Multiple (mathematics)

Short description: Product with an integer

In mathematics, a multiple is the product of any quantity and an integer.[1] 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 multiplier. If a is not zero, this is equivalent to saying that b/a 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.

## Contents

* 1 Examples
* 2 Properties
* 3 Submultiple
* 4 See also
* 5 References

## 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:

14=7×2;
49=7×7;
−21=7×(−3);
0=7×0;
3=7×(3/7),3/7 is not an integer;
−6=7×(−6/7),−6/7 is not an integer.

## Properties

* 0 is a multiple of every number (0=0⋅b).
* The product of any integer n and any integer is a multiple of n. In particular, n, which is equal to n×1, is a multiple of n (every integer is a multiple of itself), since 1 is an integer.
* If a and b are multiples of x, then a+b and a−b are also multiples of x.

## Submultiple

In some texts, "a is a submultiple of b" has the meaning of "a being a unit fraction of b" (a=1/b) or, equivalently, "b being an integer multiple n of a" (b=n a). This terminology is also used with units of measurement (for example by the BIPM[2] and NIST[3]), where a unit submultiple is obtained by prefixing the main unit, defined as the quotient of the main unit by an integer, mostly a power of 103. For example, a millimetre is the 1000-fold submultiple of a metre.[2][3] As another example, one inch may be considered as a 12-fold submultiple of a foot, or a 36-fold submultiple of a yard.

## See also

* Unit fraction
* Ideal (ring theory)
* Decimal and SI prefix
* Multiplier (linguistics)

## References

1. ↑ Weisstein, Eric W.. "Multiple". http://mathworld.wolfram.com/Multiple.html.
2. ↑ 2.0 2.1 International Bureau of Weights and Measures (2006), The International System of Units (SI) (8th ed.), ISBN 92-822-2213-6, http://www.bipm.org/utils/common/pdf/si_brochure_8_en.pdf .
3. ↑ 3.0 3.1 "NIST Guide to the SI". Nist. 2 July 2009. http://physics.nist.gov/Pubs/SP811/sec04.html. Section 4.3: Decimal multiples and submultiples of SI units: SI prefixes.

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