Multiple (mathematics)
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 [math]\displaystyle{ 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]\displaystyle{ 14 = 7 \times 2; }[/math]
- [math]\displaystyle{ 49 = 7 \times 7; }[/math]
- [math]\displaystyle{ -21 = 7 \times (-3); }[/math]
- [math]\displaystyle{ 0 = 7 \times 0; }[/math]
- [math]\displaystyle{ 3 = 7 \times (3/7), \quad 3/7 }[/math] is not an integer;
- [math]\displaystyle{ -6 = 7 \times (-6/7), \quad -6/7 }[/math] is not an integer.
Properties
- 0 is a multiple of every number ([math]\displaystyle{ 0=0\cdot b }[/math]).
- The product of any integer [math]\displaystyle{ n }[/math] and any integer is a multiple of [math]\displaystyle{ n }[/math]. In particular, [math]\displaystyle{ n }[/math], which is equal to [math]\displaystyle{ n \times 1 }[/math], is a multiple of [math]\displaystyle{ n }[/math] (every integer is a multiple of itself), since 1 is an integer.
- If [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] are multiples of [math]\displaystyle{ x, }[/math] then [math]\displaystyle{ a + b }[/math] and [math]\displaystyle{ a - b }[/math] are also multiples of [math]\displaystyle{ x }[/math].
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
References
- ↑ Weisstein, Eric W.. "Multiple". http://mathworld.wolfram.com/Multiple.html.
- ↑ 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.
- ↑ 3.0 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.
 |  |
