In arithmetic, 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 ratio (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 $6\backslash tfrac\{2\}\{3\}$ 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.
$$\backslash dfrac\{1\}\{2\}\; \backslash quad\; \backslash begin\{align\}\; \&\; \backslash leftarrow\; \backslash text\{dividend\; or\; numerator\}\; \backslash \backslash \; \&\; \backslash leftarrow\; \backslash text\{divisor\; or\; denominator\}\; \backslash end\{align\}\; \backslash Biggr\; \backslash \}\; \backslash leftarrow\; \backslash text\{quotient\}$$
Integer part definition
The quotient is also less commonly defined as the greatest
Natural 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
(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 $r\; =\; \backslash tfrac\; a\; b$ and $b\; \backslash neq\; 0$.
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
.
See also