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Multiple (mathematics)

{{Short description|Product with an integer}}

{{wiktionary|multiple|submultiple}}

In mathematics, a multiple is the product of any quantity and an integer.{{MathWorld|urlname=Multiple|title=Multiple}} 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.

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 \times 2;

: 49 = 7 \times 7;

: -21 = 7 \times (-3);

: 0 = 7 \times 0;

: 3 = 7 \times (3/7), \quad 3/7 is not an integer;

: -6 = 7 \times (-6/7), \quad -6/7 is not an integer.

Properties

  • 0 is a multiple of every number (0=0\cdot b).
  • The product of any integer n and any integer is a multiple of n. In particular, n, which is equal to n \times 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{{Which|date=November 2024}}, "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{{nbsp}}a). This terminology is also used with units of measurement (for example by the BIPM and NIST), 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.{{SIbrochure8th}}.{{cite journal |url=http://physics.nist.gov/Pubs/SP811/sec04.html |title=NIST Guide to the SI|journal=NIST |date=2 July 2009 }} Section 4.3: Decimal multiples and submultiples of SI units: SI prefixes. 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

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Category:Arithmetic

Category:Multiplication

Category:Integers