Editing Transcendental number theory

{{Short description|Study of numbers that are not solutions of polynomials with rational coefficients}}
{{Use American English|date = January 2019}}
{{Ring theory sidebar}}

'''Transcendental number theory''' is a branch of [[number theory]] that investigates [[transcendental number]]s (numbers that are not solutions of any [[polynomial equation]] with [[rational number|rational]] [[coefficient]]s), in both qualitative and quantitative ways.

==Transcendence==
{{Main|Transcendental number}}

The [[fundamental theorem of algebra]] tells us that if we have a non-constant [[polynomial]] with rational coefficients (or equivalently, by [[clearing denominators]], with [[integer]] coefficients) then that polynomial will have a [[zero of a function|root]] in the [[complex number]]s.  That is, for any non-constant polynomial <math>P</math> with rational coefficients there will be a complex number <math>\alpha</math> such that <math>P(\alpha)=0</math>.  Transcendence theory is concerned with the converse question: given a complex number <math>\alpha</math>, is there a polynomial <math>P</math> with rational coefficients such that <math>P(\alpha)=0?</math> If no such polynomial exists then the number is called transcendental.

More generally the theory deals with [[algebraic independence]] of numbers.  A set of numbers {α<sub>1</sub>, α<sub>2</sub>, …, α<sub>''n''</sub>} is called algebraically independent over a [[field (mathematics)|field]] ''K'' if there is no non-zero polynomial ''P'' in ''n'' variables with coefficients in ''K'' such that ''P''(α<sub>1</sub>, α<sub>2</sub>, …, α<sub>''n''</sub>) = 0.  So working out if a given number is transcendental is really a special case of algebraic independence where ''n''&nbsp;=&thinsp;1 and the field ''K'' is the field of [[rational number]]s.

A related notion is whether there is a [[closed-form expression]] for a number, including exponentials and logarithms as well as algebraic operations. There are various definitions of "closed-form", and questions about closed-form can often be reduced to questions about transcendence.

==History==
===Approximation by rational numbers: Liouville to Roth===

Use of the term ''transcendental'' to refer to an object that is not algebraic dates back to the seventeenth century, when [[Gottfried Leibniz]] proved that the [[sine function]] was not an [[algebraic function]].<ref>N. Bourbaki, ''Elements of the History of Mathematics'' Springer (1994).</ref>  The question of whether certain classes of numbers could be transcendental dates back to 1748<ref>{{Harvnb|Gelfond|1960|p=2}}.</ref> when [[Euler]] asserted<ref>{{cite book |first=L. |last=Euler |title=Introductio in analysin infinitorum |url=https://archive.org/details/bub_gb_jQ1bAAAAQAAJ |location=Lausanne |year=1748 }}</ref> that the number log<sub>''a''</sub>''b'' was not [[algebraic number|algebraic]] for [[rational number]]s ''a'' and ''b'' provided ''b'' is not of the form ''b''&nbsp;=&nbsp;''a''<sup>''c''</sup> for some rational ''c''.

Euler's assertion was not proved until the twentieth century, but almost a hundred years after his claim [[Joseph Liouville]] did manage to prove the existence of numbers that are not algebraic, something that until then had not been known for sure.<ref>The existence proof based on the different [[cardinalities]] of the [[real number|real]] and the [[algebraic number|algebraic]] numbers was not possible before [[Cantor's first set theory article]] in 1874.</ref>  His original papers on the matter in the 1840s sketched out arguments using [[simple continued fraction]]s to construct transcendental numbers.  Later, in the 1850s, he gave a [[Necessary and sufficient condition|necessary condition]] for a number to be algebraic, and thus a sufficient condition for a number to be transcendental.<ref>{{cite journal | last1 = Liouville | first1 = J. | year = 1844 | title = Sur les classes très étendues de quantités dont la valeur n'est ni algébrique ni même réductible à des irrationelles algébriques | journal = Comptes rendus de l'Académie des Sciences de Paris | volume = 18 | pages = [http://gallica.bnf.fr/ark:/12148/bpt6k2977n/f883.image 883–885], [http://gallica.bnf.fr/ark:/12148/bpt6k2977n/f910.image 910–911]}}; ''Journal Math. Pures et Appl.'' '''16''', (1851), pp.133–142.</ref>  This transcendence criterion was not strong enough to be necessary too, and indeed it fails to detect that the number [[Euler's number|''e'']] is transcendental.  But his work did provide a larger class of transcendental numbers, now known as [[Liouville number]]s in his honour.

