In mathematics, a multiple is the product of any quantity and an integer.[1] In other words, for the quantities
a
{\displaystyle a}
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:
-
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
- 0 is a multiple of every number (
0
=
0
⋅
a
{\displaystyle 0=0\cdot a}
).
- The product of any integer
n
{\displaystyle n}
and any integer is a multiple of n {\displaystyle n}
, is a multiple of n {\displaystyle n}
- If
a
{\displaystyle a}
then a + b {\displaystyle a+b}
and a − b {\displaystyle a-b}
are also multiples of x {\displaystyle x}
.
Submultiple
In some texts, "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
References
- Weisstein, Eric W. "Multiple". MathWorld.
- 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.
- "NIST Guide to the SI". NIST. 2 July 2009. Section 4.3: Decimal multiples and submultiples of SI units: SI prefixes.