University of Isfahan Mathematics and Society 2345-6493 1 3 2016 11 21 The missing place of the number 'e' in secondary school books The missing place of the number 'e' in secondary school books 13 23 7547 10.22108/msci.2016.7547 FA Mehdi Hassani University of Zanjan Azizeh Ahmadi University of Zanjan Journal Article 2014 04 29 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. 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. https://math-sci.ui.ac.ir/article_7547_ff9daeffa9086b3eab13b6273d4697ab.pdf