Modular Equivalents and Digital Roots of Irrational and Complex Numbers
Keywords:
Digital roots, Integral Equivalents, Irrational Numbers, Complex Numbers, Fermat's Last TheoremAbstract
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.
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