Dízimas periódicas: números cíclicos e teorema de Midy

Abstract

The present work aims to present the cyclic numbers, Midy's theorem and the relationships between them and the recurring decimals. Initially, we show the possible decimal forms of a rational number a/b. Next, we present the definition of cyclic numbers along with examples and, later, we show curiosities about the cyclic number 142857. Next, we relate the repeating decimals to these numbers showing that if a fraction 1/b generates a repeating decimal whose length of period is b -1, so b is prime and the number representing the period is cyclic. Subsequently, we demonstrate Midy's Theorem and present its applications in the determination of decimal periods. Finally, we present suggestions for didactic activities applicable in Basic Education and that use the previously mentioned concepts.