Which statement best captures what it means for a convergent pₙ/qₙ to be a 'best rational approximation' to α?
Apₙ/qₙ is closer to α than any other fraction with denominator strictly less than or equal to qₙ
Bpₙ/qₙ is the fraction with the smallest absolute error among all rationals
Cpₙ/qₙ is closer to α than any fraction whose denominator is a multiple of qₙ
Dpₙ/qₙ is the fraction with the smallest numerator that falls within 1/qₙ² of α
The formal claim is precise: no fraction p/q with 0 < q ≤ qₙ approximates α better than the nth convergent does. This is stronger than saying convergents are 'very good' approximations — it is an optimality guarantee. Option B is wrong because the claim is not global across all rationals, just among fractions with denominator up to qₙ. Options C and D describe unrelated properties.
Question 2 Multiple Choice
A gear designer needs to approximate the ratio √2 ≈ 1.41421 using integers, and the largest gear has at most 100 teeth (denominator ≤ 100). The convergents of √2 include 1/1, 3/2, 7/5, 17/12, 41/29, 99/70. Which fraction should she use, and why?
A141/100 — closest decimal truncation within the constraint
B99/70 — as the convergent with largest denominator ≤ 100, it is provably the best approximation for that budget
C41/29 — smaller denominators are more mechanically reliable
DAny fraction within 0.001 of √2 is equally valid for engineering purposes
The optimality theorem guarantees that 99/70, as a convergent, is the best rational approximation among all fractions with denominator ≤ 100 — not just the best among convergents. 141/100 seems closer as a decimal but actually has larger error (1.41 vs √2 ≈ 1.41421). Option D ignores the optimality: the continued fraction algorithm provides a provably optimal answer, making the search unnecessary.
Question 3 True / False
A fraction that is not a convergent of α can never be a best rational approximation to α.
TTrue
FFalse
Answer: True
By the optimality theorem, every best rational approximation to α must be a convergent. Non-convergents may approximate α tolerably well, but there always exists a convergent with a denominator no larger that does at least as well. This is why the continued fraction algorithm is not merely one method for finding good approximations — it is the algorithm that produces all best rational approximations.
Question 4 True / False
It is possible to find a fraction p/q with q < qₙ that approximates α more closely than the nth convergent pₙ/qₙ.
TTrue
FFalse
Answer: False
This is precisely what the best-approximation theorem rules out. No fraction with denominator smaller than qₙ can beat the nth convergent — that is the definition of 'best rational approximation.' The algebraic identity pₙqₙ₋₁ − pₙ₋₁qₙ = (−1)ⁿ, which shows consecutive convergents straddle α from opposite sides, is the key to proving this impossibility.
Question 5 Short Answer
Why is it useful to know that convergents are the *best* rational approximations, rather than merely *good* ones?
Think about your answer, then reveal below.
Model answer: The optimality guarantee means that no improvement is possible for a given denominator budget — no search is required. In applications such as gear ratios, frequency synthesis, or lattice reduction algorithms, the continued fraction algorithm hands you the provably optimal approximation at every step, making it the correct tool rather than a convenient heuristic.
The distinction between 'good' and 'best' is practically significant. Without optimality, you would need to exhaustively search fractions with small denominators to find the best one. With optimality, you run the continued fraction algorithm and stop — the convergent is guaranteed to be optimal. This is why continued fractions underlie efficient algorithms in signal processing, cryptanalysis (LLL), and numerical mathematics.