“Quotient”的意思、由来-开放百科全书

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Quotient

释义

Notation
Integer part definition
Quotient of two integers
More general "quotients"
See also
References

{{Short description|Mathematical result of division}}{{other uses}}{{See Also|Division (mathematics)|Fraction (mathematics)|Ratio}}{{Calculation results}}

In arithmetic, a quotient (from {{lang-lat|quotiens}} "how many times", pronounced {{IPAc-en|ˈ|k|w|oʊ|ʃ|ən|t}}) is the quantity produced by the division of two numbers.^[1] The quotient has widespread use throughout mathematics, and is commonly referred to as a fraction or a ratio. For example, when dividing twenty (the dividend) by three (the divisor), the quotient is six and two thirds. In this sense, a quotient is 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.

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 without the remainder becoming negative. For example, the divisor three may be subtracted up to six times from the dividend twenty 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.^[2]

Quotient of two integers

The definition of a rational number is the quotient of two integers (as long as the denominator is not a zero).

More formal definitions:^[3]

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.

Even more formally:

if r is a real number, then r is rational ⇔ ∃ integers a and b such that and

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 of a square to the side.

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.

References

1. ^{{Cite web|title=Quotient|website=Dictionary.com|url=http://dictionary.reference.com/browse/quotient}}
2. ^{{MathWorld|urlname=Quotient|title=Quotient}}
3. ^{{Cite book|url=https://www.worldcat.org/oclc/970542319|title=Discrete mathematics with applications|last=S.|first=Epp, Susanna|date=2011-01-01|publisher=Brooks/Cole|isbn=9780495391326|oclc=970542319|pages=163}}

2 : Real numbers|Division (mathematics)

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Notation

Integer part definition

Quotient of two integers

More general "quotients"

See also

References