Representatives, diagonals and a new irrational cut
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 180 minutes, as an unvalidated planning estimate. This collection contains 3 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Use positive rational tolerances until the constructed field and its order are available.
Distinguish a rational sequence, its equivalence class, and a sequence of classes.
Do not invoke completeness, a real supremum, or the desired root before proving its existence.
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Bisect rational brackets for a positive square root of 5, starting with [2,3]; let aₙ and cₙ be the lower and upper endpoints after n bisections, n≥0. Construct x=[aₙ] and prove x²=5 without assuming sqrt(5) already exists. Define bₙ=0 for n≤3 and bₙ=aₙ+(−1)ⁿ2⁻ⁿ for n≥4. Prove [bₙ]=x. Define dₙ=1 for n≤3 and dₙ=1/bₙ afterward. Prove [dₙ]=x/5, and prove 2<x<9/4. Explain why the four initial zeros and the alternating perturbation cannot determine the class order or prevent an inverse.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For each positive integer n define a rational sequence a⁽ⁿ⁾ₖ=1 when k≤n and 0 when k>n. Let xₙ=[a⁽ⁿ⁾]. Identify xₙ and the class of the naive diagonal a⁽ⁿ⁾ₙ. Explain precisely why taking the nth term of the nth representative is not a valid completeness proof. Now let (xₙ) be any Cauchy sequence of classes and choose rationals rₙ,sₙ with |xₙ−rₙ|<2⁻ⁿ and |xₙ−sₙ|<1/n. Prove both rational sequences are Cauchy, represent the same class, and that this class is the limit of (xₙ). Do not assume the diagonal class is already its limit.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Construct a positive cube root of 2 by rational bisection of [1,2], using midpoint cubes. Prove the endpoint sequences define the same class w and w³=2. Prove this class is irrational. Finally prove w is the least upper bound of S={q∈Q:q≥0 and q³<2}, viewed inside the constructed field. Use rational density to prove leastness, and make clear where the construction avoids circular use of a pre-existing supremum or cube root.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Bisect rational brackets for a positive square root of 5, starting with [2,3]; let aₙ and cₙ be the lower and upper endpoints after n bisections, n≥0. Construct x=[aₙ] and prove x²=5 without assuming sqrt(5) already exists. Define bₙ=0 for n≤3 and bₙ=aₙ+(−1)ⁿ2⁻ⁿ for n≥4. Prove [bₙ]=x. Define dₙ=1 for n≤3 and dₙ=1/bₙ afterward. Prove [dₙ]=x/5, and prove 2<x<9/4. Explain why the four initial zeros and the alternating perturbation cannot determine the class order or prevent an inverse.
For each positive integer n define a rational sequence a⁽ⁿ⁾ₖ=1 when k≤n and 0 when k>n. Let xₙ=[a⁽ⁿ⁾]. Identify xₙ and the class of the naive diagonal a⁽ⁿ⁾ₙ. Explain precisely why taking the nth term of the nth representative is not a valid completeness proof. Now let (xₙ) be any Cauchy sequence of classes and choose rationals rₙ,sₙ with |xₙ−rₙ|<2⁻ⁿ and |xₙ−sₙ|<1/n. Prove both rational sequences are Cauchy, represent the same class, and that this class is the limit of (xₙ). Do not assume the diagonal class is already its limit.
Construct a positive cube root of 2 by rational bisection of [1,2], using midpoint cubes. Prove the endpoint sequences define the same class w and w³=2. Prove this class is irrational. Finally prove w is the least upper bound of S={q∈Q:q≥0 and q³<2}, viewed inside the constructed field. Use rational density to prove leastness, and make clear where the construction avoids circular use of a pre-existing supremum or cube root.
