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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Algebraic number ： ウィキペディア英語版 Algebraic number An algebraic number is a possibly complex number that is a root of a finite,〔In order for a number to be algebraic, it has to be the root of a finite, non-zero polynomial. Pi, commonly known to be transcendental, is a root of $\backslash sin(x)\; =\; 0$ which is analytic (meaning that it is equal to its infinite Taylor series). Thus transcendental numbers can be roots of polynomials, but only if those polynomials are infinite.〕 non-zero polynomial in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients). Numbers such as that are not algebraic are said to be transcendental. All but a countable set of real and complex numbers are transcendental.〔See Properties.〕 ==Examples== *The rational numbers, expressed as the quotient of two integers ''a'' and ''b'', ''b'' not equal to zero, satisfy the above definition because $x=a/b$ is the root of $bx-a$.〔Some of the following examples come from Hardy and Wright 1972:159–160 and pp. 178–179〕 *The quadratic surds (irrational roots of a quadratic polynomial $ax^2\; +\; bx\; +\; c$ with integer coefficients $a$, $b$, and $c$) are algebraic numbers. If the quadratic polynomial is monic $(a\; =\; 1)$ then the roots are quadratic integers. *The constructible numbers are those numbers that can be constructed from a given unit length using straightedge and compass. These include all quadratic surds, all rational numbers, and all numbers that can be formed from these using the basic arithmetic operations and the extraction of square roots. (Note that by designating cardinal directions for 1, −1, $i$, and $-i$, complex numbers such as $3+\backslash sqrti$ are considered constructible.) *Any expression formed from algebraic numbers using any combination of the basic arithmetic operations and extraction of ''n''th roots gives another algebraic number. *Polynomial roots that ''cannot'' be expressed in terms of the basic arithmetic operations and extraction of ''n''th roots (such as the roots of $x^5\; -\; x\; +\; 1$). This happens with many, but not all, polynomials of degree 5 or higher. *Gaussian integers: those complex numbers $a+bi$ where both $a$ and $b$ are integers are also quadratic integers. *Trigonometric functions of rational multiples of $\backslash pi$ (except when undefined): that is, the trigonometric numbers. For example, each of $\backslash cos(\backslash pi/7)$, $\backslash cos(3\backslash pi/7)$, $\backslash cos(5\backslash pi/7)$ satisfies $8x^3\; -\; 4x^2\; -\; 4x\; +\; 1\; =\; 0$. This polynomial is irreducible over the rationals, and so these three cosines are ''conjugate'' algebraic numbers. Likewise, $\backslash tan(3\backslash pi/16)$, $\backslash tan(7\backslash pi/16)$, $\backslash tan(11\backslash pi/16)$, $\backslash tan(15\backslash pi/16)$ all satisfy the irreducible polynomial $x^4\; -\; 4x^3\; -\; 6x^2\; +\; 4x\; +\; 1$, and so are conjugate algebraic integers. *Some irrational numbers are algebraic and some are not: * *The numbers $\backslash sqrt$ and $\backslash sqrt()/2$ are algebraic since they are roots of polynomials $x^2\; -\; 2$ and $8x^3\; -\; 3$, respectively. * *The golden ratio $\backslash phi$ is algebraic since it is a root of the polynomial $x^2\; -\; x\; -\; 1$. * *The numbers $\backslash pi$ and $e$ are not algebraic numbers (see the Lindemann–Weierstrass theorem);〔Also Liouville's theorem can be used to "produce as many examples of transcendentals numbers as we please," cf Hardy and Wright p. 161ff〕 hence they are transcendental. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Algebraic number」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.