Declared Sign-Bracket Updates, Early Stops and Exact Widths
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261021-r05-bisection-method-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=0/1, right=4/1, left sign=1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(0, 4, 1, 1, No)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 4, 1, 1, No).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
2. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=1/1, right=5/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(1, 5, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 5, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
3. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=2/1, right=6/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(2, 6, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, 6, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
4. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=3/1, right=7/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(3, 7, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3, 7, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
5. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=4/1, right=8/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(4, 8, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (4, 8, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
6. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=5/1, right=9/1, left sign=1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(5, 9, 1, 1, No)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 9, 1, 1, No).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
7. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=6/1, right=10/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(6, 10, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 10, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
8. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=7/1, right=11/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(7, 11, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 11, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
9. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=8/1, right=12/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(8, 12, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 12, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
10. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=9/1, right=13/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(9, 13, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 13, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
11. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=10/1, right=14/1, left sign=1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(10, 14, 1, 1, No)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 14, 1, 1, No).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
12. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=11/1, right=15/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(11, 15, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (11, 15, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
13. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=12/1, right=16/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(12, 16, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (12, 16, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
14. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=13/1, right=17/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(13, 17, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (13, 17, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
15. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=14/1, right=18/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(14, 18, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (14, 18, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
16. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=15/1, right=19/1, left sign=1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(15, 19, 1, 1, No)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (15, 19, 1, 1, No).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
17. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=16/1, right=20/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(16, 20, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (16, 20, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
18. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=17/1, right=21/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(17, 21, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (17, 21, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
19. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=18/1, right=22/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(18, 22, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (18, 22, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
20. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=19/1, right=23/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(19, 23, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (19, 23, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
21. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=20/1, right=24/1, left sign=1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(20, 24, 1, 1, No)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (20, 24, 1, 1, No).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
22. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=21/1, right=25/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(21, 25, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (21, 25, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
23. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=22/1, right=26/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(22, 26, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (22, 26, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
24. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=23/1, right=27/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(23, 27, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (23, 27, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
25. Qualify Declared Sign Data without an Existence Claim
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=24/1, right=28/1, left sign=-1, right sign=1; require left<right. Give (left,right,left sign,right sign,at least one endpoint sign is zero or the endpoint signs are opposite?).
