Ordered Integer-Pair Complex Arithmetic and Real Embedding
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261022-r03-complex-number-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Ordered-Pair Complex Addition
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]).
View answer and explanation
([-12, -3], [-2, -9], [-14, -12])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
2. Ordered-Pair Complex Addition
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]).
View answer and explanation
([-11, -2], [-1, -7], [-12, -9])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
3. Ordered-Pair Complex Addition
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]).
View answer and explanation
([-10, -1], [0, -5], [-10, -6])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
4. Ordered-Pair Complex Addition
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]).
View answer and explanation
([-9, 0], [1, -3], [-8, -3])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
5. Ordered-Pair Complex Addition
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]).
View answer and explanation
([-8, 1], [2, -1], [-6, 0])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
6. Ordered-Pair Complex Addition
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]).
View answer and explanation
([-7, 2], [-2, 1], [-9, 3])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
7. Ordered-Pair Complex Addition
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]).
View answer and explanation
([-6, 3], [-1, 3], [-7, 6])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
8. Ordered-Pair Complex Addition
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]).
View answer and explanation
([-5, -3], [0, 5], [-5, 2])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
9. Ordered-Pair Complex Addition
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]).
View answer and explanation
([-4, -2], [1, 7], [-3, 5])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
10. Ordered-Pair Complex Addition
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]).
View answer and explanation
([-3, -1], [2, 9], [-1, 8])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
11. Ordered-Pair Complex Addition
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]).
View answer and explanation
([-2, 0], [-2, 11], [-4, 11])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
12. Ordered-Pair Complex Addition
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]).
View answer and explanation
([-1, 1], [-1, 13], [-2, 14])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
13. Ordered-Pair Complex Addition
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]).
View answer and explanation
([0, 2], [0, 15], [0, 17])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
14. Ordered-Pair Complex Addition
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]).
View answer and explanation
([1, 3], [1, 17], [2, 20])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
15. Ordered-Pair Complex Addition
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]).
View answer and explanation
([2, -3], [2, 19], [4, 16])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
16. Ordered-Pair Complex Addition
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]).
View answer and explanation
([3, -2], [-2, 21], [1, 19])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
17. Ordered-Pair Complex Addition
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]).
View answer and explanation
([4, -1], [-1, 23], [3, 22])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
18. Ordered-Pair Complex Addition
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]).
View answer and explanation
([5, 0], [0, 25], [5, 25])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
19. Ordered-Pair Complex Addition
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]).
View answer and explanation
([6, 1], [1, 27], [7, 28])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
20. Ordered-Pair Complex Addition
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]).
View answer and explanation
([7, 2], [2, 29], [9, 31])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
21. Ordered-Pair Complex Addition
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]).
View answer and explanation
([8, 3], [-2, 31], [6, 34])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
22. Ordered-Pair Complex Addition
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]).
View answer and explanation
([9, -3], [-1, 33], [8, 30])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
23. Ordered-Pair Complex Addition
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]).
View answer and explanation
([10, -2], [0, 35], [10, 33])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
24. Ordered-Pair Complex Addition
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]).
View answer and explanation
([11, -1], [1, 37], [12, 36])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
25. Ordered-Pair Complex Addition
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]).
View answer and explanation
([12, 0], [2, 39], [14, 39])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-addition · candidate
26. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([-12, -3], [-2, -9], [-10, 6])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
27. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([-11, -2], [-1, -7], [-10, 5])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
28. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([-10, -1], [0, -5], [-10, 4])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
29. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([-9, 0], [1, -3], [-10, 3])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
30. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([-8, 1], [2, -1], [-10, 2])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
31. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([-7, 2], [-2, 1], [-5, 1])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
32. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([-6, 3], [-1, 3], [-5, 0])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
33. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([-5, -3], [0, 5], [-5, -8])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
34. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([-4, -2], [1, 7], [-5, -9])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
35. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([-3, -1], [2, 9], [-5, -10])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
36. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([-2, 0], [-2, 11], [0, -11])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
37. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([-1, 1], [-1, 13], [0, -12])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
38. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([0, 2], [0, 15], [0, -13])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
39. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([1, 3], [1, 17], [0, -14])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
40. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([2, -3], [2, 19], [0, -22])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
41. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([3, -2], [-2, 21], [5, -23])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
42. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([4, -1], [-1, 23], [5, -24])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
43. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([5, 0], [0, 25], [5, -25])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
44. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([6, 1], [1, 27], [5, -26])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
45. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([7, 2], [2, 29], [5, -27])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
46. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([8, 3], [-2, 31], [10, -28])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
47. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([9, -3], [-1, 33], [10, -36])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
48. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([10, -2], [0, 35], [10, -37])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
49. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([11, -1], [1, 37], [10, -38])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
50. Ordered-Pair Complex Subtraction
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]).
