# On Transcendental Numbers: New Results and a Little History

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## Abstract

**:**

## 1. Introduction

## 2. Transcendence and Transcendental Numbers in Mathematics

## 3. Transcendental Numbers: Identities, Inequalities and Open Problems

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

- (i)
- If L is the length of the given curve, then$$\frac{L}{D}\le \pi \le \frac{L}{d}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}.$$
- (ii)
- Moreover, the first inequality becomes equality if and only if the second inequality becomes equality if and only if the given curve is a circle.
- (iii)
- If the area of the domain inside the given curve is A, then $d\phantom{\rule{4pt}{0ex}}D\phantom{\rule{4pt}{0ex}}>A\phantom{\rule{4pt}{0ex}}$.
- (iv)
- The equation ${x}^{2}-\frac{L}{2}x+A=0$ is not completely solved; for example, if the given curve is an ellipse, this equation cannot be solved.

**Remark**

**5.**

## 4. Transcendental Numbers in Mathematical Physics

**Remark**

**6.**

**Theorem**

**1.**

**Proof.**

**Remark**

**7.**

**Theorem**

**2.**

**Proof.**

**Remark**

**8.**

**Remark**

**9.**

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Marcus, S.; Nichita, F.F.
On Transcendental Numbers: New Results and a Little History. *Axioms* **2018**, *7*, 15.
https://doi.org/10.3390/axioms7010015

**AMA Style**

Marcus S, Nichita FF.
On Transcendental Numbers: New Results and a Little History. *Axioms*. 2018; 7(1):15.
https://doi.org/10.3390/axioms7010015

**Chicago/Turabian Style**

Marcus, Solomon, and Florin F. Nichita.
2018. "On Transcendental Numbers: New Results and a Little History" *Axioms* 7, no. 1: 15.
https://doi.org/10.3390/axioms7010015