The missing place of the number 'e' in secondary school books

Document Type : Research Paper

Authors

University of Zanjan

Abstract

The limited information about the number $\mathrm{e}$ found in secondary school books prompted us to write this article in defense of the possibility of providing more comprehensive explanations about this number at the secondary education level. We begin with presenting a historical background of this number and then proceed to prove the inequality, $0<\mathrm{e}-\sum_{k=0}^{n}{1}/{k!}<{1}/{(n.n!)}$ which holds for every $n\geq 1$. By utilizing this inequality, we infer the enigmatic nature of $\mathrm{e}$ and calculate the sum of $\sum_{k=0}^{n} P(n,k)$. This sum leads us to the counting of distinct paths between two arbitrary vertices of a complete graph. Finally, we demonstrate the proof of the inequality involving arithmetic, geometric, and harmonic means based on the inequality Image. All the proofs and deductions are based on the materials covered in the final year of secondary education.

Keywords

References

[1] A. Azarang, The Persian Tarof in Mathematics, Math. Intelligencer, 33 no. 4 (2011) pp. 1.
[2] M. Hassani, Cycles in graphs and derangements, Math. Gaz., 88 (2004) 123–126.
[3] M. Hassani, Derangements and Applications, J. Integer Seq. (JIS), 6 (2003) no. 1 pp. 8, Article 03.1.2.
[4] M. Hassani, On the Arithmetic–Geometric mean inequality, Tamkang J. Math., 44 (2013) 453–456.
[5] M. Hassani and A. Sofo, Sharp bounds for the constant e, Vietnam Journal of Mathematics, to appear.
[6] O. A. S. Karamzadeh, One-line proof of the AM-GM inequality, Math. Intelligencer, 33 no.2 (2011) pp. 3.
[7] N. Schaumberger, The AM-GM Inequality via x۱/x , College Mathematics Journal, 20 (1989) 320.
[8] J. J. O’Connor and Edmund F. Robertson, The MacTutor History of Mathematics archive:
http://www-history.mcs.st-andrews.ac.uk/index.html
Mathematics and Society
Volume 1, Issue 3 - Serial Number 3
September 2016
Pages 13-23

Files

History

  • Receive Date: 29 April 2014
  • Revise Date: 22 December 2014
  • Accept Date: 23 December 2014
  • Publish Date: 21 November 2016

Share

How to cite

Statistics