Modular Equivalents and Digital Roots of Irrational and Complex Numbers

Authors

  • Mohammad Asfaque St. Xavier’s College, Maitighar, Kathmandu 44600, Nepal

Keywords:

Digital roots, Integral Equivalents, Irrational Numbers, Complex Numbers, Fermat's Last Theorem

Abstract

Modular arithmetic, by definition, is arithmetic for integers. Since rational numbers are just ratios of two integers, the integral equivalents of rational numbers can be calculated using modular arithmetic for any modulus. But, the existing literature does not talk about integral equivalents of irrational numbers and complex numbers. This article proposes the idea of calculating them and finally proves a theorem related to an extension of Fermat’s last theorem which could hold true only if the integral equivalents of irrational numbers and complex numbers existed. Also, it talks about a pattern that exists when the number of variables used in the extension of Fermat’s last theorem is increased in a certain manner from 3 to infinity. Similarly, digital root (dr) is defined just for positive numbers and for non-terminating fractions. This article proposes the extension of its definition from positive integers to real numbers and even to complex numbers and thus calculates the digital root of irrational numbers and some complex numbers.

References

. A. Bonnie and C. Orin. Problem Solving Through Recreational Mathematics. Dover Books on Mathematics (reprinted ed.). Mineola. NY: Courier Dover Publications, 1999, pp. 125–127.

. F.M. Hall. An Introduction into Abstract Algebra, 1 (2nd ed.), Cambridge, U.K.: CUP Archive, 1980, p. 101

. J.L. Berggren. "modular arithmetic". Encyclopædia Britannica.

. M. Bullynck. "Modular Arithmetic before C.F. Gauss. Systematisations and discussions on remainder problems in 18th-century Germany"

. T. Watkins. “Digit Sums for Repeating Decimals”. applet-magic.com

. E. W. Weisstein. "Modular Inverse". MathWorld.

. K.H. Rosen. Discrete Mathematics and its Applications (6th ed.). New York, NY: McGraw-Hill. pp. 105, 158–160

. W. W. R. Ball and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, 1987, pp. 69-73.

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Published

2020-08-31

How to Cite

Asfaque, M. . (2020). Modular Equivalents and Digital Roots of Irrational and Complex Numbers. International Journal of Formal Sciences: Current and Future Research Trends, 5(01), 1–10. Retrieved from https://ijfscfrtjournal.isrra.org/index.php/Formal_Sciences_Journal/article/view/548

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Articles