Fraction formula - fraction formula

Let A and B be rational polynomials (polynomials) for some characters (especially variables), and let B be identically non-zero (at least one of the coefficients is non-zero). The equation A/B is called a fractional equation (or rational equation) for these characters. A is called the numerator and B the denominator. If the degree of the numerator is smaller than the degree of the denominator, it is called a proper fractional equation, otherwise it is called an improper fractional equation. Additionally, if the numerator and denominator have no common factors other than a constant, they are called irreducible, otherwise they are called reducible. When an equation is reducible, eliminating the common factors of the numerator and denominator is called reducing. For example,
( ^x2 +2x-3)/( ^x2 -1)
Simplify to (x+3)/(x+1)
is obtained.

The equality and four arithmetic operations of fractional expressions are defined as follows:

A complex fractional expression whose numerator and denominator are also fractional expressions is called a complex fractional expression. Complex fractional expressions can be reduced to ordinary fractional expressions using the four basic arithmetic operations. Expressing a given fractional expression as the sum of multiple fractional expressions is called partial fraction decomposition. In general, when the denominator of an irreducible proper fractional expression is decomposed into factors that have no common factors, the given fractional expression can be expressed in a single way as the sum of irreducible proper fractional expressions with each of these factors as denominators. For example,

Partial fraction decomposition is commonly used in the method of undetermined coefficients and is also used in integrating fractional expressions.

[Yoshio Takeuchi]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

A、Bをそれぞれいくつかの文字（とくに変数）についての有理整式（多項式）とし、Bは恒等的にゼロでない（係数のうち少なくとも一つはゼロでない）とする。式A/Bをこれら文字についての分数式（または有理式）という。Aを分子、Bを分母といい、分子の次数が分母の次数より小さいとき真分数式、そうでないとき仮分数式という。また、分母と分子が定数以外に公約数をもたないとき、既約(きやく)、そうでないとき可約であるという。可約な場合、分母分子の公約数を消し去ることを約分する（約す）という。たとえば、
　　(x²+2x-3)/(x²-1)
を約分して
　　(x+3)/(x+1)
が得られる。

　分数式の相等・四則は次のように定義される。

　分母分子がまた分数式であるものを繁分数式(はんぶんすうしき)という。繁分数式は四則計算により通常の分数式に帰着される。与えられた分数式を複数の分数式の和で表すことを部分分数分解という。一般に既約真分数式の分母が、互いに公約数をもたない因数に分解されるとき、与えられた分数式はこれら因数をそれぞれ分母とする既約真分数式の和として、一通りに表される。たとえば

部分分数分解には一般に未定係数法が用いられる。また部分分数分解は分数式の積分法に用いられる。

［竹内芳男］

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