division
Division is the operation which assigns to every two numbers (or more generally, elements of a field) and their quotient
or ratio, provided that the latter, , is distinct from zero.
The quotient (or ratio) of and may be defined as such a number (or element of the field) that . Thus,
which is the “fundamental property of quotient”.
The quotient of the numbers and () is auniquely determined number, since if one had
then we could write
from which the supposition would imply , i.e..
The explicit general expression for is
where is the inverse number (the multiplicative inverse) of , because
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For positive numbers the quotient may be obtained by performing the division algorithm
with and . If , then indicates how many times fits in .
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The quotient of and does not change if both numbers (elements) are multiplied (or divided, which is called reduction
) by any :
So we have the method for getting the quotient of complex numbers
,
where is the complex conjugate of , and the quotient of http://planetmath.org/SquareRootOfSquareRootBinomialsquare root polynomials, e.g.
in the first case one aspires after a real and in the second case after a rational denominator.
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The division is neither associative nor commutative
, but it is right distributive over addition
: