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Division is also not, in general, associative, meaning that when dividing multiple times, the order of division can change the result. [7] For example, (24 / 6) / 2 = 2, but 24 / (6 / 2) = 8 (where the use of parentheses indicates that the operations inside parentheses are performed before the operations outside parentheses).
The combination of these two symbols is sometimes known as a long division symbol or division bracket. [8] It developed in the 18th century from an earlier single-line notation separating the dividend from the quotient by a left parenthesis. [9] [10] The process is begun by dividing the left-most digit of the dividend by the divisor.
Fink protocol (also known as successive pairs or lone chooser) is a discrete protocol that can be used for online division: given a proportional division for n − 1 partners, when a new partner enters the party, the protocol modifies the existing division so that both the new partner and the existing partners remain with 1/n. The disadvantage ...
In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than the absolute value of the divisor. A fundamental property is that the quotient and the remainder ...
Originally, one of these marks (or a plain line) was used in ancient manuscripts to mark passages that were suspected of being corrupted or spurious; the practice of adding such marginal notes became known as obelism. The dagger symbol †, also called an obelisk, [10] is derived from the obelus, and continues to be used for this purpose.
The similarity between Euclidean division for integers and that for polynomials motivates the search for the most general algebraic setting in which Euclidean division is valid. The rings for which such a theorem exists are called Euclidean domains, but in this generality, uniqueness of the quotient and remainder is not guaranteed. [8]
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
Wedderburn's little theorem states that if D is a finite division algebra, then D is a finite field. [1] Over an algebraically closed field K (for example the complex numbers C), there are no finite-dimensional associative division algebras, except K itself. [2] Associative division algebras have no nonzero zero divisors.