref> The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in the case of Euclidean division)<ref>Template:Cite web</ref> or a fraction or 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 <math>6+\tfrac{2}{3}=6.66...</math> (a repeating decimal) in the second sense.

Template:AnchorIn metrology (International System of Quantities and the International System of Units), "quotient" refers to the general case with respect to the units of measurement of physical quantities.<ref name="ISO 80000-1"/><ref>Template:Cite book</ref> <ref name="International Electrotechnical Vocabulary e707">Template:Cite web</ref> Ratios is the special case for dimensionless quotients of two quantities of the same kind.<ref name="ISO 80000-1">Template:Cite web</ref><ref name="International Electrotechnical Vocabulary g891">Template:Cite web</ref> Quotients with a non-trivial dimension and compound units, especially when the divisor is a duration (e.g., "per second"), are known as rates.<ref name="International Electrotechnical Vocabulary x558">Template:Cite web</ref> For example, density (mass divided by volume, in units of kg/m³) is said to be a "quotient", whereas mass fraction (mass divided by mass, in kg/kg or in percent) is a "ratio".<ref>Template:Cite book</ref> Specific quantities are intensive quantities resulting from the quotient of a physical quantity by mass, volume, or other measures of the system "size".<ref name="ISO 80000-1"/>

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.

<math display="block"> \dfrac{1}{2} \quad \begin{align} & \leftarrow \text{dividend or numerator} \\ & \leftarrow \text{divisor or denominator} \end{align} \Biggr \} \leftarrow \text{quotient} </math>

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.<ref>Template:MathWorld</ref>

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:<ref>Template:Cite book</ref>

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 <math>r = \tfrac a b</math> and <math>b \neq 0</math>.

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.<ref>Template:Cite web</ref>

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

• Product (mathematics)
• Quotient category
• Quotient graph
• Integer division
• Quotient module
• Quotient object
• Quotient of a formal language, also left and right quotient
• Quotient ring
• Quotient set
• Quotient space (topology)
• Quotient type
• Quotition and partition

External links

• Template:Commonscatinline