Question 28277: Principal Square Root of a Perfect Square

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0055-28277 · q-20261003-r01-power-root-0052

Original question and givens

Using the principal square root on the nonnegative reals, find the exact nonnegative value of √1.

G001 — Using the principal square root on the nonnegative reals, find the exact nonnegative value of √1.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

mko-power-root

Step-by-step derivation

S001 · implies

Before

source roles = [qf-principal-square-root-exact, principal_sqrt]

After

derived = [[single_nonnegative_exact_integer_root]]

Conditions: none

List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged.

Step identifiers and dependencies

Rule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

Legacy principal ORIGINAL radicand exact midpoint square0 := mul(1, 1)

After

Legacy principal ORIGINAL radicand exact midpoint square0 = 1

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

source roles = [Legacy principal ORIGINAL radicand, 1, 0, [0, 2], 1]

After

derived = [1, Yes, 1, 2]

Conditions: none

Squaring is monotone on nonnegative integers;maintain low²<=n<high².The midpoint is BigInt floor((low+high)/2),not floating sqrt,and strictly shrinks the integer bracket.Derived values do not inherit SAFE bounds.

Step identifiers and dependencies

Rule: Legacy_isqrt_EACH_BigInt_integer_midpoint_bracket · Uses: G001, S002

S004 · calculation

Before

Legacy principal ORIGINAL radicand terminal root square := mul(1, 1)

After

Legacy principal ORIGINAL radicand terminal root square = 1

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Legacy principal ORIGINAL radicand terminal successor square := mul(2, 2)

After

Legacy principal ORIGINAL radicand terminal successor square = 4

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [Legacy principal ORIGINAL radicand, 1, 1, 2]

After

derived = [1, 4]

Conditions: none

Adjacent integer bounds certify floor sqrt=low;an exact root separately needs low²=n,and the principal root is the single nonnegative branch.

Step identifiers and dependencies

Rule: Legacy_isqrt_terminal_floor_and_single_nonnegative_branch · Uses: G001, S005

S007 · implies

Before

source roles = [1]

After

derived = [1, Yes]

Conditions: none

Check root²=r and root>=0.Monotonicity of nonnegative squaring gives uniqueness;two signed equation roots are not the principal-root value.

Step identifiers and dependencies

Rule: Legacy_PERFECT_square_exact_NONNEGATIVE_principal_root · Uses: G001, S006

Reverse checks

S007 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
S006 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
S005 · premise_and_result_audit — Recompute Legacy principal ORIGINAL radicand terminal successor square with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute Legacy principal ORIGINAL radicand terminal root square with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
S002 · premise_and_result_audit — Recompute Legacy principal ORIGINAL radicand exact midpoint square0 with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.

Result

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: 640e9198a544bbb6396db32066c8eeab28fa43525d369fcc2407b0c291a9d65d
candidate record SHA-256: 810e6276fd7cf9c4349792583431098e94a27ddde4a0780f35551a1d4a9e9497
semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683
question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e
Lean: NOT_RUN · human review: NOT_PERFORMED

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