Dual Structure of the Discrete Number Line: A Coexistence of Binary and Decimal Systems

The discrete number line ℕ₀ = {0, 1, 2, 3,... } is conventionally defined as a homogeneous sequence. This paper demonstrates that by applying a parity filter, two disjoint and complementary subsets emerge, exhibiting distinct algebraic structures that underpin the binary and decimal numeral systems, respectively.

The set of even numbers P= {2k ∣ k ∈ ℕ₀} is bijective to ℕ₀ under the mapping p ↦ p/2 establishing the natural representation of bit positions in the binary system. Conversely, the set of odd numbers I= {2k + 1 ∣ k ∈ ℕ₀} maps analogously to ℕ₀ under a unitary shift, generating the odd numbers, whose last decimal digit belongs to {1, 3, 5, 7, 9}, thereby linking to the decimal system and more generally to any odd-base system.

We show that the strict alternation P( k ) + 1 = I( k ) and I( k ) + 1 = P( k + 1 ) enables the isomorphic translation of operations between both systems. Furthermore, we explore the computational potential of this duality, proposing Binary−Decimal combinatorial representations, Parity−based parallel algorithms, and high−efficiency hybrid counters.

The primary result reveals an unexploited structural property: the number line is not a mere homogeneous continuum, but rather the synergistic interleaving of two numeral systems capable of operating both independently and cooperatively.