# 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 It might be outdated or ideologically biased.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 ?
An object's momentum is equal to its -- multiplied by its --.
In either case, the resulting percentage is multiplied by the total mixed service department expenditures to determine the amount includible in the simplified production method formula mentioned above.
Option 3: The owner has the option of paying the tenant a stipend equal to the difference between the tenant's current rent and an amount to be calculated by using the demolition stipend chart, per room per month, multiplied by the actual number of rooms in the tenant's housing accommodation, but no less than three rooms.
They divided it by 10s to get smaller units and multiplied it by 10s to get larger lengths.
The result being economic productivity is the difference between employee productivity (EPR) and average cost per person (ACP), multiplied by the number of people employed (P), or:
Differentiating, manipulating the resulting expression, and equating same to MRP (in order to satisfy the necessary condition for maximum profit), we obtain an expression that implies the firm's optimal wage is equal to the MRP of its optimal amount of labor multiplied by an expression equal to: the wage elasticity of supply at its optimal amount of labor divided by that elasticity plus 1.
Interobserver agreement for lecture material difficulty was 100% (agreements divided by agreements plus disagreements multiplied by 100%).
They divided it by 10's to get smaller units and multiplied it by 10's to get larger lengths.
When I served on the admissions subcommittees and later on the Comprehensive Exam Committee, I was able to have the exam problems structured so that examinees who multiplied factors also got the "school solutions," which at the time were found by adding percentages.
[Computed as \$1,000 multiplied by 15 percent--as under Sec.
Thus our specification (+/- .006"/ft) is multiplied by the conversion factor (7.3) which equals to +/- .044" (+/- .006"/ft X 7.3 ft = +/- .044").
For example, the failure to properly credit employees one-half hour per day for time spent performing a government function could mushroom into millions of dollars of liability when that one-half hour is multiplied by the number of employees performing the function and by the number of days the function was performed over a period of 2 to 3 years.

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