Combinatorial meaning of Euler`s number
Role of Euler`s number in differential and integral calculus, in algebra. Consideration of a new limit for the number e and giving its rigorous proof using the apparatus of mathematical analysis. Combinatorial interpretation is given for Euler’s number.
| Рубрика | Математика |
| Вид | статья |
| Язык | английский |
| Дата добавления | 28.08.2020 |
| Размер файла | 788,9 K |
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East-Siberian branch of FSUE «VNIIFTRI»
COMBINATORIAL MEANING OF EULER'S NUMBER
Perfileev M.S.
Irkutsk
Аннотация
Восточно-Сибирский филиал ФГУП «ВНИИФТРИ»
КОМБИНАТОРНОЕ ЗНАЧЕНИЕ ЧИСЛА ЭЙЛЕРА
Перфилеев М.С. Иркутск
В настоящей статье рассматривается новый предел числа e и приводится его научное доказательство с помощью аппарата математического анализа. Используя этот предел, мы проводим комбинаторную интерпретацию числа Эйлера. Это означает, что число Эйлера является отношением количества перестановок (или комбинаций) на n к числу перестановок (или комбинаций) на n с бесконечно большим числом элементов n.
Ключевые слова: число Эйлера, предел функции, формула Стирлинга, число перестановок, количество комбинаций.
Abstract
In this paper, we consider a new limit for the number e and give its rigorous proof using the apparatus of mathematical analysis. With the help of this limit, a combinatorial interpretation is given for Euler's number. It means that Euler's number is the ratio of the number of permutations (or combinations) of by n to the number of permutations (or combinations) of by n with an infinitely large number of elements n.
Keywords: Euler's number, limit of a function, Stirling's formula, number of permutations, number of combinations.
Introduction
differential euler number algebra
The number e is one of the most important mathematical constants [1], and it plays a huge role in differential and integral calculus, in algebra and the laws of physics and chemistry. Euler's number has various representations, for example, sums of infinite series, limits, infinite multiplications, and infinite continued fractions [2], [3].
Let's consider the limit
(1)
where p, q, and r are real numbers, and p is more than zero:
Proof:
Let's replace n with
We denote that . Then we get the limit
(2)
Provided (using Stirling's formula provided [5]) we get
(Note: during the transformations, we used the negligible smallness of compared to n provided and the second remarkable limit [6]).
Similarly, we can get
Then
Q.E.D.
(Note: during the transformations, we used the negligible smallness of compared to provided and the second remarkable limit).
Combinatorial meaning of Euler's number
Let's consider some properties of the obtained limit (1).
If we take p = 1, and (Theodorus' constant), and (Pythagoras' constant) [1], then we get the limit
(3)
Provided p = 1, q = 1, r = 0, the limit (1) takes the form of
(4)
Replacing n with we get the expression
(5)
Using highly accurate computer calculations for the expression (5), we can get the following results, presented in this format:
{numeric value n ; deviation of calculated results from the exact value of the number e}:
It is easy to see that as n increases by one order of magnitude, the accuracy of the approximation increases by two orders of magnitude.
Using the formula for the number of permutations of k elements of m elements,
[7] can be trivially obtained
(6)
then Euler's number can be represented as
(7)
Thus, Euler's number also has a combinatorial meaning, which is the ratio of the number of permutations of n elements from elements to the number of permutations of n elements from elements with an infinitely large number of elements n.
Also, Euler's number can be interpreted as the ratio of the number of combinations of n elements from elements to the number of combinations of n elements from elements with an infinitely large number of elements n.
Indeed, using the formula for the number of combinations
we can obtain
(8)
Then we get the result
(9)
If in the limit (5) we replace n with , we get
(10)
(Note: during the transformations, we used the negligible smallness of compared to provided ).
Conclusion
Thus, this paper presents and proves a new limit for Euler's number. The properties of this limit have also been considered here. Using the formulas for the number of permutations and combinations of m through k, it is shown that the number e has its combinatorial meaning.
References
1. Булыко А.Н. Современный словарь иностранных слов / Булыко А.Н. М.: Мартин, 2005. С. 215
2. Steven R. Finch. Mathematical Constants / Steven R. Cambridge, 2003. P. 12-16.
3. Representations of e [Electronic resourse]. URL: https://www.wolframalpha.com/input/?i=Representations+of+e (accessed: 18.07.2019)
4. Wolfram MathWorld [Electronic resourse]. URL: http://mathworld.wolfram.com/e.html (accessed: 18.07.2019)
5. Colloquium-journal, Warszawa, Poland, №4 (28), Michael Perfileev “The quotient of the number pi and the number e”. P. 60-62.
6. David H. Bailey Pi: The Next Generation: A Sourcebook on the Recent History of Pi and Its Computation / David H. Bailey, Jonathan M. Borwein.Springer, 2016. 389 p.
7. Taktarov N.G. Handbook of higher mathematics for students / Taktarov N.G. Moscow, Librocom, 2008. 149 p.
8. Novikov F.A. Discrete mathematics for programmers / Novikov F.A. St. Petersburg, Piter, 2009. P. 181-182.
9. Erosh I.L. Discrete Math. Combinatorics: Tutorial / Erosh I.L. St. Petersburg, SPbGUAP, 2001.11 p.
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