{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0001","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa3fcca0f890e50f89cfc661cf3d37be70714cbddaa02a6ac4deef8b8e3a1f96","family_id":"qf-str22-complex-addition","seed":202610223001,"semantic_template":"qf-str22-complex-addition/{\"left\":[-12,-3],\"right\":[-2,-9]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-12, -3], right=[-2, -9]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([-12, -3], [-2, -9], [-14, -12])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-12, -3], [-2, -9], [-14, -12])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0002","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6276a783355854b67dcd6da2258ef3df9bfc1c9502c518d97a76b4bbcabe743","family_id":"qf-str22-complex-addition","seed":202610223002,"semantic_template":"qf-str22-complex-addition/{\"left\":[-11,-2],\"right\":[-1,-7]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-11, -2], right=[-1, -7]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([-11, -2], [-1, -7], [-12, -9])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-11, -2], [-1, -7], [-12, -9])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0003","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1597694c14312e4b93e58039ab5145d807522c9c5386a31e42f1b1c8a5712968","family_id":"qf-str22-complex-addition","seed":202610223003,"semantic_template":"qf-str22-complex-addition/{\"left\":[-10,-1],\"right\":[0,-5]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-10, -1], right=[0, -5]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([-10, -1], [0, -5], [-10, -6])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-10, -1], [0, -5], [-10, -6])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0004","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49a06814c0e38acff0645751e5754df7061d67bd1bfca90c30ac967b09b7e037","family_id":"qf-str22-complex-addition","seed":202610223004,"semantic_template":"qf-str22-complex-addition/{\"left\":[-9,0],\"right\":[1,-3]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-9, 0], right=[1, -3]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([-9, 0], [1, -3], [-8, -3])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-9, 0], [1, -3], [-8, -3])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0005","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1d33ef78b37e189290a810e2ba6f6966b515f9ed87f812027a796c6fc6123be","family_id":"qf-str22-complex-addition","seed":202610223005,"semantic_template":"qf-str22-complex-addition/{\"left\":[-8,1],\"right\":[2,-1]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-8, 1], right=[2, -1]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([-8, 1], [2, -1], [-6, 0])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-8, 1], [2, -1], [-6, 0])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0006","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41f1923cb8c2c9a74c742238e44d7e29c3cd0b2046ea3bb95cbbd76b14e2c014","family_id":"qf-str22-complex-addition","seed":202610223006,"semantic_template":"qf-str22-complex-addition/{\"left\":[-7,2],\"right\":[-2,1]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-7, 2], right=[-2, 1]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([-7, 2], [-2, 1], [-9, 3])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-7, 2], [-2, 1], [-9, 3])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0007","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b11e36eceee20f91a9c91d6d1cd1de56f733e6385f493623e0bb171404c25995","family_id":"qf-str22-complex-addition","seed":202610223007,"semantic_template":"qf-str22-complex-addition/{\"left\":[-6,3],\"right\":[-1,3]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-6, 3], right=[-1, 3]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([-6, 3], [-1, 3], [-7, 6])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-6, 3], [-1, 3], [-7, 6])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0008","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ccff37c17e7ea4f357bf0f982af1edea1de231b458c57383d483114b2f2ef10b","family_id":"qf-str22-complex-addition","seed":202610223008,"semantic_template":"qf-str22-complex-addition/{\"left\":[-5,-3],\"right\":[0,5]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-5, -3], right=[0, 5]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([-5, -3], [0, 5], [-5, 2])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-5, -3], [0, 5], [-5, 2])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0009","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a4649b96d4a1ed4e2e13262d3cea474df0a59bb30e8f20c2603079b9ea25cc9","family_id":"qf-str22-complex-addition","seed":202610223009,"semantic_template":"qf-str22-complex-addition/{\"left\":[-4,-2],\"right\":[1,7]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-4, -2], right=[1, 7]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([-4, -2], [1, 7], [-3, 5])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-4, -2], [1, 7], [-3, 5])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0010","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9baee6456feb32691a78e4f7fc4fae6736d55513accb58be649e8d231a4817be","family_id":"qf-str22-complex-addition","seed":202610223010,"semantic_template":"qf-str22-complex-addition/{\"left\":[-3,-1],\"right\":[2,9]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-3, -1], right=[2, 9]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([-3, -1], [2, 9], [-1, 8])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-3, -1], [2, 9], [-1, 8])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0011","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51644e1ed1652e0b03f7abd172a0921f13bb133a9f9d0358ce60af685c8f6109","family_id":"qf-str22-complex-addition","seed":202610223011,"semantic_template":"qf-str22-complex-addition/{\"left\":[-2,0],\"right\":[-2,11]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-2, 0], right=[-2, 11]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([-2, 0], [-2, 11], [-4, 11])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-2, 0], [-2, 11], [-4, 11])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0012","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b93d91175a00d34f9d4475a9c0c632a5492c8bee205c1889c270263d93c47d4","family_id":"qf-str22-complex-addition","seed":202610223012,"semantic_template":"qf-str22-complex-addition/{\"left\":[-1,1],\"right\":[-1,13]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-1, 1], right=[-1, 13]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([-1, 1], [-1, 13], [-2, 14])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-1, 1], [-1, 13], [-2, 14])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0013","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24390f628863deb9f9601e13ef45f2c17ed78bf98f5431efdca95c40fcce2efa","family_id":"qf-str22-complex-addition","seed":202610223013,"semantic_template":"qf-str22-complex-addition/{\"left\":[0,2],\"right\":[0,15]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[0, 2], right=[0, 