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In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with rational coefficients. The best-known transcendental numbers are π and ''e''. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation . ==History== The name "transcendental" comes from Leibniz in his 1682 paper where he proved that is not an algebraic function of .〔()〕 Euler was probably the first person to define transcendental ''numbers'' in the modern sense. Johann Heinrich Lambert conjectured that and were both transcendental numbers in his 1761 paper proving the number is irrational. Joseph Liouville first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant : in which the th digit after the decimal point is if is equal to ( factorial) for some and otherwise.〔(Weisstein, Eric W. "Liouville's Constant", MathWorld )〕 In other words, the nth digit of this number is 1 only if is one of the numbers , etc. Liouville showed that this number is what we now call a Liouville number; this essentially means that it can be more closely approximated by rational numbers than can any irrational algebraic number. Liouville showed that all Liouville numbers are transcendental. The first number to be proven transcendental without having been specifically constructed for the purpose was , by Charles Hermite in 1873. In 1874, Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable. He also gave a new method for constructing transcendental numbers. In 1878, Cantor published a construction that proves there are as many transcendental numbers as there are real numbers.〔 (Cantor's construction builds a one-to-one correspondence between the set of transcendental numbers and the set of real numbers. In this article, Cantor only applies his construction to the set of irrational numbers. See p. 254.)〕 Cantor's work established the ubiquity of transcendental numbers. In 1882, Ferdinand von Lindemann published a proof that the number is transcendental. He first showed that is transcendental when is algebraic and not zero. Then, since is algebraic (see Euler's identity), and therefore must be transcendental. This approach was generalized by Karl Weierstrass to the Lindemann–Weierstrass theorem. The transcendence of allowed the proof of the impossibility of several ancient geometric constructions involving compass and straightedge, including the most famous one, squaring the circle. In 1900, David Hilbert posed an influential question about transcendental numbers, Hilbert's seventh problem: If is an algebraic number, that is not zero or one, and is an irrational algebraic number, is necessarily transcendental? The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem. This work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms (of algebraic numbers).〔J J O'Connor and E F Robertson: (Alan Baker ). The MacTutor History of Mathematics archive 1998.〕 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「Transcendental number」の詳細全文を読む スポンサード リンク
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