View answer and explanation
(24, 28, -1, 1, Yes)
Check only the declared sign-bracket predicate. Opposite signs alone do not establish root existence without an appropriate additional theorem. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (24, 28, -1, 1, Yes).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-bracket-qualification · candidate
26. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=0/1, right=4/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(0, 4, -1, -1, 1, 2, [2, 4, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 4, -1, -1, 1, 2, [2, 4, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
27. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=1/1, right=5/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(1, 5, -1, 1, 1, 3, [1, 3, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 5, -1, 1, 1, 3, [1, 3, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
28. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=2/1, right=6/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(2, 6, -1, 0, 1, 4, [4, 4, 0, 0])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, 6, -1, 0, 1, 4, [4, 4, 0, 0]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
29. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=3/1, right=7/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(3, 7, -1, -1, 1, 5, [5, 7, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3, 7, -1, -1, 1, 5, [5, 7, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
30. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=4/1, right=8/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(4, 8, -1, 1, 1, 6, [4, 6, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (4, 8, -1, 1, 1, 6, [4, 6, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
31. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=5/1, right=9/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(5, 9, -1, 0, 1, 7, [7, 7, 0, 0])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 9, -1, 0, 1, 7, [7, 7, 0, 0]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
32. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=6/1, right=10/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(6, 10, -1, -1, 1, 8, [8, 10, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 10, -1, -1, 1, 8, [8, 10, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
33. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=7/1, right=11/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(7, 11, -1, 1, 1, 9, [7, 9, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 11, -1, 1, 1, 9, [7, 9, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
34. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=8/1, right=12/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(8, 12, -1, 0, 1, 10, [10, 10, 0, 0])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 12, -1, 0, 1, 10, [10, 10, 0, 0]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
35. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=9/1, right=13/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(9, 13, -1, -1, 1, 11, [11, 13, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 13, -1, -1, 1, 11, [11, 13, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
36. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=10/1, right=14/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(10, 14, -1, 1, 1, 12, [10, 12, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 14, -1, 1, 1, 12, [10, 12, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
37. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=11/1, right=15/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(11, 15, -1, 0, 1, 13, [13, 13, 0, 0])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (11, 15, -1, 0, 1, 13, [13, 13, 0, 0]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
38. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=12/1, right=16/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(12, 16, -1, -1, 1, 14, [14, 16, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (12, 16, -1, -1, 1, 14, [14, 16, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
39. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=13/1, right=17/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(13, 17, -1, 1, 1, 15, [13, 15, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (13, 17, -1, 1, 1, 15, [13, 15, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
40. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=14/1, right=18/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(14, 18, -1, 0, 1, 16, [16, 16, 0, 0])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (14, 18, -1, 0, 1, 16, [16, 16, 0, 0]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
41. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=15/1, right=19/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(15, 19, -1, -1, 1, 17, [17, 19, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (15, 19, -1, -1, 1, 17, [17, 19, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
42. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=16/1, right=20/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(16, 20, -1, 1, 1, 18, [16, 18, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (16, 20, -1, 1, 1, 18, [16, 18, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
43. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=17/1, right=21/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(17, 21, -1, 0, 1, 19, [19, 19, 0, 0])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (17, 21, -1, 0, 1, 19, [19, 19, 0, 0]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
44. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=18/1, right=22/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(18, 22, -1, -1, 1, 20, [20, 22, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (18, 22, -1, -1, 1, 20, [20, 22, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
45. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=19/1, right=23/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(19, 23, -1, 1, 1, 21, [19, 21, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (19, 23, -1, 1, 1, 21, [19, 21, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
46. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=20/1, right=24/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(20, 24, -1, 0, 1, 22, [22, 22, 0, 0])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (20, 24, -1, 0, 1, 22, [22, 22, 0, 0]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
47. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=21/1, right=25/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(21, 25, -1, -1, 1, 23, [23, 25, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (21, 25, -1, -1, 1, 23, [23, 25, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
48. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=22/1, right=26/1, signs left=-1, midpoint=1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(22, 26, -1, 1, 1, 24, [22, 24, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (22, 26, -1, 1, 1, 24, [22, 24, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
49. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=23/1, right=27/1, signs left=-1, midpoint=0, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(23, 27, -1, 0, 1, 25, [25, 25, 0, 0])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (23, 27, -1, 0, 1, 25, [25, 25, 0, 0]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
50. One Sign-Bracket Update with Explicit Endpoint-Zero Priority
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. left=24/1, right=28/1, signs left=-1, midpoint=-1, right=1. Require a qualified ordered bracket. Give (left,right,left sign,midpoint sign,right sign,midpoint,next [left,right,left sign,right sign]). Test a zero left endpoint first, then a zero right endpoint, then a zero midpoint; a detected zero collapses to that point. Otherwise retain the half with opposite endpoint signs.
View answer and explanation
(24, 28, -1, -1, 1, 26, [26, 28, -1, 1])
Follow the declared sign data and zero-priority convention exactly. The updated record does not by itself prove root existence. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (24, 28, -1, -1, 1, 26, [26, 28, -1, 1]).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-single-update · candidate
51. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=0/1, right=4/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(0, 4, -1, 1, [1], [[0, 4, 2, 1, 0, 2]], 0, 2)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 4, -1, 1, [1], [[0, 4, 2, 1, 0, 2]], 0, 2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
52. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=1/1, right=5/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(1, 5, -1, 1, [1, 1], [[1, 5, 3, 1, 1, 3], [1, 3, 2, 1, 1, 2]], 1, 2)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 5, -1, 1, [1, 1], [[1, 5, 3, 1, 1, 3], [1, 3, 2, 1, 1, 2]], 1, 2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
53. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=2/1, right=6/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(2, 6, -1, 1, [1, 1, 1], [[2, 6, 4, 1, 2, 4], [2, 4, 3, 1, 2, 3], [2, 3, 5/2, 1, 2, 5/2]], 2, 5/2)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, 6, -1, 1, [1, 1, 1], [[2, 6, 4, 1, 2, 4], [2, 4, 3, 1, 2, 3], [2, 3, 5/2, 1, 2, 5/2]], 2, 5/2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
54. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=3/1, right=7/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1, -1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(3, 7, -1, 1, [1, 1, 1, -1], [[3, 7, 5, 1, 3, 5], [3, 5, 4, 1, 3, 4], [3, 4, 7/2, 1, 3, 7/2], [3, 7/2, 13/4, -1, 13/4, 7/2]], 13/4, 7/2)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3, 7, -1, 1, [1, 1, 1, -1], [[3, 7, 5, 1, 3, 5], [3, 5, 4, 1, 3, 4], [3, 4, 7/2, 1, 3, 7/2], [3, 7/2, 13/4, -1, 13/4, 7/2]], 13/4, 7/2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
55. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=4/1, right=8/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(4, 8, -1, 1, [1], [[4, 8, 6, 1, 4, 6]], 4, 6)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (4, 8, -1, 1, [1], [[4, 8, 6, 1, 4, 6]], 4, 6).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
56. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=5/1, right=9/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(5, 9, -1, 1, [1, 1], [[5, 9, 7, 1, 5, 7], [5, 7, 6, 1, 5, 6]], 5, 6)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 9, -1, 1, [1, 1], [[5, 9, 7, 1, 5, 7], [5, 7, 6, 1, 5, 6]], 5, 6).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
57. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=6/1, right=10/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(6, 10, -1, 1, [1, 1, 1], [[6, 10, 8, 1, 6, 8], [6, 8, 7, 1, 6, 7], [6, 7, 13/2, 1, 6, 13/2]], 6, 13/2)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 10, -1, 1, [1, 1, 1], [[6, 10, 8, 1, 6, 8], [6, 8, 7, 1, 6, 7], [6, 7, 13/2, 1, 6, 13/2]], 6, 13/2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
58. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=7/1, right=11/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1, -1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(7, 11, -1, 1, [1, 1, 1, -1], [[7, 11, 9, 1, 7, 9], [7, 9, 8, 1, 7, 8], [7, 8, 15/2, 1, 7, 15/2], [7, 15/2, 29/4, -1, 29/4, 15/2]], 29/4, 15/2)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 11, -1, 1, [1, 1, 1, -1], [[7, 11, 9, 1, 7, 9], [7, 9, 8, 1, 7, 8], [7, 8, 15/2, 1, 7, 15/2], [7, 15/2, 29/4, -1, 29/4, 15/2]], 29/4, 15/2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
59. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=8/1, right=12/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(8, 12, -1, 1, [1], [[8, 12, 10, 1, 8, 10]], 8, 10)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 12, -1, 1, [1], [[8, 12, 10, 1, 8, 10]], 8, 10).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
60. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=9/1, right=13/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(9, 13, -1, 1, [1, 1], [[9, 13, 11, 1, 9, 11], [9, 11, 10, 1, 9, 10]], 9, 10)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 13, -1, 1, [1, 1], [[9, 13, 11, 1, 9, 11], [9, 11, 10, 1, 9, 10]], 9, 10).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
61. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=10/1, right=14/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(10, 14, -1, 1, [1, 1, 1], [[10, 14, 12, 1, 10, 12], [10, 12, 11, 1, 10, 11], [10, 11, 21/2, 1, 10, 21/2]], 10, 21/2)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 14, -1, 1, [1, 1, 1], [[10, 14, 12, 1, 10, 12], [10, 12, 11, 1, 10, 11], [10, 11, 21/2, 1, 10, 21/2]], 10, 21/2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
62. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=11/1, right=15/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1, -1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(11, 15, -1, 1, [1, 1, 1, -1], [[11, 15, 13, 1, 11, 13], [11, 13, 12, 1, 11, 12], [11, 12, 23/2, 1, 11, 23/2], [11, 23/2, 45/4, -1, 45/4, 23/2]], 45/4, 23/2)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (11, 15, -1, 1, [1, 1, 1, -1], [[11, 15, 13, 1, 11, 13], [11, 13, 12, 1, 11, 12], [11, 12, 23/2, 1, 11, 23/2], [11, 23/2, 45/4, -1, 45/4, 23/2]], 45/4, 23/2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
63. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=12/1, right=16/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(12, 16, -1, 1, [1], [[12, 16, 14, 1, 12, 14]], 12, 14)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (12, 16, -1, 1, [1], [[12, 16, 14, 1, 12, 14]], 12, 14).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
64. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=13/1, right=17/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(13, 17, -1, 1, [1, 1], [[13, 17, 15, 1, 13, 15], [13, 15, 14, 1, 13, 14]], 13, 14)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (13, 17, -1, 1, [1, 1], [[13, 17, 15, 1, 13, 15], [13, 15, 14, 1, 13, 14]], 13, 14).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
65. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=14/1, right=18/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(14, 18, -1, 1, [1, 1, 1], [[14, 18, 16, 1, 14, 16], [14, 16, 15, 1, 14, 15], [14, 15, 29/2, 1, 14, 29/2]], 14, 29/2)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (14, 18, -1, 1, [1, 1, 1], [[14, 18, 16, 1, 14, 16], [14, 16, 15, 1, 14, 15], [14, 15, 29/2, 1, 14, 29/2]], 14, 29/2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
66. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=15/1, right=19/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1, -1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(15, 19, -1, 1, [1, 1, 1, -1], [[15, 19, 17, 1, 15, 17], [15, 17, 16, 1, 15, 16], [15, 16, 31/2, 1, 15, 31/2], [15, 31/2, 61/4, -1, 61/4, 31/2]], 61/4, 31/2)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (15, 19, -1, 1, [1, 1, 1, -1], [[15, 19, 17, 1, 15, 17], [15, 17, 16, 1, 15, 16], [15, 16, 31/2, 1, 15, 31/2], [15, 31/2, 61/4, -1, 61/4, 31/2]], 61/4, 31/2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
67. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=16/1, right=20/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(16, 20, -1, 1, [1], [[16, 20, 18, 1, 16, 18]], 16, 18)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (16, 20, -1, 1, [1], [[16, 20, 18, 1, 16, 18]], 16, 18).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
68. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=17/1, right=21/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(17, 21, -1, 1, [1, 1], [[17, 21, 19, 1, 17, 19], [17, 19, 18, 1, 17, 18]], 17, 18)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (17, 21, -1, 1, [1, 1], [[17, 21, 19, 1, 17, 19], [17, 19, 18, 1, 17, 18]], 17, 18).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
69. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=18/1, right=22/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(18, 22, -1, 1, [1, 1, 1], [[18, 22, 20, 1, 18, 20], [18, 20, 19, 1, 18, 19], [18, 19, 37/2, 1, 18, 37/2]], 18, 37/2)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (18, 22, -1, 1, [1, 1, 1], [[18, 22, 20, 1, 18, 20], [18, 20, 19, 1, 18, 19], [18, 19, 37/2, 1, 18, 37/2]], 18, 37/2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
70. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=19/1, right=23/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1, -1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(19, 23, -1, 1, [1, 1, 1, -1], [[19, 23, 21, 1, 19, 21], [19, 21, 20, 1, 19, 20], [19, 20, 39/2, 1, 19, 39/2], [19, 39/2, 77/4, -1, 77/4, 39/2]], 77/4, 39/2)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (19, 23, -1, 1, [1, 1, 1, -1], [[19, 23, 21, 1, 19, 21], [19, 21, 20, 1, 19, 20], [19, 20, 39/2, 1, 19, 39/2], [19, 39/2, 77/4, -1, 77/4, 39/2]], 77/4, 39/2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
71. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=20/1, right=24/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(20, 24, -1, 1, [1], [[20, 24, 22, 1, 20, 22]], 20, 22)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (20, 24, -1, 1, [1], [[20, 24, 22, 1, 20, 22]], 20, 22).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
72. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=21/1, right=25/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(21, 25, -1, 1, [1, 1], [[21, 25, 23, 1, 21, 23], [21, 23, 22, 1, 21, 22]], 21, 22)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (21, 25, -1, 1, [1, 1], [[21, 25, 23, 1, 21, 23], [21, 23, 22, 1, 21, 22]], 21, 22).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
73. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=22/1, right=26/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(22, 26, -1, 1, [1, 1, 1], [[22, 26, 24, 1, 22, 24], [22, 24, 23, 1, 22, 23], [22, 23, 45/2, 1, 22, 45/2]], 22, 45/2)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (22, 26, -1, 1, [1, 1, 1], [[22, 26, 24, 1, 22, 24], [22, 24, 23, 1, 22, 23], [22, 23, 45/2, 1, 22, 45/2]], 22, 45/2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
74. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=23/1, right=27/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1, 1, 1, -1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(23, 27, -1, 1, [1, 1, 1, -1], [[23, 27, 25, 1, 23, 25], [23, 25, 24, 1, 23, 24], [23, 24, 47/2, 1, 23, 47/2], [23, 47/2, 93/4, -1, 93/4, 47/2]], 93/4, 47/2)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (23, 27, -1, 1, [1, 1, 1, -1], [[23, 27, 25, 1, 23, 25], [23, 25, 24, 1, 23, 24], [23, 24, 47/2, 1, 23, 47/2], [23, 47/2, 93/4, -1, 93/4, 47/2]], 93/4, 47/2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
75. A Finite Trace Stops at the First Declared Zero Midpoint
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=24/1, right=28/1, nonzero opposite endpoint signs=-1, 1; midpoint-sign list=[1]. Give (initial left,initial right,initial left sign,initial right sign,entire supplied sign list,every performed [left,right,midpoint,midpoint sign,next left,next right] row,final left,final right). Stop at the first zero midpoint and retain unused signs only in the original input-list field; an empty list performs no update.