Liouville's criterion essentially said that algebraic numbers cannot be very well approximated by rational numbers.  So if a number can be very well approximated by rational numbers then it must be transcendental.  The exact meaning of "very well approximated" in Liouville's work relates to a certain exponent.  He showed that if α is an [[algebraic number]] of degree ''d''&nbsp;≥&nbsp;2 and ε is any number greater than zero, then the expression
:<math>\left|\alpha-\frac{p}{q}\right|<\frac{1}{q^{d+\varepsilon}}</math>
can be satisfied by only finitely many rational numbers ''p''/''q''.  Using this as a criterion for transcendence is not trivial, as one must check whether there are infinitely many solutions ''p''/''q'' for every ''d''&nbsp;≥&nbsp;2.

In the twentieth century work by [[Axel Thue]],<ref>{{cite journal |first=A. |last=Thue |title=Über Annäherungswerte algebraischer Zahlen |journal=[[Journal für die reine und angewandte Mathematik|J. Reine Angew. Math.]] |volume=1909 |year=1909 |issue= 135|pages=284–305 |doi=10.1515/crll.1909.135.284 |s2cid=125903243 }}</ref> [[Carl Ludwig Siegel|Carl Siegel]],<ref>{{cite journal |first=C. L. |last=Siegel |title=Approximation algebraischer Zahlen |journal=[[Mathematische Zeitschrift]] |volume=10 |year=1921 |issue=3–4 |pages=172–213 |doi=10.1007/BF01211608 | doi-access=free |url=https://zenodo.org/record/1538156 }}</ref> and [[Klaus Roth]]<ref>{{cite journal |first=K. F. |last=Roth |title=Rational approximations to algebraic numbers |journal=[[Mathematika]] |volume=2 |year=1955 |issue=1 |pages=1–20 |doi=10.1112/S0025579300000644 }} And "Corrigendum", p. 168, {{doi|10.1112/S002559300000826}}.</ref> reduced the exponent in Liouville's work from ''d''&nbsp;+&nbsp;ε to ''d''/2&nbsp;+&nbsp;1&nbsp;+&nbsp;ε, and finally, in 1955, to 2&nbsp;+&nbsp;ε.  This result, known as the [[Thue–Siegel–Roth theorem]], is ostensibly the best possible, since if the exponent 2&nbsp;+&nbsp;ε is replaced by just 2 then the result is no longer true.  However, [[Serge Lang]] conjectured an improvement of Roth's result; in particular he conjectured that ''q''<sup>2+ε</sup> in the denominator of the right-hand side could be reduced to <math>q^{2}(\log q)^{1+ \epsilon}</math>.

Roth's work effectively ended the work started by Liouville, and his theorem allowed mathematicians to prove the transcendence of many more numbers, such as the [[Champernowne constant]].  The theorem is still not strong enough to detect ''all'' transcendental numbers, though, and many famous constants including ''e'' and π either are not or are not known to be very well approximable in the above sense.<ref>{{cite journal |first=K. |last=Mahler |title=On the approximation of π |journal=Proc. Akad. Wetensch. Ser. A |volume=56 |year=1953 |pages=30–42 }}</ref>

