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In the following examples, the use of the distributive law on the set of real numbers is illustrated. When multiplication is mentioned in elementary mathematics, it usually refers to this kind of multiplication.
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.
Galileo's law of odd numbers. A ramification of the difference of consecutive squares, Galileo's law of odd numbers states that the distance covered by an object falling without resistance in uniform gravity in successive equal time intervals is linearly proportional to the odd numbers. That is, if a body falling from rest covers a certain ...
It also satisfies the distributive law, meaning that (+) = +. These properties may be summarized by saying that the dot product is a bilinear form . Moreover, this bilinear form is positive definite , which means that a ⋅ a {\displaystyle \mathbf {a} \cdot \mathbf {a} } is never negative, and is zero if and only if a = 0 {\displaystyle ...
For two elements a 1 + b 1 i + c 1 j + d 1 k and a 2 + b 2 i + c 2 j + d 2 k, their product, called the Hamilton product (a 1 + b 1 i + c 1 j + d 1 k) (a 2 + b 2 i + c 2 j + d 2 k), is determined by the products of the basis elements and the distributive law. The distributive law makes it possible to expand the product so that it is a sum of ...
Consider now the exterior product of and : = (+) (+) = + + + = (), where the first step uses the distributive law for the exterior product, and the last uses the fact that the exterior product is an alternating map, and in particular = (). (The fact that the exterior product is an alternating map also forces = =
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
These constructs surfaced again in the 1980s: in a 1982 paper by David Joyce [5] (where the term quandle, an arbitrary nonsense word, was coined), [6] in a 1982 paper by Sergei Matveev (under the name distributive groupoids) [7] and in a 1986 conference paper by Egbert Brieskorn (where they were called automorphic sets). [8]