Multiple (mathematics)

Last updated May 14, 2024

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 $b/a$ 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, "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.

Related Research Articles

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<span class="mw-page-title-main">Imaginary unit</span> Principal square root of −1

The imaginary unit or unit imaginary number is a solution to the quadratic equation $x 2 + 1 = 0.$ Although there is no real number with this property, $i$ can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of $i$ in a complex number is $2 + 3 i .$

<span class="mw-page-title-main">Division (mathematics)</span> Arithmetic operation

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<span class="mw-page-title-main">Ratio</span> Relationship between two numbers of the same kind

In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six. Similarly, the ratio of lemons to oranges is 6:8 and the ratio of oranges to the total amount of fruit is 8:14.

In mathematics, taking the nth root is an operation involving two numbers, the radicand and the index or degree. Taking the nth root is written as $, where x is the radicand and n is the index. This is pronounced as "the nth root of x". The definition then of an n th root of a number x is a number r which, when raised to the power of the positive integer n, yields x :$

<span class="mw-page-title-main">Quotient</span> Mathematical result of division

In arithmetic, a quotient is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division or a fraction or ratio. For example, when dividing 20 by 3, the quotient is 6 in the first sense and $in the second sense.$

A power of two is a number of the form $2 n$ where $n$ is an integer, that is, the result of exponentiation with number two as the base and integer $n$ as the exponent.

Positional notation usually denotes the extension to any base of the Hindu–Arabic numeral system. More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early numeral systems, such as Roman numerals, a digit has only one value: I means one, X means ten and C a hundred. In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string.

In computing, fixed-point is a method of representing fractional (non-integer) numbers by storing a fixed number of digits of their fractional part. Dollar amounts, for example, are often stored with exactly two fractional digits, representing the cents. More generally, the term may refer to representing fractional values as integer multiples of some fixed small unit, e.g. a fractional amount of hours as an integer multiple of ten-minute intervals. Fixed-point number representation is often contrasted to the more complicated and computationally demanding floating-point representation.

In mathematics, the lexicographic or lexicographical order is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set.

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In algebra, the greatest common divisor of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers.

A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic ; if this sequence consists only of zeros, the decimal is said to be terminating, and is not considered as repeating. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating. For example, the decimal representation of 1/3 becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is 3227/555, whose decimal becomes periodic at the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... Another example of this is 593/53, which becomes periodic after the decimal point, repeating the 13-digit pattern "1886792452830" forever, i.e. 11.18867924528301886792452830....

In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator $p$ and a non-zero denominator $q$ . For example, is a rational number, as is every integer. The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface $Q$ , or blackboard bold

In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length, no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.

The digits of some specific integers permute or shift cyclically when they are multiplied by a number n. Examples are:

The square root of 6 is the positive real number that, when multiplied by itself, gives the natural number 6. It is more precisely called the principal square root of 6, to distinguish it from the negative number with the same property. This number appears in numerous geometric and number-theoretic contexts. It can be denoted in surd form as:

References

↑ Weisstein, Eric W. "Multiple". MathWorld .
1 2 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.
1 2 "NIST Guide to the SI". NIST. 2 July 2009. Section 4.3: Decimal multiples and submultiples of SI units: SI prefixes.

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[1] Weisstein, Eric W. "Multiple". MathWorld .

[SI8-2] 1 2 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-3] 1 2 "NIST Guide to the SI". NIST. 2 July 2009. Section 4.3: Decimal multiples and submultiples of SI units: SI prefixes.

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