15]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([0, 2], [0, 15], [0, 17])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 2], [0, 15], [0, 17])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0014","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e544e2261e85aaf77d46a3fbdf57cab9e17439e1969c04ed5f1da36cfe44492e","family_id":"qf-str22-complex-addition","seed":202610223014,"semantic_template":"qf-str22-complex-addition/{\"left\":[1,3],\"right\":[1,17]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[1, 3], right=[1, 17]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([1, 3], [1, 17], [2, 20])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 3], [1, 17], [2, 20])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0015","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcdf9138c503e64f1ba4663986d0ada0ee51e0d7874c47f227c0f9df21d19195","family_id":"qf-str22-complex-addition","seed":202610223015,"semantic_template":"qf-str22-complex-addition/{\"left\":[2,-3],\"right\":[2,19]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[2, -3], right=[2, 19]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([2, -3], [2, 19], [4, 16])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, -3], [2, 19], [4, 16])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0016","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5c66c8f06584fea3a37ea32df7eff5cdf1a5c2b23e4c54d6ce6b2c4ee0e0204","family_id":"qf-str22-complex-addition","seed":202610223016,"semantic_template":"qf-str22-complex-addition/{\"left\":[3,-2],\"right\":[-2,21]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[3, -2], right=[-2, 21]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([3, -2], [-2, 21], [1, 19])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, -2], [-2, 21], [1, 19])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0017","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8acb9374d2ec2737847adb3fefaee07ae64292269b10c0893a4ac168928cb560","family_id":"qf-str22-complex-addition","seed":202610223017,"semantic_template":"qf-str22-complex-addition/{\"left\":[4,-1],\"right\":[-1,23]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[4, -1], right=[-1, 23]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([4, -1], [-1, 23], [3, 22])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, -1], [-1, 23], [3, 22])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0018","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e8aeaad26cd0fa898840862929547799139a8024fde9de93a5b928e76cfd20c","family_id":"qf-str22-complex-addition","seed":202610223018,"semantic_template":"qf-str22-complex-addition/{\"left\":[5,0],\"right\":[0,25]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[5, 0], right=[0, 25]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([5, 0], [0, 25], [5, 25])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 0], [0, 25], [5, 25])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0019","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72f63b8efb7dff7b0e99af22f677b5e1df6fed4f1e1e9d659e03f4d33df96475","family_id":"qf-str22-complex-addition","seed":202610223019,"semantic_template":"qf-str22-complex-addition/{\"left\":[6,1],\"right\":[1,27]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[6, 1], right=[1, 27]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([6, 1], [1, 27], [7, 28])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 1], [1, 27], [7, 28])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0020","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13c12a61e02e58c5ce6814594420909f4046f649f6c8e41dcc02b5627fd193ee","family_id":"qf-str22-complex-addition","seed":202610223020,"semantic_template":"qf-str22-complex-addition/{\"left\":[7,2],\"right\":[2,29]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[7, 2], right=[2, 29]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([7, 2], [2, 29], [9, 31])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 2], [2, 29], [9, 31])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0021","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92de4d43f239cfb4764ece45337c3e878830f72a8809b6b9004308097e35eae6","family_id":"qf-str22-complex-addition","seed":202610223021,"semantic_template":"qf-str22-complex-addition/{\"left\":[8,3],\"right\":[-2,31]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[8, 3], right=[-2, 31]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([8, 3], [-2, 31], [6, 34])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 3], [-2, 31], [6, 34])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0022","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cce22957526b2b2a2cb64ffed0d1cb7fe23348b9a5a44b4447240ae02fe5ff2","family_id":"qf-str22-complex-addition","seed":202610223022,"semantic_template":"qf-str22-complex-addition/{\"left\":[9,-3],\"right\":[-1,33]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[9, -3], right=[-1, 33]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([9, -3], [-1, 33], [8, 30])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, -3], [-1, 33], [8, 30])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0023","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f317e7a1941e5f8c466506788d4db5da31d1c73d9b88924db6752d8e5c962709","family_id":"qf-str22-complex-addition","seed":202610223023,"semantic_template":"qf-str22-complex-addition/{\"left\":[10,-2],\"right\":[0,35]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[10, -2], right=[0, 35]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([10, -2], [0, 35], [10, 33])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, -2], [0, 35], [10, 33])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0024","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bdf9cc4c3592bb8287866e54a4887be8d791630377f00eea60c7532824dad81","family_id":"qf-str22-complex-addition","seed":202610223024,"semantic_template":"qf-str22-complex-addition/{\"left\":[11,-1],\"right\":[1,37]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[11, -1], right=[1, 37]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([11, -1], [1, 37], [12, 36])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, -1], [1, 37], [12, 36])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0025","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebe130b7f2c612b859381b07fb72c58565c25d64f78fe17b468b60371b7e5807","family_id":"qf-str22-complex-addition","seed":202610223025,"semantic_template":"qf-str22-complex-addition/{\"left\":[12,0],\"right\":[2,39]}"},"fields":{"title":"Ordered-Pair Complex Addition","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[12, 0], right=[2, 39]. Give (left,right,[sum of real parts,sum of imaginary coefficients]).","answer":{"kind":"expression","value":"([12, 0], [2, 39], [14, 39])","explanation":"Add corresponding components, preserving their semantic order. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 0], [2, 39], [14, 39])."