===Auxiliary functions: Hermite to Baker===
Fortunately other methods were pioneered in the nineteenth century to deal with the algebraic properties of ''e'', and consequently of π through [[Euler's identity]].  This work centred on use of the so-called [[auxiliary function]].  These are [[Function (mathematics)|functions]] which typically have many zeros at the points under consideration.  Here "many zeros" may mean many distinct zeros, or as few as one zero but with a high [[Multiplicity (mathematics)#Multiplicity of a zero of a function|multiplicity]], or even many zeros all with high multiplicity.  [[Charles Hermite]] used auxiliary functions that approximated the functions <math>e^{kx}</math> for each [[natural number]] <math>k</math> in order to prove the transcendence of <math>e</math> in 1873.<ref>{{cite journal |first=C. |last=Hermite |title=Sur la fonction exponentielle |journal=C. R. Acad. Sci. Paris |volume=77 |year=1873 }}</ref>  His work was built upon by [[Ferdinand von Lindemann]] in the 1880s<ref>{{cite journal |first=F. |last=Lindemann |title=Ueber die Zahl π  |journal=[[Mathematische Annalen]] |volume=20 |year=1882 |issue=2 |pages=213–225 |doi=10.1007/BF01446522 | doi-access=free |url=https://zenodo.org/record/1428234 }}</ref> in order to prove that ''e''<sup>α</sup> is transcendental for nonzero algebraic numbers α.  In particular this proved that π is transcendental since ''e''<sup>π''i''</sup> is algebraic, and thus answered in the negative the [[Compass and straightedge constructions|problem of antiquity]] as to whether it was possible to [[Squaring the circle|square the circle]].  [[Karl Weierstrass]] developed their work yet further and eventually proved the [[Lindemann–Weierstrass theorem]] in 1885.<ref>{{cite journal |first=K. |last=Weierstrass |title=Zu Hrn. Lindemann's Abhandlung: 'Über die Ludolph'sche Zahl' |journal=Sitzungber. Königl. Preuss. Akad. Wissensch. Zu Berlin |volume=2 |year=1885 |pages=1067–1086 }}</ref>

In 1900 [[David Hilbert]] posed his famous [[Hilbert's problems|collection of problems]].  The [[Hilbert's seventh problem|seventh of these]], and one of the hardest in Hilbert's estimation, asked about the transcendence of numbers of the form ''a''<sup>''b''</sup> where ''a'' and ''b'' are algebraic, ''a'' is not zero or one, and ''b'' is [[irrational number|irrational]].  In the 1930s [[Alexander Gelfond]]<ref>{{cite journal |first=A. O. |last=Gelfond |title=Sur le septième Problème de D. Hilbert |journal=Izv. Akad. Nauk SSSR |volume=7 |year=1934 |pages=623–630 }}</ref> and [[Theodor Schneider]]<ref>{{cite journal |first=T. |last=Schneider |title=Transzendenzuntersuchungen periodischer Funktionen. I. Transzendend von Potenzen |journal=[[Journal für die reine und angewandte Mathematik]] |volume=1935 |year=1935 |issue= 172|pages=65–69 |doi=10.1515/crll.1935.172.65 |s2cid=115310510 }}</ref> proved that all such numbers were indeed transcendental using a non-explicit auxiliary function whose existence was granted by [[Siegel's lemma]].  This result, the [[Gelfond–Schneider theorem]], proved the transcendence of numbers such as [[Gelfond's constant|''e''<sup>π</sup>]] and the [[Gelfond–Schneider constant]].

