In the 7-adic metric, which of the following numbers is closest to 0?
A3
B7
C49
D343
The 7-adic absolute value is |x|₇ = 7^(−v₇(x)), where v₇(x) counts how many times 7 divides x. For 3: v₇(3) = 0, so |3|₇ = 1. For 7: v₇(7) = 1, so |7|₇ = 1/7. For 49 = 7²: v₇(49) = 2, so |49|₇ = 1/49. For 343 = 7³: v₇(343) = 3, so |343|₇ = 1/343. Numbers highly divisible by 7 are *small* in the 7-adic metric — a complete inversion of ordinary intuition about size.
Question 2 Multiple Choice
A student claims that the sequence 5, 25, 125, 625, … (powers of 5) diverges to infinity in the 5-adic metric. Is this correct?
AYes — powers of 5 grow without bound, so they diverge in any metric
BNo — in the 5-adic metric, |5ⁿ|₅ = 5^(−n) → 0, so the sequence converges to 0
CNo — the sequence converges to −1 in ℚ₅
DYes — the 5-adic metric agrees with the ordinary absolute value for positive integers
In the 5-adic metric, |5ⁿ|₅ = 5^(−v₅(5ⁿ)) = 5^(−n), which tends to 0 as n → ∞. So the sequence converges to 0, not infinity. This is the core reversal: large powers of p become *small* in the p-adic world because they are highly divisible by p. This directly contradicts Archimedean intuition, where larger numbers are farther from 0.
Question 3 True / False
In the 3-adic numbers ℚ₃, the infinite series 2 + 2·3 + 2·3² + 2·3³ + ··· converges, and its sum equals −1.
TTrue
FFalse
Answer: True
The partial sums Sₙ = 2(1 + 3 + ··· + 3^(n−1)) = 3ⁿ − 1. In the 3-adic metric, |Sₙ − (−1)|₃ = |3ⁿ|₃ = 3^(−n) → 0. So the series converges to −1 in ℚ₃. The 3-adic representation of −1 is an infinite string of 2's in base 3 — extending infinitely to the left. This illustrates how p-adic numbers represent familiar quantities (even negative integers) through infinite expansions in powers of p, and how convergence in the p-adic metric is entirely governed by divisibility.
Question 4 True / False
The p-adic numbers ℚ_p are just the rational numbers ℚ with a different notation — they introduce no new mathematical objects.
TTrue
FFalse
Answer: False
ℚ_p is a proper extension of ℚ, constructed by completing ℚ under the p-adic metric. It contains elements that are not rational numbers — limits of p-adic Cauchy sequences that do not converge in ℚ. The construction is exactly analogous to how ℝ extends ℚ by adding limits of ordinary Cauchy sequences (like √2 and π). ℚ_p adds new elements like infinite p-adic expansions that represent quantities not in ℚ. By Ostrowski's theorem, ℝ and all the ℚ_p are the *only* completions of ℚ.
Question 5 Short Answer
How does the construction of ℚ_p parallel the construction of ℝ, and what does this analogy reveal about the significance of p-adic numbers?
Think about your answer, then reveal below.
Model answer: ℝ is the completion of ℚ under the ordinary absolute value — it adds all limits of rational Cauchy sequences that were missing from ℚ under the usual metric. ℚ_p is the completion of ℚ under the p-adic absolute value |x|_p = p^(−v_p(x)) — it adds all limits of rational Cauchy sequences that are Cauchy under the p-adic metric but don't converge in ℚ. The analogy reveals that ℝ and the family {ℚ_p : p prime} are all equally valid completions of the rationals, each capturing a different notion of nearness. Ostrowski's theorem shows these are the *only* completions, making them together a complete picture of the ways ℚ can be metrically extended.
This parallel is philosophically important: ℝ is not uniquely 'natural' as a number system containing ℚ. The p-adic completions are just as natural from a purely algebraic standpoint, and arise inevitably when studying congruences and divisibility in number theory. The p-adic absolute value is non-Archimedean (satisfying the ultrametric inequality), giving ℚ_p a radically different geometry from ℝ.