View answer and explanation
(24, 28, -1, 1, [1], [[24, 28, 26, 1, 24, 26]], 24, 26)
Update signs and endpoints step by step, recording only performed updates. An early zero collapses the bracket; no unperformed later midpoint is evaluated. This trace is not a root-existence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (24, 28, -1, 1, [1], [[24, 28, 26, 1, 24, 26]], 24, 26).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-finite-trace · candidate
76. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=-3/1, right=5/1, halvings=0≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(-3, 5, 0, 8)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-3, 5, 0, 8).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
77. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=-2/1, right=6/1, halvings=1≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(-2, 6, 1, 4)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-2, 6, 1, 4).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
78. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=-1/1, right=7/1, halvings=2≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(-1, 7, 2, 2)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (-1, 7, 2, 2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
79. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=0/1, right=8/1, halvings=3≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(0, 8, 3, 1)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 8, 3, 1).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
80. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=1/1, right=9/1, halvings=4≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(1, 9, 4, 1/2)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 9, 4, 1/2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
81. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=2/1, right=10/1, halvings=5≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(2, 10, 5, 1/4)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, 10, 5, 1/4).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
82. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=3/1, right=11/1, halvings=6≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(3, 11, 6, 1/8)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3, 11, 6, 1/8).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
83. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=4/1, right=12/1, halvings=0≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(4, 12, 0, 8)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (4, 12, 0, 8).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
84. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=5/1, right=13/1, halvings=1≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(5, 13, 1, 4)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (5, 13, 1, 4).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
85. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=6/1, right=14/1, halvings=2≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(6, 14, 2, 2)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 14, 2, 2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
86. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=7/1, right=15/1, halvings=3≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(7, 15, 3, 1)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7, 15, 3, 1).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
87. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=8/1, right=16/1, halvings=4≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(8, 16, 4, 1/2)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8, 16, 4, 1/2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
88. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=9/1, right=17/1, halvings=5≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(9, 17, 5, 1/4)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 17, 5, 1/4).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
89. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=10/1, right=18/1, halvings=6≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(10, 18, 6, 1/8)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (10, 18, 6, 1/8).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
90. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=11/1, right=19/1, halvings=0≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(11, 19, 0, 8)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (11, 19, 0, 8).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
91. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=12/1, right=20/1, halvings=1≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(12, 20, 1, 4)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (12, 20, 1, 4).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
92. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=13/1, right=21/1, halvings=2≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(13, 21, 2, 2)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (13, 21, 2, 2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
93. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=14/1, right=22/1, halvings=3≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(14, 22, 3, 1)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (14, 22, 3, 1).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
94. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=15/1, right=23/1, halvings=4≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(15, 23, 4, 1/2)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (15, 23, 4, 1/2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
95. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=16/1, right=24/1, halvings=5≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(16, 24, 5, 1/4)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (16, 24, 5, 1/4).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
96. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=17/1, right=25/1, halvings=6≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(17, 25, 6, 1/8)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (17, 25, 6, 1/8).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
97. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=18/1, right=26/1, halvings=0≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(18, 26, 0, 8)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (18, 26, 0, 8).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
98. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=19/1, right=27/1, halvings=1≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(19, 27, 1, 4)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (19, 27, 1, 4).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
99. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=20/1, right=28/1, halvings=2≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
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(20, 28, 2, 2)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (20, 28, 2, 2).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
100. Geometric Width after a Declared Number of Halvings
Use only the supplied exact rational endpoints and declared signs in {-1,0,1}. Normalize rational results to reduced fractions with positive denominators, omitting /1 for integers. Sign data and width-halving do not establish root existence or convergence to a root; a separate theorem and its hypotheses would be required. initial left=21/1, right=29/1, halvings=3≥0. Give (left,right,halvings,(right−left)/2^halvings). This is the geometric width formula without an earlier zero-stop, not a function-value error.
View answer and explanation
(21, 29, 3, 1)
Divide the exact initial width by the specified power of 2. Interpreting it as a root-location bound separately requires a root in the bracket. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (21, 29, 3, 1).
Reviewed step-by-step derivation
Finite sign-bracket updates in the bisection method · qf-app21-bisection-width-after-steps · candidate
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