# partial fractions

Every fractional number, i. e. such a rational number^{} $\frac{m}{n}$ that the integer $m$ is not divisible by the integer $n$, can be decomposed to a sum of partial fractions as follows:

$$\frac{m}{n}=\frac{{m}_{1}}{{p}_{1}^{{\nu}_{1}}}+\frac{{m}_{2}}{{p}_{2}^{{\nu}_{2}}}+\mathrm{\cdots}+\frac{{m}_{t}}{{p}_{t}^{{\nu}_{t}}}$$ |

Here, the ${p}_{i}$’s are distinct positive prime numbers^{}, the ${\nu}_{i}$’s positive integers and the ${m}_{i}$’s some integers. Cf. the partial fractions of expressions.

Examples:

$$\frac{6}{289}=\frac{6}{{17}^{2}}$$ |

$$-\frac{1}{24}=-\frac{3}{{2}^{3}}+\frac{1}{{3}^{1}}$$ |

$$\frac{1}{504}=-\frac{1}{{2}^{3}}+\frac{32}{{3}^{2}}-\frac{24}{{7}^{1}}$$ |

How to get the numerators ${m}_{i}$ for decomposing a fractional number $\frac{1}{n}$ to partial fractions? First one can take the highest power ${p}^{\nu}$ of a prime $p$ which divides the denominator $n$. Then $n={p}^{\nu}u$, where $\mathrm{gcd}(u,{p}^{\nu})=1$. Euclid’s algorithm gives some integers $x$ and $y$ such that

$$1=xu+y{p}^{\nu}.$$ |

Dividing this equation by ${p}^{\nu}u$ gives the

$$\frac{1}{n}=\frac{1}{{p}^{\nu}u}=\frac{x}{{p}^{\nu}}+\frac{y}{u}.$$ |

If $u$ has more than one distinct prime factors, a similar procedure can be made for the fraction $\frac{y}{u}$, and so on.

Note. The numerators ${m}_{1}$, ${m}_{2}$, …, ${m}_{t}$ in the decomposition are not unique. E. g., we have also

$$-\frac{1}{24}=-\frac{11}{{2}^{3}}+\frac{4}{{3}^{1}}.$$ |

Cf. the programme “Murto” (in Finnish) or “Murd” (in Estonian) or “Bruch” (in German) or “Bråk” (in Swedish) or “Fraction”(in French) http://www.wakkanet.fi/ 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 |