The next big result in this field occurred in the 1960s, when [[Alan Baker (mathematician)|Alan Baker]] made progress on a problem posed by Gelfond on [[linear forms in logarithms]].  Gelfond himself had managed to find a non-trivial lower bound for the quantity
:<math>|\beta_1\log\alpha_1 +\beta_2\log\alpha_2|\,</math>
where all four unknowns are algebraic, the αs being neither zero nor one and the βs being irrational.  Finding similar lower bounds for the sum of three or more logarithms had eluded Gelfond, though.  The proof of [[Baker's theorem]] contained such bounds, solving Gauss' [[class number problem]] for class number one in the process.  This work won Baker the [[Fields medal]] for its uses in solving [[Diophantine equation]]s.  From a purely transcendental number theoretic viewpoint, Baker had proved that if α<sub>1</sub>, ..., α<sub>''n''</sub> are algebraic numbers, none of them zero or one, and β<sub>1</sub>, ..., β<sub>''n''</sub> are algebraic numbers such that 1, β<sub>1</sub>, ..., β<sub>''n''</sub> are [[linearly independent]] over the rational numbers, then the number
:<math>\alpha_1^{\beta_1}\alpha_2^{\beta_2}\cdots\alpha_n^{\beta_n}</math>
is transcendental.<ref>A. Baker, ''Linear forms in the logarithms of algebraic numbers. I, II, III'', Mathematika '''13''' ,(1966), pp.204–216; ibid. '''14''', (1967), pp.102–107; ibid. '''14''', (1967), pp.220–228, {{MathSciNet | id = 0220680}}</ref>

===Other techniques: Cantor and Zilber===

In the 1870s, [[Georg Cantor]] started to develop [[set theory]] and, in 1874, published a [[Georg Cantor's first set theory article|paper]] proving that the algebraic numbers could be put in [[Bijection|one-to-one correspondence]] with the set of [[natural number]]s, and thus that the set of transcendental numbers must be [[uncountable set|uncountable]].<ref>{{cite journal |first=G. |last=Cantor |url=http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=266194 |title=Ueber eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen |journal=[[Journal für die reine und angewandte Mathematik|J. Reine Angew. Math.]] |volume=1874 |year=1874 |issue= 77|pages=258–262 |doi=10.1515/crll.1874.77.258 |s2cid=199545885 |language=de}}</ref>  Later, in 1891, Cantor used his more familiar [[Cantor's diagonal argument|diagonal argument]] to prove the same result.<ref>{{cite journal |first=G. |last=Cantor |title=Ueber eine elementare Frage der Mannigfaltigkeitslehre |journal=Jahresbericht der Deutschen Mathematiker-Vereinigung |volume=1 |year=1891 |pages=75–78 |url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002113910 |language=de}}</ref>  While Cantor's result is often quoted as being purely existential and thus unusable for constructing a single transcendental number,<ref>{{cite book |first1=M. |last1=Kac |first2=U. |last2=Stanislaw |title=Mathematics and Logic |url=https://archive.org/details/mathematicslogic0000kacm |url-access=registration |publisher=Fredering A. Praeger |year=1968 |page=[https://archive.org/details/mathematicslogic0000kacm/page/13 13] }}</ref><ref>{{cite book |first=E. T. |last=Bell |title=Men of Mathematics |location=New York |publisher=Simon & Schuster |year=1937 |page=[https://archive.org/details/menofmathematics0041bell/page/569 569] |title-link=Men of Mathematics }}</ref> the proofs in both the aforementioned papers give methods to construct transcendental numbers.<ref>{{cite journal |first=R. |last=Gray |title=Georg Cantor and Transcendental Numbers |url = http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf |journal=[[American Mathematical Monthly]] |volume=101 |year=1994 |issue=9 |pages=819–832 |jstor=2975129 |doi=10.1080/00029890.1994.11997035 }}</ref>