},"skills":["Ordered-Pair Complex Addition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0026","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a447b30683ec74b09ef2deec7f1258acf4c881790461c246950166c3022bcbb0","family_id":"qf-str22-complex-subtraction","seed":202610223026,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[-12,-3],\"right\":[-2,-9]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-12, -3], right=[-2, -9]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([-12, -3], [-2, -9], [-10, 6])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-12, -3], [-2, -9], [-10, 6])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0027","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca77fc0a4efbf90684ebcdbe12543a770bd46e7ce82124cfdcc5d6c54ae5e2e7","family_id":"qf-str22-complex-subtraction","seed":202610223027,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[-11,-2],\"right\":[-1,-7]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-11, -2], right=[-1, -7]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([-11, -2], [-1, -7], [-10, 5])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-11, -2], [-1, -7], [-10, 5])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0028","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20e309597f4676114e7b45baf746ce6cf59c27d61ace937422f3e26794615d36","family_id":"qf-str22-complex-subtraction","seed":202610223028,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[-10,-1],\"right\":[0,-5]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-10, -1], right=[0, -5]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([-10, -1], [0, -5], [-10, 4])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-10, -1], [0, -5], [-10, 4])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0029","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"820ee1f255d7340d7b795870513d56c5269126e6b1a8624568300b7d15aea95e","family_id":"qf-str22-complex-subtraction","seed":202610223029,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[-9,0],\"right\":[1,-3]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-9, 0], right=[1, -3]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([-9, 0], [1, -3], [-10, 3])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-9, 0], [1, -3], [-10, 3])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0030","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12b657cb4b89e49a91ba34021219e1e3857e4f6688d9a821063b7b0d6fc40dcf","family_id":"qf-str22-complex-subtraction","seed":202610223030,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[-8,1],\"right\":[2,-1]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-8, 1], right=[2, -1]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([-8, 1], [2, -1], [-10, 2])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-8, 1], [2, -1], [-10, 2])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0031","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eea0525d02a06b8b364ef16b19746c6c47fa3b34a56f1a12e6314c5b4bf26f6a","family_id":"qf-str22-complex-subtraction","seed":202610223031,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[-7,2],\"right\":[-2,1]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-7, 2], right=[-2, 1]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([-7, 2], [-2, 1], [-5, 1])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-7, 2], [-2, 1], [-5, 1])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0032","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61e20a67f3ba8862245faae4e807c5a41c2408213e18282599bed9991b5344c3","family_id":"qf-str22-complex-subtraction","seed":202610223032,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[-6,3],\"right\":[-1,3]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-6, 3], right=[-1, 3]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([-6, 3], [-1, 3], [-5, 0])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-6, 3], [-1, 3], [-5, 0])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0033","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28f6d345fc52b6e91ace2fa388fffc7189473f4d85bde54708a55ec9616bfe2a","family_id":"qf-str22-complex-subtraction","seed":202610223033,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[-5,-3],\"right\":[0,5]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-5, -3], right=[0, 5]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([-5, -3], [0, 5], [-5, -8])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-5, -3], [0, 5], [-5, -8])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0034","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b18128e88d13e1eb0c952e53752c679af2f507e816cf23e345728f23337695e","family_id":"qf-str22-complex-subtraction","seed":202610223034,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[-4,-2],\"right\":[1,7]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-4, -2], right=[1, 7]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([-4, -2], [1, 7], [-5, -9])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-4, -2], [1, 7], [-5, -9])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0035","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"585c2ac4ac2c97c787e4d3d5ecdcc3a26e44329e45082ee0ece21800d7c5fcdd","family_id":"qf-str22-complex-subtraction","seed":202610223035,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[-3,-1],\"right\":[2,9]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-3, -1], right=[2, 9]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([-3, -1], [2, 9], [-5, -10])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-3, -1], [2, 9], [-5, -10])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0036","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff0f13aa8f0c53feedc3987d99150afe760c45c0eadfd123588510507f10adfd","family_id":"qf-str22-complex-subtraction","seed":202610223036,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[-2,0],\"right\":[-2,11]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-2, 0], right=[-2, 11]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([-2, 0], [-2, 11], [0, -11])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-2, 0], [-2, 11], [0, -11])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0037","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f36d6ebd85d0e0dec1a0c5b119deb751be5e78eb05dd9c58febd92754ed1b0d","family_id":"qf-str22-complex-subtraction","seed":202610223037,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[-1,1],\"right\":[-1,13]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[-1, 1], right=[-1, 13]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([-1, 1], [-1, 13], [0, -12])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-1, 1], [-1, 13], [0, -12])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0038","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53313f400ecaf69cc09f440faffab18903d5c711a58a7060b94e3ebf44cb9ec5","family_id":"qf-str22-complex-subtraction","seed":202610223038,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[0,2],\"right\":[0,15]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[0, 2], right=[0, 15]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([0, 2], [0, 15], [0, -13])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 