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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 ...
R is a monoid under multiplication, meaning that: (a · b) · c = a · (b · c) for all a, b, c in R (that is, ⋅ is associative). There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R (that is, 1 is the multiplicative identity). [b] Multiplication is distributive with respect to addition, meaning that:
In the second step, the distributive law is used to simplify each of the two terms. Note that this process involves a total of three applications of the distributive property. In contrast to the FOIL method, the method using distributivity can be applied easily to products with more terms such as trinomials and higher.
The grid method uses the distributive property twice to expand the product, once for the horizontal factor, and once for the vertical factor. Historically the grid calculation (tweaked slightly) was the basis of a method called lattice multiplication , which was the standard method of multiple-digit multiplication developed in medieval Arabic ...
An element x is called a dual distributive element if ∀y,z: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). In a distributive lattice, every element is of course both distributive and dual distributive. In a non-distributive lattice, there may be elements that are distributive, but not dual distributive (and vice versa).
To see that the expanded product equals the sum on the first line, apply the distributive law to the product. This expands the product into a sum of monomials of the form x a 1 x 2 a 2 x 3 a 3 ⋯ {\displaystyle x^{a_{1}}x^{2a_{2}}x^{3a_{3}}\cdots } for some sequence of coefficients a i {\displaystyle a_{i}} , only finitely many of which can be ...
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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.