# multiplication

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## multiplication,

fundamental operation in arithmetic and algebra. Multiplication by a whole number can be interpreted as successive addition. For example, a number N multiplied by 3 is N + N + N. In general, multiplying positive numbers N and M gives the area of the rectangle with sides N and M. The result of a multiplication is known as the product. Numbers that give a product when multiplied together are called factors of that product. The symbol of the operation is × or · and, in algebra, simple juxtaposition (e.g., xy means x×y or x·y). Like addition, multiplication, in arithmetic and elementary algebra, obeys the associative lawassociative law,
in mathematics, law holding that for a given operation combining three quantities, two at a time, the initial pairing is arbitrary; e.g., using the operation of addition, the numbers 2, 3, and 4 may be combined (2+3)+4=5+4=9 or 2+(3+4)=2+7=9.
, the commutative lawcommutative law,
in mathematics, law holding that for a given binary operation (combining two quantities) the order of the quantities is arbitrary; e.g., in addition, the numbers 2 and 5 can be combined as 2+5=7 or as 5+2=7.
, and, in combination with addition, the distributive lawdistributive law.
In mathematics, given any two operations, symbolized by * and +, the first operation, *, is distributive over the second, +, if a*(b+c)=(a*b)+(a*c) for all possible choices of a, b, and c.
. Multiplication in abstract algebra, as between vectors or other mathematical objects, does not always obey these rules. Quantities with unlike units may sometimes be multiplied, resulting in such units as foot-pounds, gram-centimeters, and kilowatt-hours. See also divisiondivision,
fundamental operation in arithmetic; the inverse of multiplication. Division may be indicated by the symbol ÷, as in 15 ÷ 3, or simply by a fraction, 15/3. The number that is being divided, e.g.
.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Multiplication

a binary operation that associates to objects a, b an object c; a and b are called factors, and c is called their product. Multiplication is indicated by the symbol × or by the symbol. The first symbol was introduced by the English mathematician W. Oughtred in 1631, and the second by the German savant G. von Leibniz in 1698. When multiplying letters rather than numbers, we omit these symbols and write ab instead of a × b or a · b. The concrete sense of a multiplication depends on the nature of the factors and the definition of the multiplication. Multiplication of positive integers is the operation that associates to positive integers a and b the positive integer c = ab = a + a + . . . + a, where a is taken b times. Multiplication of fractions m/n and p/q is defined by the equation

The product of fractions is a fraction whose absolute value is the product of the absolute values of the factors. The product of fractions is positive if both factors have the same sign and is negative otherwise. Multiplication of irrational numbers is defined in terms of multiplication of rational approximations of these numbers. Multiplication of complex numbers α and β given as α = a + bi and β = c + di is defined by means of the equation

αβ = (acbd) + (ad + bc)i

If α and β are given in polar form,

α = r1(cos φ1) + i sin φ1)

β = r2(cos φ2 + i sin φ2)

then αβ is defined as

αβ = r1,r2 {cos (φ1 + φ2) + i sin (φ1 + φ2)}

that is, the modulus of the product is the product of the moduli of the factors and the argument of the product is the sum of the arguments of the factors.

Multiplication of numbers has the following properties: (1) ab = ba (commutativity), (2) a(bc) = (ab)c (associativity), and (3) a(b + c) = ab + ac (distributivity of multiplication over addition). We have a · 0 = 0 and a · 1 = a. The techniques for multiplying multivalued expressions rely on these properties.

Further generalization of multiplication relies on the possibility of viewing numbers as operators on vectors in the plane. Thus, to the complex number r(cos φ + i sin φ) we associate the operator of dilation of all vectors by a factor r and their rotation through an angle φ about the origin. Here, to the product of complex numbers there corresponds the product of the operators associated with these numbers, that is, the operator that is the result of successive application of the operators associated with the numbers in question. Such multiplication of operators can be extended to operators that cannot be represented by numbers, for example, to linear operators. In this way, we are led to define multiplication of matrices, of quarternions viewed as dilations and rotations in 3-space, and of kernels of integral operators. In these generalizations some of the properties of multiplication of numbers may not hold. The property that fails to hold most frequently is commutativity.

The study of the general properties of multiplication is part of algebra, in particular, group theory and ring theory.

## multiplication

[‚məl·tə·pli′kā·shən]
(electronics)
An increase in current flow through a semiconductor because of increased carrier activity.
(mathematics)
Any algebraic operation analogous to multiplication of real numbers.
(nucleonics)
The ratio of neutron flux in a subcritical reactor to that supplied by a neutron source; it is the factor by which, in effect, the reactor multiplies the source strength.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

## multiplication

1. an arithmetical operation, defined initially in terms of repeated addition, usually written a × b, a.b, or ab, by which the product of two quantities is calculated: to multiply a by positive integral b is to add a to itself b times. Multiplication by fractions can then be defined in the light of the associative and commutative properties; multiplication by 1/n is equivalent to multiplication by 1 followed by division by n: for example 0.3 × 0.7 = 0.3 × 7/10 = (0.3 × 7)/10 = 2.1/10 = 0.21
2. the act or process in animals, plants, or people of reproducing or breeding
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005
References in periodicals archive ?
We show that (3) together with the multiplicative set [??] is really the right inverse of (2).
The multiplicative situation can be conceptualised in the following ways:
The linear form in Equation (10) is a natural generalization of the two extreme cases of a purely additive ([beta] = 0) risk and a purely multiplicative ([alpha] = 0) risk.
Note that, compared with the previous results, we consider the complex network phenomena (multiplicative noises, stochastic nonlinearity functions, correlated noises, and missing measurements) and design the locally optimal filter to solve the state estimation problem for complex network systems based on the projection theory.
The simulation circuit computing step (2) has the multiplicative degree at most [mathematical expression not reproducible].
Several authors have previously applied the alternating (or multiplicative) Schwarz method to the continuous problem based on the partitioning of the domain into overlapping subdomains and subsequently discretized by introducing uniform meshes on each subdomain; see, e.g., [6, 7, 15, 16, 17, 18, 19].
An IPR R = [([r.sub.ik]).sub.nxn] is multiplicative consistent with [r.sub.ik] = ([[mu].sub.ik], [v.sub.ik])(i, k = 1, 2, ..., n), if
In [14] a (multiplicative) Hom-triple system is defined as a (multiplicative) ternary Hom-algebra (A, [,,],[alpha]) such that (9) and (10) are satisfied (thus a multiplicative Hom-Lts is seen as a Hom-triple system in which identity (11) holds; observe that this definition of a Hom-triple system is different from the one formerly given in [21], where a Hom-triple system is just the Hom-algebra (A, [,,], [alpha])).
The IIR procedures considered here are the OS-EM (or equivalently the multiplicative Euler method) in (21), a rescaled OS-EM, and the multiplicative third-order RK method of the CIR.
In this article, an out space accelerating branch-and-bound algorithm is presented for globally solving the generalized affine multiplicative programs problem (GAMP).
The first multiplicative Zagreb index of [mathematical expression not reproducible] is
Seasonal component can be additive or multiplicative with the trend.

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