2], [0, 15], [0, -13])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0039","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45cd10675c886b1b1051cd4ab64971bab20dfd21f3166f704e0c501a936f33dd","family_id":"qf-str22-complex-subtraction","seed":202610223039,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[1,3],\"right\":[1,17]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[1, 3], right=[1, 17]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([1, 3], [1, 17], [0, -14])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 3], [1, 17], [0, -14])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0040","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e9544549857ff1f57a7eed496f7b3535ffdd2baf9e78717d6b09421d061ea6a","family_id":"qf-str22-complex-subtraction","seed":202610223040,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[2,-3],\"right\":[2,19]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[2, -3], right=[2, 19]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([2, -3], [2, 19], [0, -22])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, -3], [2, 19], [0, -22])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0041","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb44915f954014b90c8a3c099b1656af3e01e795b7ca1f1036920d3c5a82367b","family_id":"qf-str22-complex-subtraction","seed":202610223041,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[3,-2],\"right\":[-2,21]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[3, -2], right=[-2, 21]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([3, -2], [-2, 21], [5, -23])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, -2], [-2, 21], [5, -23])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0042","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb9302c5ea40e2e2827ec8077a3078cd1c88e37456a239f0fb20e3d149f71168","family_id":"qf-str22-complex-subtraction","seed":202610223042,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[4,-1],\"right\":[-1,23]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[4, -1], right=[-1, 23]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([4, -1], [-1, 23], [5, -24])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, -1], [-1, 23], [5, -24])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0043","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70738365defda8f0bbd79d879991191eb14f505627e995f4afa1db87a7f84b79","family_id":"qf-str22-complex-subtraction","seed":202610223043,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[5,0],\"right\":[0,25]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[5, 0], right=[0, 25]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([5, 0], [0, 25], [5, -25])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 0], [0, 25], [5, -25])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0044","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51524fe9fd752bbcf212150b47b58c0e9050f51c30ee0d663b416d4563d1509c","family_id":"qf-str22-complex-subtraction","seed":202610223044,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[6,1],\"right\":[1,27]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[6, 1], right=[1, 27]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([6, 1], [1, 27], [5, -26])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 1], [1, 27], [5, -26])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0045","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1c97523b3d8f10fce37f9e6e088555d89e18025f670ef056a7c18ba21d39812","family_id":"qf-str22-complex-subtraction","seed":202610223045,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[7,2],\"right\":[2,29]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[7, 2], right=[2, 29]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([7, 2], [2, 29], [5, -27])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 2], [2, 29], [5, -27])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0046","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1c74fca9bf03a1f5460648dd1b537cbe7260864515c89fe415e1a7bc3af74a7","family_id":"qf-str22-complex-subtraction","seed":202610223046,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[8,3],\"right\":[-2,31]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[8, 3], right=[-2, 31]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([8, 3], [-2, 31], [10, -28])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 3], [-2, 31], [10, -28])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0047","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2e069b78a84e52c0803f44673ad2c13f60121bd8cb687157d4b04537c17ad03","family_id":"qf-str22-complex-subtraction","seed":202610223047,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[9,-3],\"right\":[-1,33]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[9, -3], right=[-1, 33]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([9, -3], [-1, 33], [10, -36])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, -3], [-1, 33], [10, -36])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0048","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7b04a8fc2f8384f2de3ebe4d84cc21c6261f3c38377d77245011f66cdd4c1a4","family_id":"qf-str22-complex-subtraction","seed":202610223048,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[10,-2],\"right\":[0,35]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[10, -2], right=[0, 35]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([10, -2], [0, 35], [10, -37])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, -2], [0, 35], [10, -37])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0049","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"391b7ac651859c28c180d4d3a983300f5e3b1227df1c94dc89c558600f47a6af","family_id":"qf-str22-complex-subtraction","seed":202610223049,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[11,-1],\"right\":[1,37]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[11, -1], right=[1, 37]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([11, -1], [1, 37], [10, -38])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, -1], [1, 37], [10, -38])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0050","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec17c1b2fed06ce768c35507d5b41853ddf7d876ff7efaf01fa9dc2b17a3ce87","family_id":"qf-str22-complex-subtraction","seed":202610223050,"semantic_template":"qf-str22-complex-subtraction/{\"left\":[12,0],\"right\":[2,39]}"},"fields":{"title":"Ordered-Pair Complex Subtraction","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[12, 0], right=[2, 39]. Give (left,right,[left real−right real,left imaginary coefficient−right imaginary coefficient]).","answer":{"kind":"expression","value":"([12, 0], [2, 39], [10, -39])","explanation":"Subtract in the declared operand order rather than swapping components or taking absolute differences. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 0], [2, 39], [10, -39])."