Abstract
The main aim of the article is to investigate the irrational and transcendental properties of certain real numbers by means of the factorial series and the factorial number system. The difference between the factorial series and the factorial system is that the factorial series does not set an upper bound at a given place after the radix point, while in the factorial system (i – 1) is the maximal possible value for ri after the radix point. I give an extended definition of periodic numbers, and show the relationship between periodic and irrational numbers. I prove the transcendence of e by means of the factorial series and the factorial number system.
Highlights
Factorials have been known for thousands of years [1] [2]
The difference between the factorial series and the factorial system is that the factorial series does not set an upper bound at a given place after the radix point, while in the factorial system (i – 1) is the maximal possible value for ri after the radix point
I prove the transcendence of e by means of the factorial series and the factorial number system
Summary
Factorials have been known for thousands of years [1] [2]. The factorial number system [4] [5], though known for centuries, is much less used in the study of irrational numbers than another tool, the theory of continued fractions [6]. Wikipedia deals with the factorial number system in 8 languages, while continued fractions are described in 42 languages. The reason for this discrepancy may be the great achievements of using continuous fractions by Bombelli, Wallis, Huygens, and especially by Lambert’s proof of the irrationality of π [7]
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