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 a uniquely 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
For positive numbers the quotient may be obtained by performing the division algorithm with and . If , then indicates how many times fits in .
The quotient of and does not change if both numbers (elements) are multiplied (or divided, which action 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 square root polynomials, e.g.
in the first case one aspires after a real and in the second case after a rational denominator.
The division is neither associative nor commutative, but it is right distributive over addition:
See also the proportion equation.