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17 is divided into 3 groups of 5, with 2 as leftover. Here, the dividend is 17, the divisor is 3, the quotient is 5, and the remainder is 2 (which is strictly smaller than the divisor 3), or more symbolically, 17 = (3 × 5) + 2. In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the ...
The theorem which underlies the definition of the Euclidean division ensures that such a quotient and remainder always exist and are unique. [20] In Euclid's original version of the algorithm, the quotient and remainder are found by repeated subtraction; that is, r k−1 is subtracted from r k−2 repeatedly until the remainder r k is smaller ...
In his original paper, Gauss made another choice, by choosing the unique associate such that the remainder of its division by 2 + 2i is one. In fact, as N(2 + 2i) = 8, the norm of the remainder is not greater than 4. As this norm is odd, and 3 is not the norm of a Gaussian integer, the norm of the remainder is one, that is, the remainder is a unit.
In this case, s is called the least absolute remainder. [3] As with the quotient and remainder, k and s are uniquely determined, except in the case where d = 2n and s = ± n. For this exception, we have: a = k⋅d + n = (k + 1)d − n. A unique remainder can be obtained in this case by some convention—such as always taking the positive value ...
In numerical analysis and linear algebra, lower–upper ( LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition ). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination.
In numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes through the points of the dataset. [ 1] Given a set of n + 1 data points , with no two the same, a polynomial function is said to interpolate the data if for each .
Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones ...
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the ...