While Cantor used set theory to prove the plenitude of transcendental numbers, a recent development has been the use of [[model theory]] in attempts to prove an [[unsolved problem]] in transcendental number theory.  The problem is to determine the [[transcendence degree]] of the field
:<math>K=\mathbb{Q}(x_1,\ldots,x_n,e^{x_1},\ldots,e^{x_n})</math>
for complex numbers ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> that are linearly independent over the rational numbers.  [[Stephen Schanuel]] [[Schanuel's conjecture|conjectured]] that the answer is at least ''n'', but no proof is known.  In 2004, though, [[Boris Zilber]] published a paper that used model theoretic techniques to create a structure that behaves very much like the [[complex number]]s equipped with the operations of addition, multiplication, and exponentiation.  Moreover, in this abstract structure Schanuel's conjecture does indeed hold.<ref>{{cite journal |first=B. |last=Zilber |title=Pseudo-exponentiation on algebraically closed fields of characteristic zero |journal=Annals of Pure and Applied Logic |volume=132 |year=2005 |issue=1 |pages=67–95 |mr=2102856 |doi=10.1016/j.apal.2004.07.001 |doi-access=free }}</ref>  Unfortunately it is not yet known that this structure is in fact the same as the complex numbers with the operations mentioned; there could exist some other abstract structure that behaves very similarly to the complex numbers but where Schanuel's conjecture doesn't hold.  Zilber did provide several criteria that would prove the structure in question was '''C''', but could not prove the so-called Strong Exponential Closure axiom.  The simplest case of this axiom has since been proved,<ref>{{cite journal |first=D. |last=Marker |title=A remark on Zilber's pseudoexponentiation |journal=Journal of Symbolic Logic |volume=71 |issue=3 |year=2006 |pages=791–798 |mr=2250821 |jstor=27588482 |doi=10.2178/jsl/1154698577|s2cid=1477361 }}</ref> but a proof that it holds in full generality is required to complete the proof of the conjecture.

==Approaches==

A typical problem in this area of mathematics is to work out whether a given number is transcendental. [[Georg Cantor|Cantor]] used a [[cardinality]] argument to show that there are only [[countable set|countably]] many algebraic numbers, and hence [[almost all]] numbers are transcendental.  Transcendental numbers therefore represent the typical case; even so, it may be extremely difficult to prove that a given number is transcendental (or even simply irrational).

For this reason transcendence theory often works towards a more quantitative approach.  So given a particular complex number α one can ask how close α is to being an algebraic number.  For example, if one supposes that the number α is algebraic then can one show that it must have very high degree or a minimum polynomial with very large coefficients?  Ultimately if it is possible to show that no finite degree or size of coefficient is sufficient then the number must be transcendental.  Since a number α is transcendental if and only if ''P''(α)&nbsp;≠&nbsp;0 for every non-zero polynomial ''P'' with integer coefficients, this problem can be approached by trying to find lower bounds of the form
:<math> |P(a)| > F(A,d) </math>

where the right hand side is some positive function depending on some measure ''A'' of the size of the [[coefficient]]s of ''P'', and its [[Degree of a polynomial|degree]] ''d'', and such that these lower bounds apply to all ''P'' ≠ 0. Such a bound is called a '''transcendence measure'''.

The case of ''d''&nbsp;=&thinsp;1 is that of "classical" [[diophantine approximation]] asking for lower bounds for
:<math>|ax + b|</math>.

The methods of transcendence theory and diophantine approximation have much in common: they both use the [[auxiliary function]] concept.

==Major results==

The [[Gelfond–Schneider theorem]] was the major advance in transcendence theory in the period 1900–1950. In the 1960s the method of [[Alan Baker (mathematician)|Alan Baker]] on [[linear forms in logarithms]] of [[algebraic number]]s reanimated transcendence theory, with applications to numerous classical problems and [[diophantine equation]]s.

==Mahler's classification==
[[Kurt Mahler]] in 1932 partitioned the transcendental numbers into 3 classes, called '''S''', '''T''', and '''U'''.<ref name=Bug250>{{harvnb|Bugeaud|2012|p=250}}.</ref>  Definition of these classes draws on an extension of the idea of a [[Liouville number]] (cited above).

===Measure of irrationality of a real number===
{{Main|Irrationality measure}}
One way to define a Liouville number is to consider how small a given [[real number]] '''x''' makes linear polynomials |''qx''&nbsp;−&nbsp;''p''| without making them exactly&nbsp;0.  Here ''p'', ''q'' are integers with |''p''|, |''q''| bounded by a positive integer&nbsp;''H''.

