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In abstract algebra, given a magma with binary operation ∗ (which could nominally be termed multiplication), left division of b by a (written a \ b) is typically defined as the solution x to the equation a ∗ x = b, if this exists and is unique. Similarly, right division of b by a (written b / a) is the solution y to the equation y ∗ a = b ...
A finite-dimensional unital associative algebra (over any field) is a division algebra if and only if it has no nonzero zero divisors. Whenever A is an associative unital algebra over the field F and S is a simple module over A, then the endomorphism ring of S is a division algebra over F; every associative division algebra over F arises in ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
In calculus and mathematical analysis, algebraic operation is also used for the operations that may be defined by purely algebraic methods. For example, exponentiation with an integer or rational exponent is an algebraic operation, but not the general exponentiation with a real or complex exponent.
In the division of 43 by 5, we have: 43 = 8 × 5 + 3, so 3 is the least positive remainder. We also have that: 43 = 9 × 5 − 2, and −2 is the least absolute remainder. These definitions are also valid if d is negative, for example, in the division of 43 by −5, 43 = (−8) × (−5) + 3, and 3 is the least positive remainder, while,
Notice also that the set obtained by forming all the combinations a + b √ 2, where a and b are integers, is an example of an object known in abstract algebra as a ring, and more specifically as an integral domain. The number ω is a unit in that integral domain. See also algebraic number field.
In calculus and mathematical analysis, algebraic operation is also used for the operations that may be defined by purely algebraic methods. For example, exponentiation with an integer or rational exponent is an algebraic operation, but not the general exponentiation with a real or complex exponent. Also, the derivative is an operation that is ...
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.
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