View source for Multiple (mathematics)

{{Short description|Product with an integer}}
{{wiktionary|multiple|submultiple}}
In [[mathematics]], a '''multiple''' is the [[Product (mathematics)|product]] of any quantity and an [[integer]].<ref>{{MathWorld|urlname=Multiple|title=Multiple}}</ref> In other words, for the quantities ''a'' and ''b'', it can be said that ''b'' is a multiple of ''a'' if ''b'' = ''na'' for some integer ''n'', which is called the [[Multiplication|multiplier]]. If ''a'' is not [[zero]], this is equivalent to saying that <math>b/a</math> is an integer.

When ''a'' and ''b'' are both integers, and ''b'' is a multiple of ''a'', then ''a'' is called a ''[[divisor]]'' of ''b''. One says also that ''a'' divides ''b''. If ''a'' and ''b'' are not integers, mathematicians prefer generally to use '''integer multiple''' instead of ''multiple'', for clarification. In fact, ''multiple'' is used for other kinds of product; for example, a [[polynomial]] ''p'' is a multiple of another polynomial ''q'' if there exists third polynomial ''r'' such that ''p'' = ''qr''.

==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), \quad 3/7</math> is not an integer;
: <math>  -6 = 7 \times (-6/7), \quad -6/7</math> is not an integer.

==Properties==
* 0 is a multiple of every number (<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>.

==Submultiple==
In some texts{{Which|date=November 2024}}, "''a'' is a '''submultiple''' of ''b''" has the meaning of "''a'' being a [[unit fraction]] of ''b''" (''a''{{=}}''b''/''n'') or, equivalently, "''b'' being an [[integer multiple]] ''n'' of ''a''" (''b''{{=}}''n''{{nbsp}}''a''). This terminology is also used with [[units of measurement]] (for example by the [[BIPM]]<ref name="SI8"/> and [[NIST]]<ref name="NIST"/>), where a ''[[unit submultiple]]'' is obtained by [[prefix|prefixing]] the main unit, defined as the quotient of the main unit by an integer, mostly a power of 10<sup>3</sup>. For example, a [[millimetre]] is the 1000-fold submultiple of a [[metre]].<ref name="SI8">{{SIbrochure8th}}.</ref><ref name="NIST">{{cite journal |url=http://physics.nist.gov/Pubs/SP811/sec04.html |title=NIST Guide to the SI|journal=NIST |date=2 July 2009 }} Section 4.3: ''Decimal multiples and submultiples of SI units: SI prefixes''.</ref> As another example, one [[inch]] may be considered as a 12-fold submultiple of a [[Foot (unit)|foot]], or a 36-fold submultiple of a [[yard (unit)|yard]].

==See also==
* [[Unit fraction]]
* [[Ideal (ring theory)]]
* [[Decimal]] and [[SI prefix]]
* [[Multiplier (linguistics)]]

==References==
{{reflist}}

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[[Category:Arithmetic]]
[[Category:Multiplication]]
[[Category:Integers]]