Let <math>m(x, 1, H)</math> be the minimum non-zero absolute value these polynomials take and take:

:<math>\omega(x, 1, H) = -\frac{\log m(x, 1, H)}{\log H}</math>
:<math>\omega(x, 1) = \limsup_{H\to\infty}\, \omega(x,1,H).</math>

ω(''x'',&nbsp;1) is often called the '''measure of irrationality''' of a real number&nbsp;''x''. For rational numbers, ω(''x'',&nbsp;1) = 0 and is at least 1 for irrational real numbers. A Liouville number is defined to have infinite measure of irrationality. [[Roth's theorem]] says that irrational real algebraic numbers have measure of irrationality 1.

===Measure of transcendence of a complex number===
Next consider the values of polynomials at a complex number ''x'', when these polynomials have integer coefficients, degree at most ''n'', and [[Height of a polynomial|height]] at most ''H'', with ''n'', ''H'' being positive integers.

Let <math>m(x, n, H)</math> be the minimum non-zero absolute value such polynomials take at <math>x</math> and take:
:<math>\omega(x, n, H) = -\frac{\log m(x, n, H)}{n\log H}</math>
:<math>\omega(x, n) = \limsup_{H\to\infty}\, \omega(x,n,H).</math>

Suppose this is infinite for some minimum positive integer&nbsp;''n''.  A complex number ''x'' in this case is called a '''U&nbsp;number''' of degree&nbsp;''n''.

Now we can define 
:<math>\omega (x) = \limsup_{n\to\infty}\, \omega(x,n).</math> 
ω(''x'') is often called the '''measure of transcendence''' of&nbsp;''x''. If the ω(''x'', ''n'') are bounded, then ω(''x'') is finite, and ''x'' is called an '''S number'''.  If the ω(''x'', ''n'') are finite but unbounded, ''x'' is called a '''T number'''. ''x''&nbsp;is algebraic if and only if&nbsp;ω(''x'')&nbsp;=&nbsp;0.

Clearly the Liouville numbers are a subset of the U numbers.  [[William LeVeque]] in 1953 constructed U numbers of any desired degree.<ref name=LV172>{{harvnb|LeVeque|2002|p=II:172}}.</ref> The [[Liouville numbers]] and hence the U numbers are uncountable sets. They are sets of measure 0.<ref>{{harvnb|Burger|Tubbs|2004|p=170}}.</ref>

T numbers also comprise a set of measure 0.<ref>{{harvnb|Burger|Tubbs|2004|p=172}}.</ref> It took about 35 years to show their existence. [[Wolfgang M. Schmidt]] in 1968 showed that examples exist.  However, [[almost all]] complex numbers are S numbers.<ref name=Bug251/> Mahler proved that the exponential function sends all non-zero algebraic numbers to S numbers:<ref>{{harvnb|LeVeque|2002|pp=II:174–186}}.</ref><ref>{{harvnb|Burger|Tubbs|2004|p=182}}.</ref> this shows that ''e'' is an S number and gives a proof of the transcendence of {{pi}}.  This number {{pi}} is known not to be a U number.{{sfn|Baker|1975|p=86}}  Many other transcendental numbers remain unclassified.

Two numbers ''x'', ''y'' are called '''algebraically dependent''' if there is a non-zero polynomial ''P'' in two indeterminates with integer coefficients such that ''P''(''x'',&nbsp;''y'')&nbsp;=&nbsp;0. There is a powerful theorem that two complex numbers that are algebraically dependent belong to the same Mahler class.<ref name=LV172/><ref>{{harvnb|Burger|Tubbs|2004|p=163}}.</ref>  This allows construction of new transcendental numbers, such as the sum of a Liouville number with ''e'' or&nbsp;{{pi}}.

The symbol S probably stood for the name of Mahler's teacher [[Carl Ludwig Siegel]], and T and U are just the next two letters.