},"skills":["Ordered-Pair Complex Subtraction"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0051","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c612121d2ff8f78201d51b522d11ba6043adeffe1f6f0d8c15428ef2d03029f2","family_id":"qf-str22-complex-multiplication","seed":202610223051,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[-12,-3],\"right\":[-2,-9]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[-12, -3], right=[c,d]=[-2, -9]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([-12, -3], [-2, -9], [-3, 114])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-12, -3], [-2, -9], [-3, 114])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0052","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b8b91c1a4805ce5522ceee423e491ad6a690c1a947900e03c308309ea7f41ec","family_id":"qf-str22-complex-multiplication","seed":202610223052,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[-11,-2],\"right\":[-1,-7]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[-11, -2], right=[c,d]=[-1, -7]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([-11, -2], [-1, -7], [-3, 79])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-11, -2], [-1, -7], [-3, 79])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0053","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1734ae4c7808083cdcd623a68eaf5116eaedb3918cbf14518f590b68c831127a","family_id":"qf-str22-complex-multiplication","seed":202610223053,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[-10,-1],\"right\":[0,-5]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[-10, -1], right=[c,d]=[0, -5]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([-10, -1], [0, -5], [-5, 50])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-10, -1], [0, -5], [-5, 50])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0054","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e37284aabed3edd1e6c5391f2cd0ff1b58bbdaf059a3cbb8139781b578be197a","family_id":"qf-str22-complex-multiplication","seed":202610223054,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[-9,0],\"right\":[1,-3]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[-9, 0], right=[c,d]=[1, -3]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([-9, 0], [1, -3], [-9, 27])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-9, 0], [1, -3], [-9, 27])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0055","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c039d64b41f8f21a2d5f5f8aea5528383b4e0027b8cb18e48e2579c2ad76fe7","family_id":"qf-str22-complex-multiplication","seed":202610223055,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[-8,1],\"right\":[2,-1]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[-8, 1], right=[c,d]=[2, -1]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([-8, 1], [2, -1], [-15, 10])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-8, 1], [2, -1], [-15, 10])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0056","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96899ab29ec3e067a5669fb486da401abc597c0092aec05e1d217285d991b6b5","family_id":"qf-str22-complex-multiplication","seed":202610223056,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[-7,2],\"right\":[-2,1]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[-7, 2], right=[c,d]=[-2, 1]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([-7, 2], [-2, 1], [12, -11])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-7, 2], [-2, 1], [12, -11])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0057","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd9ef007c34b225632fd939cbd8f02c9fe1d2e44b7a03aace584707276f95d98","family_id":"qf-str22-complex-multiplication","seed":202610223057,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[-6,3],\"right\":[-1,3]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[-6, 3], right=[c,d]=[-1, 3]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([-6, 3], [-1, 3], [-3, -21])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-6, 3], [-1, 3], [-3, -21])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0058","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"930f9e2ad4d0305d87e94cdfdb1f445368f741c97e6f19353ee2bea8e85ed4c1","family_id":"qf-str22-complex-multiplication","seed":202610223058,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[-5,-3],\"right\":[0,5]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[-5, -3], right=[c,d]=[0, 5]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([-5, -3], [0, 5], [15, -25])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-5, -3], [0, 5], [15, -25])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0059","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d1badd42a008f06af36730a8ce0d304b0ea684808b4c9644055e6d231484ba8","family_id":"qf-str22-complex-multiplication","seed":202610223059,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[-4,-2],\"right\":[1,7]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[-4, -2], right=[c,d]=[1, 7]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([-4, -2], [1, 7], [10, -30])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-4, -2], [1, 7], [10, -30])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0060","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"956ab2a17706afabe5fdc6aeee0e5035804b7f1d4fb95629229b706230b49606","family_id":"qf-str22-complex-multiplication","seed":202610223060,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[-3,-1],\"right\":[2,9]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[-3, -1], right=[c,d]=[2, 9]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([-3, -1], [2, 9], [3, -29])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-3, -1], [2, 9], [3, -29])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0061","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e9a6be96e68be909a33630fd7afd7f743530ab2d4f830884cc6033edee5a6b2","family_id":"qf-str22-complex-multiplication","seed":202610223061,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[-2,0],\"right\":[-2,11]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[-2, 0], right=[c,d]=[-2, 11]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([-2, 0], [-2, 11], [4, -22])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-2, 0], [-2, 11], [4, -22])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0062","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2638571756d34f93aa51289e0bd68e41b4c733cacc055aa567d7215f76c1de38","family_id":"qf-str22-complex-multiplication","seed":202610223062,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[-1,1],\"right\":[-1,13]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[-1, 1], right=[c,d]=[-1, 13]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([-1, 1], [-1, 13], [-12, -14])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-1, 1], [-1, 13], [-12, -14])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0063","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe72a151ba78bab31a21ae8d3d8ab81e5be2be82d0d81e6622499f100e798c4e","family_id":"qf-str22-complex-multiplication","seed":202610223063,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[0,2],\"right\":[0,15]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[0, 