View answer and explanation
([12, 0], [2, 39], [10, -39])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-subtraction · candidate
51. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([-12, -3], [-2, -9], [-3, 114])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
52. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([-11, -2], [-1, -7], [-3, 79])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
53. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([-10, -1], [0, -5], [-5, 50])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
54. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([-9, 0], [1, -3], [-9, 27])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
55. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([-8, 1], [2, -1], [-15, 10])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
56. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([-7, 2], [-2, 1], [12, -11])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
57. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([-6, 3], [-1, 3], [-3, -21])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
58. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([-5, -3], [0, 5], [15, -25])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
59. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([-4, -2], [1, 7], [10, -30])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
60. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([-3, -1], [2, 9], [3, -29])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
61. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([-2, 0], [-2, 11], [4, -22])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
62. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([-1, 1], [-1, 13], [-12, -14])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
63. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([0, 2], [0, 15], [-30, 0])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
64. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([1, 3], [1, 17], [-50, 20])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
65. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([2, -3], [2, 19], [61, 32])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
66. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([3, -2], [-2, 21], [36, 67])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
67. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([4, -1], [-1, 23], [19, 93])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
68. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([5, 0], [0, 25], [0, 125])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
69. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([6, 1], [1, 27], [-21, 163])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
70. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([7, 2], [2, 29], [-44, 207])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
71. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([8, 3], [-2, 31], [-109, 242])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
72. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([9, -3], [-1, 33], [90, 300])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
73. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([10, -2], [0, 35], [70, 350])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
74. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([11, -1], [1, 37], [48, 406])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
75. The Negative Real Contribution from the Product of Imaginary Parts
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]).
View answer and explanation
([12, 0], [2, 39], [24, 468])
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]).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-multiplication · candidate
76. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([-12, 1], -11, No, No)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
77. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([-11, 0], -11, Yes, Yes)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
78. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([-10, 0], -10, Yes, Yes)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
79. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([-9, 0], -8, Yes, No)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
80. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([-8, 1], -8, No, No)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
81. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([-7, 0], -7, Yes, Yes)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
82. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([-6, 0], -5, Yes, No)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
83. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([-5, 0], -5, Yes, Yes)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
84. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([-4, 1], -4, No, No)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
85. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([-3, 0], -2, Yes, No)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
86. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([-2, 0], -2, Yes, Yes)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
87. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([-1, 0], -1, Yes, Yes)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
88. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([0, 1], 1, No, No)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
89. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([1, 0], 1, Yes, Yes)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
90. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([2, 0], 2, Yes, Yes)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
91. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([3, 0], 4, Yes, No)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
92. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([4, 1], 4, No, No)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
93. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([5, 0], 5, Yes, Yes)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
94. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([6, 0], 7, Yes, No)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
95. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([7, 0], 7, Yes, Yes)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
96. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([8, 1], 8, No, No)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
97. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([9, 0], 10, Yes, No)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
98. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([10, 0], 10, Yes, Yes)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
99. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([11, 0], 11, Yes, Yes)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
100. A Complex Pair versus a Real-Embedding Candidate
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]?).
View answer and explanation
([12, 1], 13, No, No)
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).
Reviewed step-by-step derivation
Complex number · qf-str22-complex-real-embedding-audit · candidate
Canonical question JSONL · English JSONL · Validation report