partial fractions

Every fractional number, i. e. such a rational numberPlanetmathPlanetmathPlanetmath mn that the integer m is not divisible by the integer n, can be decomposed to a sum of partial fractions as follows:


Here, the pi’s are distinct positive prime numbersMathworldPlanetmath, the νi’s positive integers and the mi’s some integers.  Cf. the partial fractions of expressions.



How to get the numerators mi for decomposing a fractional number 1n to partial fractions?  First one can take the highest power pν of a prime p which divides the denominator n.  Then  n=pνu,  where  gcd(u,pν)=1.  Euclid’s algorithm gives some integers x and y such that


Dividing this equation by pνu gives the


If u has more than one distinct prime factors, a similar procedure can be made for the fraction yu, and so on.

Note.  The numerators  m1, m2, …, mt  in the decomposition are not unique.  E. g., we have also


Cf. the programme “Murto” (in Finnish) or “Murd” (in Estonian) or “Bruch” (in German) or “Bråk” (in Swedish) or “Fraction”(in French) pahio/ohjelmi.htmlhere.

Title partial fractions
Canonical name PartialFractions
Date of creation 2013-03-22 14:18:10
Last modified on 2013-03-22 14:18:10
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 34
Author pahio (2872)
Entry type Definition
Classification msc 11A41
Synonym partial fractions of fractional numbers
Related topic CategoryOfAdditiveFractions
Defines fractional number