Multiple (mathematics)

Product with an integer

Look up multiple or submultiple in Wiktionary, the free dictionary.

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 {\displaystyle 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.

## Examples

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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 ; {\displaystyle 14=7\times 2;}
49 = 7 × 7 ; {\displaystyle 49=7\times 7;}
− 21 = 7 × ( − 3 ) ; {\displaystyle -21=7\times (-3);}
0 = 7 × 0 ; {\displaystyle 0=7\times 0;}
3 = 7 × ( 3 / 7 ) , 3 / 7 {\displaystyle 3=7\times (3/7),\quad 3/7} is not an integer;
− 6 = 7 × ( − 6 / 7 ) , − 6 / 7 {\displaystyle -6=7\times (-6/7),\quad -6/7} is not an integer.

## Properties

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

## Submultiple

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In some texts[which?], "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 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

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

## References

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1. ^ Weisstein, Eric W. "Multiple". MathWorld.
2. ^ a b International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), ISBN 92-822-2213-6, archived (PDF) from the original on 2021-06-04, retrieved 2021-12-16.
3. ^ a b "NIST Guide to the SI". NIST. 2 July 2009. Section 4.3: Decimal multiples and submultiples of SI units: SI prefixes.