2], right=[c,d]=[0, 15]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([0, 2], [0, 15], [-30, 0])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 2], [0, 15], [-30, 0])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0064","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c21ff26636c1a3c8b993214d7924c2af130fcf9c652525e04c096c24168a1771","family_id":"qf-str22-complex-multiplication","seed":202610223064,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[1,3],\"right\":[1,17]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[1, 3], right=[c,d]=[1, 17]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([1, 3], [1, 17], [-50, 20])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 3], [1, 17], [-50, 20])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0065","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b2db527aa571bb6afb593c002f8da08a722de24b19b131ecdc3c02ad53e4b5f","family_id":"qf-str22-complex-multiplication","seed":202610223065,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[2,-3],\"right\":[2,19]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[2, -3], right=[c,d]=[2, 19]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([2, -3], [2, 19], [61, 32])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, -3], [2, 19], [61, 32])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0066","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5062194c0ef5360a4d135c15da0d8ca12482c6dcb810b574bb7b14d38f447da8","family_id":"qf-str22-complex-multiplication","seed":202610223066,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[3,-2],\"right\":[-2,21]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[3, -2], right=[c,d]=[-2, 21]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([3, -2], [-2, 21], [36, 67])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, -2], [-2, 21], [36, 67])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0067","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce7f6f37f938039cf05f21ee23d36200cb1e328f9f1957fe59d487739c7b0911","family_id":"qf-str22-complex-multiplication","seed":202610223067,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[4,-1],\"right\":[-1,23]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[4, -1], right=[c,d]=[-1, 23]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([4, -1], [-1, 23], [19, 93])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, -1], [-1, 23], [19, 93])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0068","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cb041654d45ac0537c30129b50a404bdb1a977b0c2ad484314c3921f2123be5","family_id":"qf-str22-complex-multiplication","seed":202610223068,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[5,0],\"right\":[0,25]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[5, 0], right=[c,d]=[0, 25]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([5, 0], [0, 25], [0, 125])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 0], [0, 25], [0, 125])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0069","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d9f0b639f6b6145cab51f5007ffb456ac563979333d298e07374d7b09cf41d4","family_id":"qf-str22-complex-multiplication","seed":202610223069,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[6,1],\"right\":[1,27]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[6, 1], right=[c,d]=[1, 27]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([6, 1], [1, 27], [-21, 163])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 1], [1, 27], [-21, 163])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0070","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e622d98d0ff29eada37b83316155d040ab7a31c1f205ffd74a0e82d3ccffba5f","family_id":"qf-str22-complex-multiplication","seed":202610223070,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[7,2],\"right\":[2,29]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[7, 2], right=[c,d]=[2, 29]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([7, 2], [2, 29], [-44, 207])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 2], [2, 29], [-44, 207])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0071","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6678363ba3b6835ff2c31239af8f4acc962fcdff453cf52938a8e11c52830cc4","family_id":"qf-str22-complex-multiplication","seed":202610223071,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[8,3],\"right\":[-2,31]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[8, 3], right=[c,d]=[-2, 31]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([8, 3], [-2, 31], [-109, 242])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 3], [-2, 31], [-109, 242])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0072","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9aca7cc05e3488b292199335d88a6cfaaf4644054c9297fba4c4eb891032982","family_id":"qf-str22-complex-multiplication","seed":202610223072,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[9,-3],\"right\":[-1,33]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[9, -3], right=[c,d]=[-1, 33]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([9, -3], [-1, 33], [90, 300])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, -3], [-1, 33], [90, 300])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0073","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a17358bfc1389db54c7d1d438980618cc5820b6413933e150d078e1352a641a6","family_id":"qf-str22-complex-multiplication","seed":202610223073,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[10,-2],\"right\":[0,35]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[10, -2], right=[c,d]=[0, 35]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([10, -2], [0, 35], [70, 350])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, -2], [0, 35], [70, 350])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0074","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a731327b54429447490cd8ad3c58e850d7fbb4ee354c3b4a034cf1ad7e9977f","family_id":"qf-str22-complex-multiplication","seed":202610223074,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[11,-1],\"right\":[1,37]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[11, -1], right=[c,d]=[1, 37]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([11, -1], [1, 37], [48, 406])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, -1], [1, 37], [48, 406])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0075","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38a8cebd607ca5ae33b8fc74f11c37ebe161693e8fa1ac5383727250b7680fc3","family_id":"qf-str22-complex-multiplication","seed":202610223075,"semantic_template":"qf-str22-complex-multiplication/{\"left\":[12,0],\"right\":[2,39]}"},"fields":{"title":"The Negative Real Contribution from the Product of Imaginary Parts","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[12, 0], right=[c,d]=[2, 39]. Give (left,right,[a*c−b*d,a*d+b*c]).","answer":{"kind":"expression","value":"([12, 0], [2, 39], [24, 468])","explanation":"Distribute the product and replace i² by −1. Pairwise component multiplication is not complex multiplication. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 0], [2, 39], [24, 468])."},"skills":["The Negative Real Contribution from the Product of Imaginary Parts"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0076","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"422fa0c1d2d9f6027ea8066a154230b46ac20f53dc3d649563ea219a2ab8d8d7","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223076,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":-11,\"value\":[-12,1]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[-12, 1], real candidate=-11. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([-12, 1], -11, No, No)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-12, 1], -11, No, No)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0077","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7573dc4d7e1f080852c5de2611776ce33709dac79112b7667b4d2e25530a47f8","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223077,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":-11,\"value\":[-11,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[-11, 0], real candidate=-11. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([-11, 0], -11, Yes, Yes)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-11, 0], -11, Yes, Yes)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0078","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d26b2f589aa723cb46ebb50165bdb0589698a0c168656473bfaaf02603a1441","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223078,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":-10,\"value\":[-10,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[-10, 0], real candidate=-10. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([-10, 0], -10, Yes, Yes)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-10, 0], -10, Yes, Yes)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0079","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9eea89edfd008d804917032f32283d3cdc3e622a03cc489924cd81633b355ca3","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223079,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":-8,\"value\":[-9,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[-9, 0], real candidate=-8. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([-9, 0], -8, Yes, No)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-9, 0], -8, Yes, No)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0080","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d49b105097ffafdbb57c645a852bcb086f94ca924355434910af195c0b02e889","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223080,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":-8,\"value\":[-8,1]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[-8, 1], real candidate=-8. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([-8, 1], -8, No, No)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-8, 1], -8, No, No)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0081","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49e26e73267e35532b868a8f5574cb85273093c5e108b5d9aed1583650b7e31f","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223081,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":-7,\"value\":[-7,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[-7, 0], real candidate=-7. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([-7, 0], -7, Yes, Yes)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-7, 0], -7, Yes, Yes)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0082","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3919819bee76bef70fc19880922a0b183a9791631686d30232efc859e094571c","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223082,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":-5,\"value\":[-6,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[-6, 0], real candidate=-5. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([-6, 0], -5, Yes, No)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-6, 0], -5, Yes, No)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0083","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6820bd863ddd88506b9c386ce72b2587328f1dc7e4aa0b5294621b5d1033fed6","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223083,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":-5,\"value\":[-5,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[-5, 0], real candidate=-5. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([-5, 0], -5, Yes, Yes)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-5, 0], -5, Yes, Yes)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0084","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b079d436a608454bd19f25113e53d176106de8c69967d8b5bb09561fda66f92","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223084,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":-4,\"value\":[-4,1]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[-4, 1], real candidate=-4. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([-4, 1], -4, No, No)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-4, 1], -4, No, No)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0085","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a250fbf31183de5876c856d07edb5d94984c8e0daed834255e7df7af891eeb1f","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223085,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":-2,\"value\":[-3,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[-3, 0], real candidate=-2. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([-3, 0], -2, Yes, No)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-3, 0], -2, Yes, No)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0086","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdc2e48ee5c1d5edf24f69d89b836ca5fe970ed081efa3087683131b4b397cef","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223086,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":-2,\"value\":[-2,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[-2, 0], real candidate=-2. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([-2, 0], -2, Yes, Yes)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-2, 0], -2, Yes, Yes)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0087","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6f67e84b284c27864f90b296ebdd42249421160cc8826829efc180a6e178bc5","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223087,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":-1,\"value\":[-1,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[-1, 0], real candidate=-1. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([-1, 0], -1, Yes, Yes)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([-1, 0], -1, Yes, Yes)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0088","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a428eceb4b20e70497e3b5ba58bf827b7975fd3264b833f7e4a9f4afa195714","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223088,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":1,\"value\":[0,1]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[0, 1], real candidate=1. