On the Residue Classes of Real Numbers and Its Topological Properties
On the Residue Classes of Real Numbers and Their Topological Properties
Abstract
The concept of congruence is associated with integers. Thus, a natural question arises. Can this concept of congruence be associated with the real numbers? A graduate thesis from MSU-IIT provides an answer to the question. [1]
Let α > 0 be a fixed real number and a, b ∈ ℝ, then a ≡ b (mod α) if and only if a − b = kα for some k ∈ ℤ. This definition parallels the concept of residue classes of integers ℤn for a fixed n ∈ ℤ. As a result, this also constitutes a residue classes of real numbers denoted as ℝα. The element [r]α ∈ ℝα is the set {r + kα : k ∈ ℤ}. With respect to addition, ℝα is an abelian group.
On the other hand, this set ℝα of residue classes can be extended to topology. Consider the mapping γ : ℝ → ℝα which is defined by γ(x) = [r]α such that x = r + kα for some k ∈ ℤ and 0 ≤ r < α. Let ε > 0, the symmetric open ball in ℝ centered at x ∈ ℝ of radius ε is defined by B(ε, [r]α) = {γ(y) : |x – y| < ε}. Through this, the basis element in ℝα centered at [r]α ∈ ℝα determined by x ∈ [r]α, can be defined by the set Bx(ε, [r]α) = {γ(y) : y ∈ B(ε, [r]α)}. The set of these basis elements generates the topology T in ℝα.
The topological structure of ℝα resembles an infinite loop or, in particular, a circle.
The concept can be applied to vectors. Consider a collection ℝu of ℝu for all 0 < u, c ∈ ℝ. Define ℝα ⊕ ℝβ = ℝα+β (or specifically [x]α + [y]β = [x+y]α+β) with cℝα = ℝcα (or specifically c[x]α = [x]cα), where 0 < α, β < u. The operations satisfy the vector properties.
Geometrically, ℝα × ℝβ resembles a toroidal (or doughnut-shaped) coordinate system, where we can graph functions f:ℝα → ℝβ, such as, f([x]α) = [x/(x–1)]β, where the curve asymptotically approaches x = 1 in the torus.
Another interesting investigation is the expansion of ℝα to complex numbers in the form ℂz = ℂa + ℂbi or Re(ℂz) + Im(ℂz) for some z ∈ ℂ.
Notes
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