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partial fractions∗
pahio†
2013-03-21 17:18:37
Every fractional number, i. e. such a rational number m
n that the integer m
is not divisible by the integer n, can be decomposed to a sum of partial fractions
as follows:
m
m1
m2
mt
= ν1 + ν2 + · · · + νt
n
p1
p2
pt
Here, the pi ’s are distinct positive prime numbers, the νi ’s positive integers and
the mi ’s some integers. Cf. the partial fractions of expressions.
Examples:
6
6
=
289
172
3
1
1
= − 3+ 1
−
24
2
3
1
1
32 24
= − 3+ 2 − 1
504
2
3
7
How to get the numerators mi for decomposing a fractional number n1 to
partial fractions? First one can take the highest power pν of a prime p which
divides the denominator n. Then n = pν u, where gcd (u, pν ) = 1. Euclid’s
algorithm gives some integers x and y such that
1 = xu+ypν .
Dividing this equation by pν u gives the decomposition
1
1
x y
= ν = ν+ .
n
p u
p
u
If u has more than one distinct prime factors, a similar procedure can be made
for the fraction uy , and so on.
∗ hPartialFractionsi
created: h2013-03-21i by: hpahioi version: h35761i Privacy setting:
h1i hDefinitioni h11A41i
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Note. The numerators m1 , m2 , . . . , mt in the decomposition are not
unique. E. g., we have also
−
4
1
11
= − 3 + 1.
24
2
3
Cf. the programme “Murto” (in Finnish) or “Murd” (in Estonian) or “Bruch”
(in German) or “Bråk” (in Swedish) or “Fraction”(in French) here.
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