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{{Unreliable sources|date=September 2009}} | |||
In [[mathematics]], a '''multiple''' is the [[Multiplication|product]] of any quantity and an [[integer]].<ref>{{MathWorld|urlname=Multiple|title=Multiple}}</ref><ref>[http://wordnetweb.princeton.edu/perl/webwn?s=multiple WordNet lexicon database, Princeton University]</ref><ref>[http://www.wordreference.com/definition/multiple WordReference.com]</ref> In other words, for the quantities ''a'' and ''b'', we say that ''b'' is a multiple of ''a'' if ''b'' = ''na'' for some integer ''n'', which is called the [[Multiplication|multiplier]] or [[coefficient]]. If ''a'' is not [[zero]], this is equivalent to saying that ''b''/''a'' is an integer with no [[remainder]].<ref>[http://www.thefreedictionary.com/multiple The Free Dictionary by Farlex]</ref><ref>[http://dictionary.reference.com/browse/multiple Dictionary.com Unabridged]</ref><ref>[http://dictionary.cambridge.org/define.asp?key=52498&dict=CALD Cambridge Dictionary Online]</ref> If ''a'' and ''b'' are both integers, and ''b'' is a multiple of ''a'', then ''a'' is called a [[divisor]] of ''b''. | |||
==Examples== | |||
14, 49, -21 and 0 are multiples of 7, whereas 3 and -6 are not. This is because there are integers that 7 may be multiplied by to reach the values of 14, 49, 0 and -21, while there are no such ''integers'' for 3 and -6. Each of the products listed below, and in particular, the products for 3 and -6, is the ''only'' way that the relevant number can be written as a product of 7 and another real number: | |||
* <math> 14 = 7 \times 2</math> | |||
* <math> 49 = 7 \times 7</math> | |||
* <math> -21 = 7 \times (-3)</math> | |||
* <math> 0 = 7 \times 0</math> | |||
* <math> 3 = 7 \times (3/7)</math>, <math>3/7</math> is a rational number, not an integer | |||
* <math> -6 = 7 \times (-6/7)</math>, <math>-6/7</math> is a rational number, not an integer. | |||
==Properties== | |||
*0 is a multiple of everything (<math>0=0\cdot b</math>). | |||
*The product of any integer <math>n</math> and any integer is a multiple of <math>n</math>. In particular, <math>n</math>, which is equal to <math>n \times 1</math>, is a multiple of <math>n</math> (every integer is a multiple of itself), since 1 is an integer. | |||
*If <math>a</math> and <math>b</math> are multiples of <math>x</math> then <math>a+b</math> and <math>a-b</math> are also multiples of <math>x</math>. | |||
==References== | |||
<references/> | |||
==See also== | |||
*[[Ideal (ring theory)]] | |||
*[[Decimal]] and [[SI prefix]] | |||
{{DEFAULTSORT:Multiple (Mathematics)}} | |||
[[Category:Arithmetic]] | |||
[[Category:Multiplication]] | |||
Revision as of 02:53, 10 January 2014
Template:Unreliable sources In mathematics, a multiple is the product of any quantity and an integer.[1][2][3] In other words, for the quantities a and b, we say that b is a multiple of a if b = na for some integer n, which is called the multiplier or coefficient. If a is not zero, this is equivalent to saying that b/a is an integer with no remainder.[4][5][6] If a and b are both integers, and b is a multiple of a, then a is called a divisor of b.
Examples
14, 49, -21 and 0 are multiples of 7, whereas 3 and -6 are not. This is because there are integers that 7 may be multiplied by to reach the values of 14, 49, 0 and -21, while there are no such integers for 3 and -6. Each of the products listed below, and in particular, the products for 3 and -6, is the only way that the relevant number can be written as a product of 7 and another real number:
- , is a rational number, not an integer
- , is a rational number, not an integer.
Properties
- 0 is a multiple of everything ().
- The product of any integer and any integer is a multiple of . In particular, , which is equal to , is a multiple of (every integer is a multiple of itself), since 1 is an integer.
- If and are multiples of then and are also multiples of .
References
- ↑
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