Editing Transcendental number (section)

==History==
The name "transcendental" comes {{ety|la|trānscendere|to climb over or beyond, surmount}},<ref>{{cite encyclopedia |title=transcendental |dictionary=[[Oxford English Dictionary]] |url=http://www.oed.com/view/Entry/204606}} ''s.v.''</ref> and was first used for the mathematical concept in [[Gottfried Leibniz|Leibniz's]] 1682 paper in which he proved that {{math|sin ''x''}} is not an [[algebraic function]] of {{mvar|x}}.<ref>{{harvnb|Leibniz|Gerhardt|Pertz|1858|pp=97–98}}; {{harvnb|Bourbaki|1994|p=74}}</ref> [[Leonhard Euler|Euler]], in the eighteenth century, was probably the first person to define transcendental ''numbers'' in the modern sense.<ref>{{harvnb|Erdős|Dudley|1983}}</ref>

[[Johann Heinrich Lambert]] conjectured that {{mvar|[[E (mathematical constant)|e]]}} and [[Pi|{{mvar|π}}]] were both transcendental numbers in his 1768 paper proving the number {{mvar|π}} is [[irrational number|irrational]], and proposed a tentative sketch proof that {{mvar|π}} is transcendental.<ref>{{harvnb|Lambert|1768}}</ref>

[[Joseph Liouville]] first proved the existence of transcendental numbers in 1844,<ref name=Kempner>{{harvnb|Kempner|1916}}</ref> and in 1851 gave the first decimal examples such as the [[Liouville number|Liouville constant]] <!-- "Decimal Liouville constant" uses 10^-n! | "Binary Liouville constant" uses 2^-n! //-->

<math display=block>
\begin{align}
  L_b &= \sum_{n=1}^\infty 10^{-n!} \\[2pt]
  &= 10^{-1} + 10^{-2} + 10^{-6} + 10^{-24} + 10^{-120} + 10^{-720} + 10^{-5040} + 10^{-40320} + \ldots \\[4pt]
  &= 0.\textbf{1}\textbf{1}000\textbf{1}00000000000000000\textbf{1}00000000000000000000000000000000000000000000000000000\ \ldots 
\end{align}</math>

in which the {{mvar|n}}th digit after the decimal point is {{math|1}} if {{mvar|n}} = {{mvar|k}}{{math|!}} ({{mvar|k}} [[factorial]]) for some {{mvar|k}} and {{math|0}} otherwise.<ref>{{cite web| url = http://mathworld.wolfram.com/LiouvillesConstant.html| title = Weisstein, Eric W. "Liouville's Constant", MathWorld}}</ref> In other words, the {{mvar|n}}th digit of this number is 1 only if {{mvar|n}} is one of {{math|1=1! = 1, 2! = 2, 3! = 6, 4! = 24}}, etc. Liouville showed that this number belongs to a class of transcendental numbers that can be more closely approximated by [[rational number]]s than can any irrational algebraic number, and this class of numbers is called the [[Liouville number]]s. Liouville showed that all Liouville numbers are transcendental.<ref>{{harvnb|Liouville|1851}}</ref>

The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers' existence was {{mvar|e}}, 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 [[Cantor's first set theory article|a new method for constructing transcendental numbers]].<ref>{{harvnb|Cantor|1874}}; {{harvnb|Gray|1994}}</ref> Although this was already implied by his proof of the countability of the algebraic numbers, Cantor also published a construction that proves there are as many transcendental numbers as there are real numbers.{{efn|
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.<ref>{{harvnb|Cantor|1878|p=254}}</ref>
}}
Cantor's work established the ubiquity of transcendental numbers.

In 1882 [[Ferdinand von Lindemann]] published the first complete proof that {{mvar|π}} is transcendental. He first proved that {{math|''e{{sup|a}}''}} is transcendental if {{mvar|a}} is a non-zero algebraic number. Then, since {{math|''e{{sup|iπ}}'' {{=}} −1}} is algebraic (see [[Euler's identity]]), {{math|''iπ''}} must be transcendental. But since {{math|''i''}} is algebraic, {{mvar|π}} must therefore be transcendental. This approach was generalized by [[Karl Weierstrass]] to what is now known as the [[Lindemann–Weierstrass theorem]]. The transcendence of {{mvar|π}} implies that geometric constructions involving [[compass and straightedge]] only cannot produce certain results, for example [[squaring the circle]].

In 1900 [[David Hilbert]] posed a question about transcendental numbers, [[Hilbert's seventh problem]]: If {{mvar|a}} is an [[algebraic number]] that is not 0 or 1, and {{mvar|b}} is an irrational algebraic number, is {{math|''a{{sup|b}}''}} necessarily transcendental? The affirmative answer was provided in 1934 by the [[Gelfond–Schneider theorem]]. This work was extended by [[Alan Baker (mathematician)|Alan Baker]] in the 1960s in his work on lower bounds for linear forms in any number of logarithms (of algebraic numbers).<ref>{{cite report |first=Alan |last=Baker |year=1998 |title=J.J. O'Connor and E.F. Robertson |type=biographies |series=The MacTutor History of Mathematics archive |publisher=[[University of St. Andrew's]] |place=[[St Andrews|St. Andrew's, Scotland]] |website=www-history.mcs.st-andrews.ac.uk |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Baker_Alan.html}}</ref>