===Koksma's equivalent classification===
[[Jurjen Koksma]] in 1939 proposed another classification based on approximation by algebraic numbers.<ref name=Bug250/><ref name="Baker, p. 87">{{harvnb|Baker|1975|p=87}}.</ref>

Consider the approximation of a complex number ''x'' by algebraic numbers of degree ≤&thinsp;''n'' and height ≤&thinsp;''H''. Let α be an algebraic number of this finite set such that |''x''&nbsp;−&nbsp;α| has the minimum positive value. Define ω*(''x'', ''H'', ''n'') and ω*(''x'', ''n'') by:

:<math>|x-\alpha| = H^{-n\omega^*(x,H,n)-1}.</math>
:<math>\omega^*(x,n) = \limsup_{H\to\infty}\, \omega^*(x,n,H).</math>

If for a smallest positive integer ''n'', ω*(''x'', ''n'') is infinite, ''x'' is called a '''U*-number''' of degree&nbsp;''n''.

If the ω*(''x'', ''n'') are bounded and do not converge to 0, ''x'' is called an '''S*-number''',

A number ''x'' is called an '''A*-number''' if the ω*(''x'', ''n'') converge to&nbsp;0.

If the ω*(''x'', ''n'') are all finite but unbounded, ''x'' is called a '''T*-number''',

Koksma's and Mahler's classifications are equivalent in that they divide the transcendental numbers into the same classes.<ref name="Baker, p. 87"/> The ''A*''-numbers are the algebraic numbers.<ref name=Bug251>{{harvnb|Bugeaud|2012|p=251}}.</ref>

===LeVeque's construction===

Let

:<math>\lambda= \tfrac{1}{3} + \sum_{k=1}^\infty 10^{-k!}.</math>

It can be shown that the ''n''th root of λ (a Liouville number) is a U-number of degree ''n''.<ref>{{harvnb|Baker|1975|p=90}}.</ref>

This construction can be improved to create an uncountable family of U-numbers of degree ''n''. Let ''Z'' be the set consisting of every other power of 10 in the series above for λ. The set of all subsets of ''Z'' is uncountable. Deleting any of the subsets of ''Z'' from the series for λ creates uncountably many distinct Liouville numbers, whose ''n''th roots are U-numbers of degree ''n''.

===Type===
The [[supremum]] of the sequence {ω(''x'',&thinsp;''n'')} is called the '''type'''. Almost all real numbers are S numbers of type 1, which is minimal for real S numbers. Almost all complex numbers are S numbers of type 1/2, which is also minimal. The claims of almost all numbers were conjectured by Mahler and in 1965 proved by Vladimir Sprindzhuk.<ref name="Baker, p. 86">{{harvnb|Baker|1975|p=86}}.</ref>

==Open problems==

While the Gelfond–Schneider theorem proved that a large class of numbers was transcendental, this class was still countable.  Many well-known mathematical constants are still not known to be transcendental, and in some cases it is not even known whether they are rational or irrational.  A partial list can be found [[Transcendental number#Conjectured transcendental numbers|here]].

A major problem in transcendence theory is showing that a particular set of numbers is algebraically independent rather than just showing that individual elements are transcendental.  So while we know that ''e'' and ''π'' are transcendental that doesn't imply that ''e''&nbsp;+&nbsp;''π'' is transcendental, nor other combinations of the two (except ''e''<sup>π</sup>, [[Gelfond's constant]], which is known to be transcendental).  Another major problem is dealing with numbers that are not related to the exponential function.  The main results in transcendence theory tend to revolve around ''e'' and the logarithm function, which means that wholly new methods tend to be required to deal with numbers that cannot be expressed in terms of these two objects in an elementary fashion.

