# Ordered Integer-Pair Complex Arithmetic and Real Embedding

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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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r03-complex-number-0100/)

Complex number · qf-str22-complex-real-embedding-audit · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261022-r03-complex-number-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261022-r03-complex-number-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261022-r03-complex-number-foundations/validation-report.json)