Frações Decimais

Abstract

This work was based on the article by professor Alfredo Wagner Martins Pinto [1]. Let x be a real number, there are only ⌊ x ⌋ integers and 0 ≤ {x} <1 real such that x = ⌊ x ⌋ + {x}, where ⌊ x ⌋ is the integer part of x and {x} mantissa of x. From this information, it is possible to develop the decimal representation of this real number, which is the first objective of this work. The mantissa can be written as a series, such that if a number is rational the mantissa of that real number is periodic. Soon after, knowledge of Euler's ϕ function will be necessary. Through algebraic manipulations and analytical concepts, he continues that every rational number is periodic. Hence the need to determine the length of the period of this rational number, the second objective of the work. That is, we will determine how many digits the period of a non-integer rational number has. Through Group Theory, it was tried to determine a less tiring way to find the size of this period. Finally, a better estimate was found for the second objective of the work. . The mantissa can be written as a series, such that if a number is rational the mantissa of that real number is periodic. Soon after, knowledge of Euler's ϕ function will be necessary.