Division is the operationMathworldPlanetmath which assigns to every two numbers (or more generally, elements of a field) a and b their quotientPlanetmathPlanetmath or ratio, provided that the latter, b, is distinct from zero.

The quotient (or ratio)  ab  of a and b may be defined as such a number (or element of the field) x that  bx=a.  Thus,


which is the “fundamental property of quotient”.

The quotient of the numbers a and b (0) is a uniquely determined number, since if one had


then we could write


from which the supposition b0 would imply x-y=0, i.e. x=y.

The explicit general expression for ab is


where b-1 is the inverse number (the multiplicative inverseMathworldPlanetmath) of a, because

  • For positive numbers the quotient may be obtained by performing the division algorithmPlanetmathPlanetmath with a and b.  If  a>b>0,  then ab indicates how many times b fits in a.

  • The quotient of a and b does not change if both numbers (elements) are multiplied (or divided, which is called reductionPlanetmathPlanetmath) by any  k0:


    So we have the method for getting the quotient of complex numbersMathworldPlanetmathPlanetmath,


    where b¯ is the complex conjugate of b, 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.

  • The division is neither associative nor commutativePlanetmathPlanetmathPlanetmathPlanetmath, but it is right distributive over additionPlanetmathPlanetmath:

Title division
Canonical name Division
Date of creation 2014-08-08 17:51:29
Last modified on 2014-08-08 17:51:29
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 29
Author pahio (2872)
Entry type Definition
Classification msc 00A05
Classification msc 12E99
Related topic InverseFormingInProportionToGroupOperation
Related topic DivisionInGroup
Related topic ConjugationMnemonic
Related topic Difference2
Related topic UniquenessOfDivisionAlgorithmInEuclideanDomain
Defines quotient
Defines ratio
Defines fundamental property of quotient
Defines reduction