A ``Carry-Over'' Approach to Explaining Cyclic Numbers and Repeating Decimals in Base-b Systems.pdf

REVISED (new version explictly states the method more clearly, see below):

A novel expository approach for explaining and generating cyclic numbers, leveraging existing knowledge in number theory.

This work is the first of many to come and presents a novel method for explaining and generating cyclic numbers, building on established principles in number theory. In the literature on cyclic numbers, many treatments focus on the algebraic or group-theoretic properties without emphasizing the "carry–over" viewpoint based on shifting digits

By linking the mechanical process of digit shifting with the cyclic behaviour observed in repeating decimals, the approach not only clarifies the underlying modular arithmetic - in particular, how the minimal exponent t (satisfying bᵗ ≡ 1 mod p) relates directly to the cycle length and fraction 1/p - but also provides practical intuitive tools for generating such numbers.

b^t/p = K + 1/p
(where K is our cyclic when (p - 1) = t )

For our minimal exponent satisfying: bᵖ⁻¹ ≡ 1 (mod p), K is thus a cyclic number of (p - 1) digits in respective base b.

Proven by the carry-over method bᵗ/p = K + 1/p, whereby (p - 1) steps completes the fraction giving bᵗ. K is thus allowed to take on the whole form of our recurring 1/p, becoming a cyclic number. This discovery is the simplest tool for proving cyclic numbers to have ever existed and it leverages already existing knowledge in number theory.

This paper emphasizes the carry–over process as a pedagogical tool. By enhancing understanding and application, this approach aims to contribute to both educational practices and further research in the field.

(As this method relies on elementary modular arithmetic and the standard division algorithm rather than deeper concepts like group orders, cyclic subgroups, or field theory. This can also make it more accessible to a broader audience, including high-school or early undergraduate learners.)

For further collaboration and questions, contact me at cs.kava@proton.me