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Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule = along with the associative, commutative, and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field with the real numbers as a subfield.
Dividing two complex numbers (when the divisor is nonzero) results in another complex number, which is found using the conjugate of the denominator: + + = (+) (+) = + + + = + + + +. This process of multiplying and dividing by r − i s {\displaystyle r-is} is called 'realisation' or (by analogy) rationalisation .
In particular, if either or in the complex domain can be computed with some complexity, then that complexity is attainable for all other elementary functions. Below, the size n {\displaystyle n} refers to the number of digits of precision at which the function is to be evaluated.
Even though contains copies of the complex numbers, it is not an associative algebra over the complex numbers. Because it is possible to divide quaternions, they form a division algebra. This is a structure similar to a field except for the non-commutativity of multiplication.
In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following: R (the real numbers) C (the complex numbers) H ...
The split-complex number = + can be represented by the matrix (). Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The squared modulus of z is given by the determinant of the corresponding matrix.
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. A finite-dimensional unital associative algebra (over any field) is a division algebra if and only if it has no nonzero zero ...
The basic principle of Karatsuba's algorithm is divide-and-conquer, using a formula that allows one to compute the product of two large numbers and using three multiplications of smaller numbers, each with about half as many digits as or , plus some additions and digit shifts.