PlanetMath: partial fractions

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partial fractions

(Definition)

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

$\displaystyle \frac{m}{n} = \frac{m_1}{p_1^{\nu_1}}+\frac{m_2}{p_2^{\nu_2}}+\cdots+\frac{m_t}{p_t^{\nu_t}}$

Here, the

's are distinct positive prime numbers, the $\nu_i$ 's positive integers and the

's some integers. Cf. the partial fractions of expressions.

Examples:

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

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

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

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

$\displaystyle 1 = xu+yp^{\nu}.$

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

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

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 , , ..., in the decomposition are not unique. E. g., we have also

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

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

"partial fractions" is owned by pahio.

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Other names:

partial fractions of fractional numbers

Also defines:

fractional number

This object's parent.

Attachments:: partial fractions in Euclidean domains (Result) by stevecheng
partial fractions for polynomials (Result) by stevecheng

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Cross-references: decomposition, fraction, similar, prime factors, equation, Euclid's algorithm, denominator, divides, numerators, prime numbers, positive, sum, divisible, integer, rational number
There are 14 references to this entry.

This is version 27 of partial fractions, born on 2004-04-14, modified 2006-09-29.
Object id is 5761, canonical name is PartialFractions.
Accessed 9118 times total.

Classification:

AMS MSC:

11A41 (Number theory :: Elementary number theory :: Primes)

Pending Errata and Addenda

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