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([0, 1], 1, No, No)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 1], 1, No, No)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0089","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"caf4de890400386635b4245c68218f80486ddf8908181acbe5503b426e9a27d6","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223089,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":1,\"value\":[1,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[1, 0], real candidate=1. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([1, 0], 1, Yes, Yes)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 0], 1, Yes, Yes)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0090","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"713a5540e89d9d3e42a8f188dd7f0ccb15213bb04538efc3ece6fe3fd727441e","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223090,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":2,\"value\":[2,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[2, 0], real candidate=2. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([2, 0], 2, Yes, Yes)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, 0], 2, Yes, Yes)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0091","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb193587b57a397e90254f57e13310333e3310171709024d2b4f5aff172b1ba7","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223091,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":4,\"value\":[3,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[3, 0], real candidate=4. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([3, 0], 4, Yes, No)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, 0], 4, Yes, No)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0092","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a9ef4de1da162d8a88149d54f0c6a0f3b1aac9fdd01ee321ce28beda936e7ef","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223092,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":4,\"value\":[4,1]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[4, 1], real candidate=4. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([4, 1], 4, No, No)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, 1], 4, No, No)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0093","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba9c8ed3e0bad9b30e62c0a73717603e544c544d66148e1118e7c66f8179b950","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223093,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":5,\"value\":[5,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[5, 0], real candidate=5. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([5, 0], 5, Yes, Yes)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 0], 5, Yes, Yes)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0094","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ab6a99c65a3001c58ca6da8ace8746c7e1e8d86dbe9a669fa0082c063f07bec","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223094,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":7,\"value\":[6,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[6, 0], real candidate=7. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([6, 0], 7, Yes, No)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 0], 7, Yes, No)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0095","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20205a0c66f4b3a55d5e27d899e254b23f6576b097237f01509be33fb98fd50a","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223095,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":7,\"value\":[7,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[7, 0], real candidate=7. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([7, 0], 7, Yes, Yes)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 0], 7, Yes, Yes)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0096","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86452402313857271c4b350e3d46e8d28df291ec87001195bf69ca68503648a2","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223096,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":8,\"value\":[8,1]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[8, 1], real candidate=8. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([8, 1], 8, No, No)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 1], 8, No, No)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0097","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"239992a6296e454b5dd35cf7bf2b53df440d1b6616cf5dab31289c5c72441436","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223097,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":10,\"value\":[9,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[9, 0], real candidate=10. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([9, 0], 10, Yes, No)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, 0], 10, Yes, No)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0098","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fa50449e4a7c2670140cb0fb4375d4e3315919d36c74fef808f63efd9ab35e7","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223098,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":10,\"value\":[10,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[10, 0], real candidate=10. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([10, 0], 10, Yes, Yes)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, 0], 10, Yes, Yes)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0099","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3be11ab6ca195531d7afbed11f2c9fe6daa2343c0edc934b97ce203f8ab169c7","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223099,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":11,\"value\":[11,0]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[11, 0], real candidate=11. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([11, 0], 11, Yes, Yes)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, 0], 11, Yes, Yes)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r03-complex-number-0100","batch_id":"batch-20261013-advance-20261022-r03-complex-number-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c1c3cbc43ddd4fa05ff54620850b21ab602e8336639ab68d317bb7440b5df91","family_id":"qf-str22-complex-real-embedding-audit","seed":202610223100,"semantic_template":"qf-str22-complex-real-embedding-audit/{\"real_candidate\":13,\"value\":[12,1]}"},"fields":{"title":"A Complex Pair versus a Real-Embedding Candidate","stem":"Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. value=[12, 1], real candidate=13. Give (value,real candidate,is the imaginary coefficient zero?,is value exactly [candidate,0]?).","answer":{"kind":"expression","value":"([12, 1], 13, No, No)","explanation":"Being in the real embedding and matching this particular real candidate are separate checks. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 1], 13, No, No)."},"skills":["A Complex Pair versus a Real-Embedding Candidate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}