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Introduction to Liouville Numbers

100%
Grabowski A., Korniłowicz A.
Formalized Mathematics
|
2017
|
tom 25
|
nr 1
39-48
EN
The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an object which can be approximated “quite closely” 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 [...] 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 [...] for a finite sequence {ak}k∈ℕ and b ∈ ℕ. Based on this definition, we also introduced the so-called Liouville number as [...] substituting in the definition of L(ak, b) the constant sequence of 1’s and b = 10. Another important examples of transcendental numbers are e and π [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 Abad’s list of “Top 100 Theorems”. 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.
2
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All Liouville Numbers are Transcendental

100%
Korniłowicz A., Naumowicz A., Grabowski A.
Formalized Mathematics
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2017
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tom 25
|
nr 1
49-54
EN
In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” 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 “quite closely” 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 [...] 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 π [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 Liouville’s theorem on Diophantine approximation.
3
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Rational points on the unit sphere

88%
Schmutz E.
Open Mathematics
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2008
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tom 6
|
nr 3
482-487
EN
It is known that the unit sphere, centered at the origin in ℝn, has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordinates: for every point ν on the unit sphere in ℝn, and every ν > 0; there is a point r = (r 1; r 2;…;r n) such that: ⊎ ‖r-v‖∞ < ε.⊎ r is also a point on the unit sphere; Σ r i 2 = 1.⊎ r has rational coordinates; $$ r_i = \frac{{a_i }} {{b_i }} $$ for some integers a i, b i.⊎ for all $$ i,0 \leqslant \left| {a_i } \right| \leqslant b_i \leqslant (\frac{{32^{1/2} \left\lceil {log_2 n} \right\rceil }} {\varepsilon })^{2\left\lceil {log_2 n} \right\rceil } $$ . One consequence of this result is a relatively simple and quantitative proof of the fact that the rational orthogonal group O(n;ℚ) is dense in O(n;ℝ) with the topology induced by Frobenius’ matrix norm. Unitary matrices in U(n;ℂ) can likewise be approximated by matrices in U(n;ℚ(i))
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