[[Schanuel's conjecture]] would solve the first of these problems somewhat as it deals with algebraic independence and would indeed confirm that ''e'' + ''π'' is transcendental.  It still revolves around the exponential function, however, and so would not necessarily deal with numbers such as [[Apéry's constant]] or the [[Euler–Mascheroni constant]].  Another extremely difficult unsolved problem is the so-called [[Constant problem|constant or identity problem]].<ref>{{cite journal |first=D. |last=Richardson |title=Some Undecidable Problems Involving Elementary Functions of a Real Variable |journal=Journal of Symbolic Logic |volume=33 |year=1968 |issue=4 |pages=514–520 |mr=0239976 |jstor=2271358 |doi=10.2307/2271358|s2cid=37066167 }}</ref>

==Notes==
{{reflist|30em}}

==References==
*{{cite book | first=Alan | last=Baker | author-link=Alan Baker (mathematician) | title=Transcendental Number Theory | publisher=[[Cambridge University Press]] | year=1975
 | others=paperback edition 1990
 | isbn=0-521-20461-5 | zbl=0297.10013 }}
*{{cite book
 | last=Bugeaud
 | first=Yann
 | title=Distribution modulo one and Diophantine approximation
 | series=Cambridge Tracts in Mathematics
 | volume=193
 | publisher=[[Cambridge University Press]]
 | year=2012
 | isbn=978-0-521-11169-0
 | zbl=1260.11001
}}
*{{cite book
 | last1=Burger
 | first1=Edward B.
 | last2=Tubbs
 | first2=Robert
 | title=Making transcendence transparent. An intuitive approach to classical transcendental number theory
 | publisher=[[Springer-Verlag|Springer]]
 | year=2004
 | isbn=978-0-387-21444-3
 | zbl=1092.11031
}}
*{{cite book |author-link=Alexander Gelfond |first=A. O. |last=Gelfond |title=Transcendental and Algebraic Numbers |publisher=Dover |year=1960 |zbl=0090.26103 }}
*{{cite book |author-link=Serge Lang |first=Serge |last=Lang |title=Introduction to Transcendental Numbers |publisher=Addison–Wesley |year=1966 | zbl=0144.04101 }}
*{{cite book
 | last=LeVeque
 | first=William J.
 | author-link=William J. LeVeque
 | title=Topics in Number Theory, Volumes I and II
 | publisher=Dover
 | year=2002
 | orig-year=1956
 | isbn=978-0-486-42539-9
 | url-access=registration
 | url=https://archive.org/details/topicsinnumberth0000leve
}}
*{{cite book
 | last1=Natarajan
 | first1=Saradha
 | author1-link=:fr:Saradha Natarajan
 | last2=Thangadurai
 | first2=Ravindranathan
 | publisher=[[Springer Verlag]]
 | title=Pillars of Transcendental Number Theory
 | year=2020
 | isbn=978-981-15-4154-4
 | zbl=
}}
*{{cite book
 | last=Sprindzhuk
 | first=Vladimir G.
 | author1-link=Vladimir Gennadievich Sprindzuk
 | title=Mahler's Problem in Metric Number Theory (1967)
 | series=AMS Translations of Mathematical Monographs
 | publisher=[[American Mathematical Society]]
 | year=1969
 | others=Translated from Russian by B. Volkmann
 | isbn=978-1-4704-4442-6
 | zbl=
}}
*{{cite book
 | last=Sprindzhuk
 | first=Vladimir G.
 | author1-link=Vladimir Gennadievich Sprindzuk
 | title=Metric theory of Diophantine approximations
 | series=Scripta Series in Mathematics
 | publisher=[[Wiley (publisher)|Wiley]]
 | year=1979
 | others=Translated from Russian by Richard A. Silverman. Foreword by Donald J. Newman.
 | isbn=0-470-26706-2
 | zbl=0482.10047
}}

==Further reading==
* [[Alan Baker (mathematician)|Alan Baker]] and [[Gisbert Wüstholz]], ''Logarithmic Forms and Diophantine Geometry'', New Mathematical Monographs '''9''', Cambridge University Press, 2007, {{isbn|978-0-521-88268-2}}

{{Number theory-footer}}

{{DEFAULTSORT:Transcendence Theory}}
[[Category:Analytic number theory]]
[[Category:Transcendental numbers| ]]