• ### Finding the Number at Any Digger After the Comma In Decimals

Remzi AKTAY

Çubuk Bilim ve Sanat Merkezi, Turkey.

Abstract: In this study, it is aimed to develop a method that enables to find the digit in any digit after the comma in the decimal form of the number 1/n for a natural number n whose prime factors are not 2 and 5, and based on this method, the fraction 1/n, a being a natural number greater than one, is followed by the decimal point. It is aimed to generalize the method to give the number in any of its digits. It started with a question “What is the digit in the 2021th digit after the comma in the decimal form of the number 1/2021?” As a method, modular arithmetic rules, Euler Function, Euler Theorem, Chinese Remainder Theorem were used. As a result of the studies, since finding the digit in k. digits from the beginning after the comma of the number 1/n actually means finding the digit in the first digit of the number 10k-1. 1/n after the comma, the form of the number 10k-1 as n.p +q, 0 < q < n has been found. When the number of n.p +q found is multiplied by the number 1/n, the number p + q/n  will be formed, and the first digit after the comma in the decimal form of the number q/n will be the result we are looking for. The coding of this study was done, and the program that gives the number in any digit after the comma was made. Thanks to this method, it was possible to generalize within the number a/n (a natural number greater than one).

Keywords: Decimals, Euler Function, Modular Arithmetic, Number Theory.