Beberapa Sifat Midy Pada Desimal Berulang Untuk Suatu Pembilang

  • Moh Affaf STKIP PGRI Bangkalan
  • Zaiful Ulum STKIP PGRI Bangkalan
Keywords: Decimal Representation, Midy’s Theorem, Repeating Decimal

Abstract

Let n  be a natural number relatively prime to 10 and m be a positive integer relatively prime to n. Let the period of m/n be ab with b greater than 1. Break the repeating block of ab digits up into b subblocks, each of length a, and let B(n_m, a,b) denote the sum of these b blocks. If n is a prime number relative prime to 10, and the period of 1/p is 2a, then (B(p_1, a,2) equal to 10^a-1. This result was known as Midy’s Theorem. In 2004, has been showed that for p is a prime greater than 5, and the period of 1/p is 3a, then (B(p_1, a,3) equal to 10^a-1. In 2005, also was showed that for p is a prime greater than 5, and the period of 1/p is ab, then (B(p_1, a,b) multiply of 10^a-1. In 2007, the study of repeating decimal of 1/n is expanded, that n don't have to prime greater than 5, but just relatively prime to 10. And then, in 2018, was investigated property of repeating decimal of m/n with m don't have to equal to 1, but enough less than n and relatively prime to n. In this paper, further investigation will be carried out on the properties of these conditions.

Published
2020-01-06