The symbol for multiplication is `` **b ''.

According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.

The multiplication operation is defined as: multiplication by 1 means no change, multiplication by 2 means shifting to the left, and multiplication by 3 means shifting to the left and then performing xor with the initial unshifted value.

* Addition and multiplication of complex numbers and quaternions is associative.

Addition of octonions is also associative, but multiplication of octonions is non-associative.

Moreover, if any coefficient is a fixed power of 2, the multiplication can be replaced by bit shifting.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

An associative R-algebra is an additive abelian group A which has the structure of both a ring and an R-module in such a way that ring multiplication is R-bilinear:

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

Ackermann's three-argument function,, is defined such that for p = 0, 1, 2, it reproduces the basic operations of addition, multiplication, and exponentiation as

In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative.

The set of all invertible elements is therefore closed under multiplication and forms a Moufang loop.

Is X a Banach space, the space B ( X ) = B ( X, X ) forms a unital Banach algebra ; the multiplication operation is given by the composition of linear maps.

In computing, a binary prefix is a specifier or mnemonic that is prepended to the units of digital information, the bit and the byte, to indicate multiplication by a power of 1024.

Ron Rivest estimated in 1977 that factoring a 125-digit number would require 40 quadrillion years, even with the highly conservative assumption that modular multiplication could be carried out in a nanosecond ; he therefore then believed that RSA-129 could never be broken in practice.

Unlike the simple counting board used in elementary schools, very efficient suanpan techniques have been developed to do multiplication, division, addition, subtraction, square root and cube root operations at high speed.

While division rings and algebras as discussed here are assumed to have associative multiplication, nonassociative division algebras such as the octonions are also of interest.

* math enthusiasts who can in less than 25 seconds mentally solve 2 x 5 digit multiplication problems ( e. g., 23 x 48, 856 ) that have been presented orally by the researcher.

They have some kind of gain, or multiplication factor relating the magnitude of the output signal to the input signal.

Scientific applications generally require no more than 40 digits of, so the primary motivation for these computations is the human desire to break records, but the extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms.

The operation is required to be associative so that for all x, y and z, but need not be commutative so that does not have to equal ( contrast to the standard multiplication operator on real numbers, where ).

This is an algebra built up from an orthonormal basis of mutually orthogonal vectors under addition and multiplication, p of which have norm + 1 and q of which have norm − 1, with the product rule for the basis vectors

; Loop nest optimization: Some pervasive algorithms such as matrix multiplication have very poor cache behavior and excessive memory accesses.

Even the inherent statistical fluctuations of neutron multiplication within the chain reaction, have implications with regard to implosion speed and symmetry.

A little before he was 4 years old, he had already written his first letter ; learned how to do multiplication and tell time when an average child would have learned to do all that after the age of 7.

Efficient multiplication algorithms have existed since the advent of the decimal system.

Indeed, given a ring R, we can define a preadditive category R to have a single object A, let Hom ( A, A ) be R, and let composition be ring multiplication.

Once we have defined multiplication for formal power series, we can define multiplicative inverses as follows.

If one does not require a ring to have a unit, then one has to add the requirement of continuity of the additive inverse, or equivalently, to define the topological ring as a ring which is a topological group ( for +) in which multiplication is continuous, too.

They are called addition and multiplication and commonly denoted by "+" and "⋅"; e. g. a + b and a ⋅ b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition ; i. e., a ⋅ ( b + c )

Since we could have put any real numbers in place of 2, 1, and 3 above, and still have obtained a true equation, we say that multiplication of real numbers distributes over addition of real numbers.

Thus, for example, groups have a signature containing two operators: the multiplication operator m, taking two arguments, and the inverse operator i, taking one argument, and the identity element e, a constant, which may be considered an operator that takes zero arguments.

However, Alon, Shpilka and Umans have recently shown that some of these conjectures implying fast matrix multiplication are incompatible with another plausible conjecture, the sunflower conjecture.

Non-degenerate singular values always have unique left-and right-singular vectors, up to multiplication by a unit-phase factor e < sup > iφ </ sup > ( for the real case up to sign ).

Addition and multiplication are pipelined and have a latency of three and five cycles, respectively.

While studying compositions of linear transformations, Arthur Cayley was led to define matrix multiplication and inverses.

Advances in the numeral system and mathematical notation eventually led to the discovery of mathematical operations such as addition, subtraction, multiplication, division, squaring, square root, and so forth.

They are led by an invisible hand to make nearly the same distribution of the necessaries of life, which would have been made, had the earth been divided into equal portions among all its inhabitants, and thus without intending it, without knowing it, advance the interest of the society, and afford means to the multiplication of the species.

Also, the concession programme, which requires the naming of all those who participated along with the accused return for a lighter sentence, has led to a multiplication of names.

The weapon uses several different bows instead of one bow with a tackle system to achieve a higher acceleration of the sinew via the multiplication with each bow's pulling effect.

This all makes it a more convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary, octal and hexadecimal systems, although the sexagesimal system ( where the reciprocals of all 5-smooth numbers terminate ) does better in this respect ( but at the cost of an unwieldy multiplication table ).

Unlike DLP systems ( where it is possible to use the same procedure for squaring and multiplication ) the EC addition is significantly different for doubling () and general addition () depending on the coordinate system used.

Contrary to floating-point arithmetic, in a logarithmic number system multiplication, division and exponentiation are easy to implement but addition and subtraction are difficult.

For example, in a decimal floating-point system with three digits, the multiplication that humans would write as

Kilo-( symbol: k, lowercase ) is a unit prefix in the metric system denoting multiplication of the unit by one thousand.

The concentration of nitrogenous wastes in the urine of mammals and some birds is dependent on an elaborate countercurrent multiplication system.

Other features were one of the first hardware-implementations of a multiplication instruction in an MPU, full 16-bit arithmetic, and an especially fast interrupt system.

A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema.

We can define a multiplication on the set S using the Steiner triple system by setting aa

In mathematics, a multiplication table ( sometimes, less formally, a times table ) is a mathematical table used to define a multiplication operation for an algebraic system.

Consider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows:

The first section introduces the Arabic numeral system, including lattice multiplication and methods for converting between different representation systems.

Consider the direct system composed of the groups Z / p < sup > n </ sup > Z and the homomorphisms Z / p < sup > n </ sup > Z → Z / p < sup > n + 1 </ sup > Z which are induced by multiplication by p. The direct limit of this system consists of all the roots of unity of order some power of p, and is called the Prüfer group Z ( p < sup >∞</ sup >).

A model for the real number system consists of a set R, two distinct elements 0 and 1 of R, two binary operations + and * on R ( called addition and multiplication, resp.

The two's-complement system has the advantage that the fundamental arithmetic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers ( as long as the inputs are represented in the same number of bits and any overflow beyond those bits is discarded from the result ).

* In electrical engineering, κ is the multiplication factor, a function of the R / X ratio of the equivalent power system network, which is used in calculating the peak short-circuit current of a system fault.