On the Residue Classes of Real Numbers and Its Topological Properties
On the Residue Classes of Real Numbers and Its Topological Properties
Abstract[edit]
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 gives an answer to the question. [1]
Let α > 0 be fixed real number and a, b ∈ ℝ, then a ≡ b mode(mod α) if and only if a − b = k α for some k ∈ ℤ. This definition is parallel to the concept of the residue of classes of integer ℤn for fixed n∈ ℤ. As a result, this also constitutes a residue classes 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 center 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 center 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 generate the topology T in ℝα.
The topological structure of ℝα looks like an indefinite 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 in particular [x]α + [y]β = [x+y]α+β) with cℝα = ℝcα (or in particular c[x]α = [x]cα), where 0 < α, β < u, the operations satisfy the vector properties.
Geometrically ℝα x ℝβ looks like a toroidal coordinates (or doughnut shaped space) where we can graph functions f:ℝα → ℝβ, such as, f([x]α) = [x/(x– 1)]β where the curve windingly approaching the asymptote x = 1 in the torus.
Another interesting investigation is, we can expand ℝα to complex numbers in the form ℂz = ℂa + ℂbi or Re(ℂz) + Im(ℂz) for some z∈ ℂ.
Notes[edit]
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