The rational numbers Q are incomplete: the sets {x ∈ Q | x² < 2} and {x ∈ Q | x² > 2} partition Q, but both are open.
A sequence is Cauchy if its entries eventually become arbitrarily close.
Two sequences are equivalent if they eventually become arbitrarily close to each other: s ∼ t if for every positive ε ∈ Q, there exists and N such that for all k ≥ N, |s_k - t_k| < ε.
equivalence relation: binary relation that is reflexive, symmetric, and transitive
equivalence class: for ≈ as equivalence relation on S, the equivalence class of a ∈ S is ⟦a⟧ = {x ∈ S | a ≈ x}.
quotient: based on the above equivalence class, is the set {⟦a⟧ | a ∈ S} (so set of all equivalence classes)
The set of real numbers is the set {s : N → Q | s is Cauchy}. It is the quotient of set of rational Cauchy sequences, with respect to equivalence. This is the completion of Q.
An alternate absolute value.
If q ≠ 0, the p-adic norm of rational q is p ^ (-(padic_val_rat p q)).
If q = 0, p-adic norm of q is 0.
The p-adic numbers are the Cauchy completion of Q with respect to the p-adic norm.