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Multiple (mathematics)

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In mathematics, a multiple is the product of any quantity and an integer.[1] In other words, for the quantities {\displaystyle a} and {\displaystyle b}, it can be said that {\displaystyle b} is a multiple of {\displaystyle a} if {\displaystyle b=n\cdot a} for some integer n, which is called the multiplier. If {\displaystyle a} is not zero, this is equivalent to saying that {\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#edit

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:

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

Properties#edit

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

Submultiple#edit

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#edit

References#edit

  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.
Categories ArithmeticMultiplicationIntegers

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