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.

This ensures that the multiplication operation is continuous.

The set of invertible elements in any unital Banach algebra is an open set, and the inversion operation on this set is continuous, ( and hence homeomorphism ) so that it forms a topological group under multiplication.

If n ≥ 1 and is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation ; it is written as ( Z / nZ )< sup >×</ sup > or Z < sub > n </ sub >< sup >*</ sup >.

The language of arithmetic has symbols for 0, 1, the successor operation, addition, and multiplication, intended to be interpreted in the usual way over the natural numbers.

Additive notation is typically used when numerical addition or a commutative operation other than multiplication interprets the group operation ;

Multiplicative notation is typically used when numerical multiplication or a noncommutative operation interprets the group operation.

Also, the multiplication operation in a field is required to be commutative.

The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold ; subtraction and division are defined implicitly in terms of the inverse operations of addition and multiplication :< ref group =" note "> That is, the axiom for addition only assumes a binary operation The axiom of inverse allows one to define a unary operation that sends an element to its negative ( its additive inverse ); this is not taken as given, but is implicitly defined in terms of addition as " is the unique b such that ", " implicitly " because it is defined in terms of solving an equation — and one then defines the binary operation of subtraction, also denoted by "−", as in terms of addition and additive inverse.

In the same way, one defines the binary operation of division ÷ in terms of the assumed binary operation of multiplication and the implicitly defined operation of " reciprocal " ( multiplicative inverse ).</ ref >

* By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers.

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.

* 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.

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.

* The algebra of all bounded real-or complex-valued functions defined on some set ( with pointwise multiplication and the supremum norm ) is a unital Banach algebra.

Although most often used for matrices whose entries are real or complex numbers, the definition of the determinant only involves addition, subtraction and multiplication, and so it can be defined for square matrices with entries taken from any commutative ring.

equipped with the group multiplication defined by ( f ∗ g )( t ) := f ( 2t ) if 0 ≤ t ≤ 1 / 2 and ( f ∗ g )( t ) := g ( 2t − 1 ) if 1 / 2 ≤ t ≤ 1.

Prime numbers give rise to two more general concepts that apply to elements of any commutative ring R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements.

Generally, any number concept leading to multiplication cannot be defined in Presburger arithmetic, since that leads to incompleteness and undecidability.

Exponentiation to the power of any natural number other than zero can be defined consistently whenever multiplication is power-associative.

More specifically, let be the set of all polynomials with integer coefficients ( such a polynomial is also referred to as a polynomial over ), with addition and multiplication defined to be natural polynomial addition and multiplication.

Looking at the definition of a vector space, we see that properties 2 and 3 above assure closure of W under addition and scalar multiplication, so the vector space operations are well defined.

The addition operation is defined by pointwise addition of functions and the multiplication operation is defined by function composition.

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

Given this, it is quite natural and convenient to designate a general sequence a < sub > n </ sub > by by the formal expression, even though the latter is not an expression formed by the operations of addition and multiplication defined above ( from which only finite sums can be constructed ).

The < tt > MixColumns </ tt > step can also be viewed as a multiplication by a particular MDS matrix in a finite field.

The ket can be computed by normal matrix multiplication.

Then the bra can be computed by normal matrix multiplication.

The algebra multiplication and the Banach space norm are required to be related by the following inequality:

Membership in co-NP is also straightforward: one can just list the prime factors of m, which the verifier can confirm to be valid by multiplication and the AKS primality test.

Among his principal miracles are: ( 1 ) procuring of food for a sick monk and curing the wife of his benefactor ; ( 2 ) escape from hurt when surrounded by wolves ; ( 3 ) obedience of a bear which evacuated a cave at his biddings ; ( 4 ) producing a spring of water near his cave ; ( 5 ) repletion of the Luxeuil granary when empty ; ( 6 ) multiplication of bread and beer for his community ; ( 7 ) curing of the sick monks, who rose from their beds at his request to reap the harvest ; ( 8 ) giving sight to a blind man at Orleans ; ( 9 ) taming a bear, and yoking it to a plough.

As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers ( whereas ten has only two non-trivial factors: 2 and 5, with neither 3 nor 4 ), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table.