In ℚ₅ (the 5-adic numbers), which pair of integers is closest under the 5-adic metric |·|₅?
A1 and 2, because they differ by 1 and are adjacent integers
B1 and 6, because they differ by 5
C1 and 126, because they differ by 5³ = 125
D1 and 3126, because they differ by 5⁵ = 3125
|1 − 3126|₅ = |3125|₅ = |5⁵|₅ = 5^{−5} = 1/3125, which is the smallest of the four distances. In the 5-adic metric, a larger power of 5 dividing the difference means the numbers are *closer*, the opposite of ordinary intuition. |1−2|₅ = 1 (no factor of 5), |1−6|₅ = 1/5, |1−126|₅ = 1/125, |1−3126|₅ = 1/3125. Numbers we think of as 'far apart' can be extremely close p-adically if their difference is highly divisible by p.
Question 2 Multiple Choice
The p-adic numbers ℚ_p and the real numbers ℝ are both completions of the rationals ℚ. What is the key difference between these two completions?
Aℝ adds algebraic numbers to ℚ, while ℚ_p adds p-adic power series
BBoth complete ℚ by adding limits of Cauchy sequences, but under different metrics — the ordinary absolute value for ℝ, and the p-adic norm for ℚ_p
Cℝ uses Dedekind cuts while ℚ_p uses equivalence classes of Cauchy sequences — they are fundamentally different constructions
Dℝ is the unique completion of ℚ; ℚ_p is a different kind of object, not a completion in the metric space sense
Both ℝ and ℚ_p are constructed by exactly the same process — taking Cauchy sequences of rationals and identifying sequences with the same limit — but using different notions of distance. The ordinary absolute value makes rationals 'close' when their numerical difference is small; the p-adic norm makes rationals close when their difference is divisible by a high power of p. Ostrowski's theorem makes this precise: every non-trivial absolute value on ℚ is either the ordinary one (giving ℝ) or a p-adic one (giving ℚ_p) for some prime p.
Question 3 True / False
In the 3-adic metric, the number 3^100 is very large — much larger than 1.
TTrue
FFalse
Answer: False
|3^100|₃ = 3^{−100}, which is extremely small — not large. In the p-adic metric, high powers of p are *close to zero*, not far from zero. This is the complete reversal of ordinary intuition: the p-adic norm measures divisibility by p, so numbers highly divisible by p are tiny. 3^100 is as small as you can get in ℚ₃.
Question 4 True / False
The infinite sum −1 = (p−1) + (p−1)p + (p−1)p² + ··· converges in the p-adic metric because each successive term is smaller under |·|_p than the previous one.
TTrue
FFalse
Answer: True
The k-th term is (p−1)p^k, and |(p−1)p^k|_p = p^{−k} → 0 as k → ∞. In a complete ultrametric space like ℚ_p, a series converges if and only if its terms tend to 0 (a much simpler criterion than in ℝ). The sum really does equal −1: partial sums are 1 + p + p² + ··· + p^n = (p^{n+1} − 1)/(p − 1) · (p − 1) ... actually the sum (p−1)(1 + p + ··· + p^n) = p^{n+1} − 1, and p^{n+1} → 0 p-adically, so the sum converges to −1.
Question 5 Short Answer
What does it mean for two integers to be 'close' in the p-adic metric, and how does this differ from ordinary closeness on the number line?
Think about your answer, then reveal below.
Model answer: Two integers are close in the p-adic metric if their difference is divisible by a high power of p. The p-adic distance is |a − b|_p = p^{−v_p(a−b)}, so high divisibility by p means small distance. This is the opposite of ordinary closeness: on the real number line, 1 and 2 are very close (differ by 1), but in ℚ₅, 1 and 3126 are far closer (differ by 5⁵). Large powers of p are p-adically tiny, not large. The p-adic metric is measuring arithmetic structure (divisibility) rather than magnitude.
This reversal — large powers of p being small, high divisibility meaning closeness — is the core intuition that makes p-adic numbers feel alien at first but becomes natural once you internalize that distance is being defined by arithmetic structure rather than geometric separation.