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Question 35516: Arithmetic Number-System Boundary

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0070-35516 · q-20260914-arithmetic-domain-boundary-0016

Original question and givens

In the number system N, including 0, how should the result of 3/2 be classified?

  • G001 — In the number system N, including 0, how should the result of 3/2 be classified?

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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-arithmetic-domain-boundary, [a, b, number_system, operator, state]]

After

derived = [arithmetic, null, null]

Conditions: none

Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules.

Step identifiers and dependencies

Rule: R70_SOURCE_bank_specific_interface · Uses: G001

S002 · implies

Before

source roles = [3, 2, /]

After

derived = [N={0,1,2,...}]

Conditions: none

FIRST fix THIS question’s N={0,1,2,...} and nonnegative integral operands.The question asks for a value WITHIN this system,not a silent change to integers/rationals.

Step identifiers and dependencies

Rule: Natural_arithmetic_GIVEN_N_includes0_and_operands · Uses: G001, S001

S003 · calculation

Before

Natural division divisor times Euclidean quotient := mul(2, 1)

After

Natural division divisor times Euclidean quotient = 2

Conditions: none

This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Natural division reconstruct dividend := sum(2, 1)

After

Natural division reconstruct dividend = 3

Conditions: none

This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [3, 2]

After

derived = [1, 1, Yes]

Conditions: none

Write a=bk+r with0≤r<b.If r=0,the natural quotient is k.If r≠0,an integer quotient would imply r=b(m−k),contradicting0<r<b.Do not use truncation or a decimal as an N-valued answer.

Step identifiers and dependencies

Rule: Division_EUCLIDEAN_exact_remainder_and_nonnegative_quotient · Uses: G001, S004

S006 · implies

Before

source roles = [not_closed_in_N, not_closed_in_N]

After

derived = [in N not closed in the stated domain]

Conditions: none

Check state metadata ONLY AFTER deriving the result;never use it instead of calculation.Keep zero divisor,nonclosure and N-valued result distinct.

Step identifiers and dependencies

Rule: Arithmetic_STATE_checked_AFTER_derivation · Uses: G001, S005

Reverse checks

  • S006 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S005 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S004 · premise_and_result_audit — Recompute Natural division reconstruct dividend with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Natural division divisor times Euclidean quotient with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S001 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.

Result

Not closed in N

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: b112f57a744c4443ae300194425e9bced03d637850fa144b88ddf68ab3efb0a2
  • candidate record SHA-256: 6462ca2ca44eba87d8c93023b8d68323a4ae71cf3bd888bae5fd13fd74fb8f6b
  • semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80
  • question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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