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We would like to explore the behavior of the expansion of ordinary fractions, the length of the non-periodic part and the period if it is an endless tithe, with the help of the function φ Euler. In addition to the decimal expansions that are the most common, we explore the expansions for a base b whatsoever. We will present some examples of expansions of ordinary fractions for different number bases generating finite decimals, but also simple and composite periodical decimals.

The Function φ of Euler and the Periodic Expansion Fractions in the Base b