Quotient
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In
). 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 (a repeating decimal) in the second sense.In
[5] Quotients with a non-trivial 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".[8]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—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
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:[10]
- 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 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 to the side in a square.[11]
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
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
References
- ^ "Quotient". Dictionary.com.
- ^ Weisstein, Eric W. "Integer Division". mathworld.wolfram.com. Retrieved 2020-08-27.
- ^ a b c "ISO 80000-1:2022(en) Quantities and units — Part 1: General". iso.org. Retrieved 2023-07-23.
- ISBN 978-0-412-99041-0.
- ^ "IEC 60050 - Details for IEV number 102-01-22: "quotient"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2023-09-13.
- ^ "IEC 60050 - Details for IEV number 102-01-23: "ratio"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2023-09-13.
- ^ "IEC 60050 - Details for IEV number 112-03-18: "rate"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2023-09-13.
- ^ Thompson, A.; Taylor, B. N. (March 4, 2020). "NIST Guide to the SI, Chapter 7: Rules and Style Conventions for Expressing Values of Quantities". Special Publication 811 | The NIST Guide for the use of the International System of Units. National Institute of Standards and Technology. Retrieved October 25, 2021.
- ^ Weisstein, Eric W. "Quotient". MathWorld.
- OCLC 970542319.
- ^ "Irrationality of the square root of 2". www.math.utah.edu. Retrieved 2020-08-27.
External links
- Media related to Quotients at Wikimedia Commons