3 research outputs found
Introduction to Liouville Numbers
The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an object which can be approximated âquite closelyâ by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and
0 <
x â
p
q
<
1
q
n
.
It is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number. It is defined in Section 6 quite generally as the sum
Xâ
k=1
ak
b
k!
for a finite sequence {ak}kâN and b â N. Based on this definition, we also introduced
the so-called Liouville number as
L =
Xâ
k=1
10âk! = 0.110001000000000000000001 . . . ,
substituting in the definition of L(ak, b) the constant sequence of 1âs and b = 10. Another important examples of transcendental numbers are e and Ï [7], [13], [6]. At the end, we show that the construction of an arbitrary Lioville constant satisfies the properties of a Liouville number [12], [1]. We show additionally, that the set of all Liouville numbers is infinite, opening the next item from Abad and Abadâs list of âTop 100 Theoremsâ. We show also some preliminary constructions linking real sequences and finite sequences, where summing formulas are involved. In the Mizar [14] proof, we follow closely https: //en.wikipedia.org/wiki/Liouville_number. The aim is to show that all Liouville
numbers are transcendental.Grabowski Adam - Institute of Informatics, University of BiaĆystok, BiaĆystok, PolandKorniĆowicz Artur - Institute of Informatics, University of BiaĆystok, BiaĆystok, PolandTom M. Apostol. Modular Functions and Dirichlet Series in Number Theory. Springer- Verlag, 2nd edition, 1997.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek and Piotr Rudnicki. Two programs for SCM. Part I - preliminaries. Formalized Mathematics, 4(1):69-72, 1993.Sophie Bernard, Yves Bertot, Laurence Rideau, and Pierre-Yves Strub. Formal proofs of transcendence for e and Ï as an application of multivariate and symmetric polynomials. In Jeremy Avigad and Adam Chlipala, editors, Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs, pages 76-87. ACM, 2016.Jesse Bingham. Formalizing a proof that e is transcendental. Journal of Formalized Reasoning, 4:71-84, 2011.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.CzesĆaw ByliĆski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.CzesĆaw ByliĆski. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.J.H. Conway and R.K. Guy. The Book of Numbers. Springer-Verlag, 1996.Manuel Eberl. Liouville numbers. Archive of Formal Proofs, December 2015. http://isa-afp.org/entries/Liouville_Numbers.shtml, Formal proof development.Adam Grabowski, Artur KorniĆowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015.JarosĆaw Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.RafaĆ Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Joseph Liouville. Nouvelle dĂ©monstration dâun thĂ©orĂšme sur les irrationnelles algĂ©briques, insĂ©rĂ© dans le Compte Rendu de la derniĂšre sĂ©ance. Compte Rendu Acad. Sci. Paris, SĂ©r.A (18):910â911, 1844.Jan PopioĆek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263-264, 1990.Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125-130, 1991.Konrad Raczkowski and Andrzej Nedzusiak. Real exponents and logarithms. Formalized Mathematics, 2(2):213-216, 1991.MichaĆ J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Wojciech A. Trybulec. Binary operations on finite sequences. Formalized Mathematics, 1 (5):979-981, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990
All Liouville Numbers are Transcendental
In this Mizar article, we complete the formalization of one of the items from Abad and Abadâs challenge list of âTop 100 Theoremsâ about Liouville numbers and the existence of transcendental numbers. It is item #18 from the âFormalizing 100 Theoremsâ list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an example of an object which can be approximated âquite closelyâ by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and
0 <
x â
p
q
<
1
q
n
.
It is easy to show that all Liouville numbers are irrational. The definition and basic notions are contained in [10], [1], and [12]. Liouvile constant, which is defined formally in [12], is the first explicit transcendental (not algebraic) number, another notable examples are e and Ï [5], [11], and [4]. Algebraic numbers were formalized with the help of the Mizar system [13] very recently, by Yasushige Watase in [23] and now we expand these techniques into the area of not only pure algebraic domains (as fields, rings and formal polynomials), but also for more settheoretic fields. Finally we show that all Liouville numbers are transcendental, based on Liouvilleâs theorem on Diophantine approximation.KorniĆowicz Artur - Institute of Informatics, University of BiaĆystok, BiaĆystok, PolandNaumowicz Adam - Institute of Informatics, University of BiaĆystok, BiaĆystok, PolandGrabowski Adam - Institute of Informatics, University of BiaĆystok, BiaĆystok, PolandTom M. Apostol. Modular Functions and Dirichlet Series in Number Theory. Springer- Verlag, 2nd edition, 1997.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Sophie Bernard, Yves Bertot, Laurence Rideau, and Pierre-Yves Strub. Formal proofs of transcendence for e and _ as an application of multivariate and symmetric polynomials. In Jeremy Avigad and Adam Chlipala, editors, Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs, pages 76-87. ACM, 2016.Jesse Bingham. Formalizing a proof that e is transcendental. Journal of Formalized Reasoning, 4:71-84, 2011.CzesĆaw ByliĆski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.CzesĆaw ByliĆski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.J.H. Conway and R.K. Guy. The Book of Numbers. Springer-Verlag, 1996.Manuel Eberl. Liouville numbers. Archive of Formal Proofs, December 2015. http://isa-afp.org/entries/Liouville_Numbers.shtml, Formal proof development.Adam Grabowski and Artur KorniĆowicz. Introduction to Liouville numbers. Formalized Mathematics, 25(1):39-48, 2017.Adam Grabowski, Artur KorniĆowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015.RafaĆ Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.Joseph Liouville. Nouvelle dĂ©monstration dâun thĂ©orĂšme sur les irrationnelles algĂ©briques, insĂ©rĂ© dans le Compte Rendu de la derniĂšre sĂ©ance. Compte Rendu Acad. Sci. Paris, SĂ©r.A (18):910â911, 1844.Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2): 265-269, 2001.Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339-346, 2001.Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391-395, 2001.Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461-470, 2001.MichaĆ Muzalewski and LesĆaw W. Szczerba. Construction of finite sequences over ring and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):97-104, 1991.Andrzej Trybulec. Function domains and FrĂŠnkel operator. Formalized Mathematics, 1 (3):495-500, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Yasushige Watase. Algebraic numbers. Formalized Mathematics, 24(4):291-299, 2016.Katarzyna Zawadzka. The sum and product of finite sequences of elements of a field. Formalized Mathematics, 3(2):205-211, 1992