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In approximate arithmetic, such as floating-point arithmetic, the distributive property of multiplication (and division) over addition may fail because of the limitations of arithmetic precision. For example, the identity 1 / 3 + 1 / 3 + 1 / 3 = ( 1 + 1 + 1 ) / 3 {\displaystyle 1/3+1/3+1/3=(1+1+1)/3} fails in decimal arithmetic , regardless of ...
The FOIL method is a special case of a more general method for multiplying algebraic expressions using the distributive law. The word FOIL was originally intended solely as a mnemonic for high-school students learning algebra. The term appears in William Betz's 1929 text Algebra for Today, where he states: [2]
A free group of rank k clearly has subgroups of every rank less than k. Less obviously, a (nonabelian!) free group of rank at least 2 has subgroups of all countable ranks. The commutator subgroup of a free group of rank k > 1 has infinite rank; for example for F(a,b), it is freely generated by the commutators [a m, b n] for non-zero m and n.
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011.
This property is also sometimes called the diamond property, after the shape of the diagram shown on the right. Some authors reserve the term diamond property for a variant of the diagram with single reductions everywhere; that is, whenever a → b and a → c, there must exist a d such that b → d and c → d. The single-reduction variant is ...
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix.It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a standard choice of basis.
The formula for the difference of two squares can be used for factoring polynomials that contain the square of a first quantity minus the square of a second quantity. For example, the polynomial can be factored as follows:
The generalized distributive law (GDL) is a generalization of the distributive property which gives rise to a general message passing algorithm. [1] It is a synthesis of the work of many authors in the information theory , digital communications , signal processing , statistics , and artificial intelligence communities.
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