# 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 It might be outdated or ideologically biased.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 ?
The objective of this paper is to present a novel architecture for implementation of vedic multiplication algorithm suitable for FPGA implementation.
For a JPEG image of 8x8 block size in spatial domain, the algorithm first decomposes the 2D DCT into one pair of 1D DCTs, and the calculation can be completed in only 24 multiplications. The 2D spatial data is a linear combination of base image obtained by the outer product of the column and row vectors of cosine functions such that the inverse DCT is as efficient.
However, the floating-point instruction set is inconvenient to realize large integer modular multiplication which is the core operation of asymmetric cryptography.
We need 13 additions and 6 multiplications. Using the schoolbook method we need 4 additions and 9 multiplications.
From the Foundation level, the Australian Curriculum: Mathematics (Australian Curriculum and Reporting Authority, 2012) prescribes student learning about multiplication and division in terms of solving problems and calculating strategies.
Hermida, "Pre-synthesis optimization of multiplications to improve circuit performance," in Proceedings of the Design, Automation and Test in Europe (DATE '06), vol.
B = The number of multiplications needed in bottom up approach, [tau] = The number of children (degree) for each node of the tree, [log.sub.[tau]]n = L = Level of the tree (0 ...
In this work, we want to reduce its computation by focusing on its bottleneck operation, scalar multiplication. The operation to compute
The number of multiplications is extremely important, since it is a relatively complex and power consuming operation, to be performed in hardware.
Participants reported using mainly additions (Median (Mdn) (3) = 40% of their operations), followed by subtractions (Mdn = 25%) and multiplications (Mdn = 20%), while divisions were reported to be used less often (Mdn = 10%).
66-81) writes, rules for multiplication are then established, along with the notion of rotation, whence complex numbers follow.
Hariri and Reyhani-Masoleh [10] proposed a number of bit-serial and bit-parallel Montgomery multipliers and showed that MM can accelerate the ECC scalar multiplication. Recently, in [11], they have considered concurrent error detection for MM over binary field.

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