In mathematics, many objects, including symbols expressed as natural numbers, are called orders. For example, there are the following: (1) When group G is a finite set (finite group), if the number of elements of G is n , then n is called the order of G. (2) For an element a of group G , the smallest positive integer m for which a m = e ( e is the identity element of group G ) is called the order of a . For example, if ω is the m- th root of 1, then the smallest positive integer m for which ω m = 1 is the order of ω. (3) The maximum number of intersections between an algebraic curve C and a line l (obtained by moving l in various ways) is sometimes called an order. (4) If z = a is a pole of a complex function f ( z ), and the principal part of the expansion of f ( z ) into a Laurent series begins with the term ( z - a ) - n , then a is said to be the nth (order) pole. Also, n is called the order of a . Similarly, when f ( a ) = 0 and the Taylor expansion at a begins with the term ( z - a ) n , the order of the zero point a of f ( z ) is called n . Source: Encyclopaedia Britannica Concise Encyclopedia About Encyclopaedia Britannica Concise Encyclopedia Information |
数学では,自然数で表わされる標識について,位数と名づけられる対象が多い。たとえば次のような例がある。 (1) 群 G が有限集合 (有限群) であるとき,G の元の個数が n であれば,この n を G の位数という。 (2) 群 G の1つの元 a に対し,am=e ( e は群 G の単位元) となる最小の正整数 m を a の位数という。たとえば,ω を1の m 乗根とすると,ωm=1 となるような最小の正の整数 m は,ω の位数である。 (3) 代数曲線 C と直線 l との交点の個数 ( l をいろいろと動かして得られる) の最大数を位数ということがある。 (4) z=a が複素関数 f(z) の極であって,f(z) をローラン級数に展開した主要部が (z-a)-n の項から始まるとき,a は n 位 (次) の極であるという。また n を a の位数という。同様に f(a)=0 で,a でのテイラー展開が (z-a)n の項から始まるとき,f(z) の零点 a の位数を n という。
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