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

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

References

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