Exact Ordered-Pair Conjugation and Modulus-Squared Invariants
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batch-20261013-advance-20261023-r01-complex-modulus-conjugate-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. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-12, -3]. Give (value,conjugate).
View answer and explanation
([-12, -3], [-12, 3])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -3], [-12, 3]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
2. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-11, -2]. Give (value,conjugate).
View answer and explanation
([-11, -2], [-11, 2])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -2], [-11, 2]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
3. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-10, -1]. Give (value,conjugate).
View answer and explanation
([-10, -1], [-10, 1])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, -1], [-10, 1]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
4. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-9, 0]. Give (value,conjugate).
View answer and explanation
([-9, 0], [-9, 0])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 0], [-9, 0]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
5. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-8, 1]. Give (value,conjugate).
View answer and explanation
([-8, 1], [-8, -1])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 1], [-8, -1]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
6. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-7, 2]. Give (value,conjugate).
View answer and explanation
([-7, 2], [-7, -2])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 2], [-7, -2]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
7. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-6, 3]. Give (value,conjugate).
View answer and explanation
([-6, 3], [-6, -3])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 3], [-6, -3]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
8. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-5, -3]. Give (value,conjugate).
View answer and explanation
([-5, -3], [-5, 3])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, -3], [-5, 3]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
9. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-4, -2]. Give (value,conjugate).
View answer and explanation
([-4, -2], [-4, 2])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, -2], [-4, 2]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
10. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-3, -1]. Give (value,conjugate).
View answer and explanation
([-3, -1], [-3, 1])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, -1], [-3, 1]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
11. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-2, 0]. Give (value,conjugate).
View answer and explanation
([-2, 0], [-2, 0])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 0], [-2, 0]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
12. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-1, 1]. Give (value,conjugate).
View answer and explanation
([-1, 1], [-1, -1])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 1], [-1, -1]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
13. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[0, 2]. Give (value,conjugate).
View answer and explanation
([0, 2], [0, -2])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2], [0, -2]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
14. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[1, 3]. Give (value,conjugate).
View answer and explanation
([1, 3], [1, -3])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 3], [1, -3]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
15. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[2, -3]. Give (value,conjugate).
View answer and explanation
([2, -3], [2, 3])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, -3], [2, 3]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
16. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[3, -2]. Give (value,conjugate).
View answer and explanation
([3, -2], [3, 2])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2], [3, 2]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
17. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[4, -1]. Give (value,conjugate).
View answer and explanation
([4, -1], [4, 1])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1], [4, 1]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
18. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[5, 0]. Give (value,conjugate).
View answer and explanation
([5, 0], [5, 0])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0], [5, 0]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
19. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[6, 1]. Give (value,conjugate).
View answer and explanation
([6, 1], [6, -1])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1], [6, -1]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
20. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[7, 2]. Give (value,conjugate).
View answer and explanation
([7, 2], [7, -2])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2], [7, -2]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
21. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[8, 3]. Give (value,conjugate).
View answer and explanation
([8, 3], [8, -3])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([8, 3], [8, -3]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
22. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[9, -3]. Give (value,conjugate).
View answer and explanation
([9, -3], [9, 3])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([9, -3], [9, 3]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
23. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[10, -2]. Give (value,conjugate).
View answer and explanation
([10, -2], [10, 2])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([10, -2], [10, 2]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
24. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[11, -1]. Give (value,conjugate).
View answer and explanation
([11, -1], [11, 1])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([11, -1], [11, 1]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
25. Conjugation of an Ordered Complex Pair
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[12, 0]. Give (value,conjugate).
View answer and explanation
([12, 0], [12, 0])
Keep the real part and negate exactly the imaginary coefficient. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([12, 0], [12, 0]).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugate · candidate
26. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-12, -3]. Give (value,modulus squared).
View answer and explanation
([-12, -3], 153)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -3], 153).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
27. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-11, -2]. Give (value,modulus squared).
View answer and explanation
([-11, -2], 125)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -2], 125).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
28. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-10, -1]. Give (value,modulus squared).
View answer and explanation
([-10, -1], 101)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, -1], 101).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
29. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-9, 0]. Give (value,modulus squared).
View answer and explanation
([-9, 0], 81)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 0], 81).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
30. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-8, 1]. Give (value,modulus squared).
View answer and explanation
([-8, 1], 65)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 1], 65).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
31. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-7, 2]. Give (value,modulus squared).
View answer and explanation
([-7, 2], 53)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 2], 53).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
32. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-6, 3]. Give (value,modulus squared).
View answer and explanation
([-6, 3], 45)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 3], 45).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
33. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-5, -3]. Give (value,modulus squared).
View answer and explanation
([-5, -3], 34)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, -3], 34).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
34. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-4, -2]. Give (value,modulus squared).
View answer and explanation
([-4, -2], 20)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, -2], 20).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
35. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-3, -1]. Give (value,modulus squared).
View answer and explanation
([-3, -1], 10)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, -1], 10).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
36. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-2, 0]. Give (value,modulus squared).
View answer and explanation
([-2, 0], 4)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 0], 4).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
37. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-1, 1]. Give (value,modulus squared).
View answer and explanation
([-1, 1], 2)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 1], 2).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
38. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[0, 2]. Give (value,modulus squared).
View answer and explanation
([0, 2], 4)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2], 4).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
39. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[1, 3]. Give (value,modulus squared).
View answer and explanation
([1, 3], 10)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 3], 10).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
40. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[2, -3]. Give (value,modulus squared).
View answer and explanation
([2, -3], 13)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, -3], 13).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
41. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[3, -2]. Give (value,modulus squared).
View answer and explanation
([3, -2], 13)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2], 13).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
42. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[4, -1]. Give (value,modulus squared).
View answer and explanation
([4, -1], 17)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1], 17).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
43. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[5, 0]. Give (value,modulus squared).
View answer and explanation
([5, 0], 25)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0], 25).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
44. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[6, 1]. Give (value,modulus squared).
View answer and explanation
([6, 1], 37)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1], 37).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
45. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[7, 2]. Give (value,modulus squared).
View answer and explanation
([7, 2], 53)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2], 53).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
46. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[8, 3]. Give (value,modulus squared).
View answer and explanation
([8, 3], 73)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([8, 3], 73).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
47. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[9, -3]. Give (value,modulus squared).
View answer and explanation
([9, -3], 90)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([9, -3], 90).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
48. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[10, -2]. Give (value,modulus squared).
View answer and explanation
([10, -2], 104)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([10, -2], 104).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
49. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[11, -1]. Give (value,modulus squared).
View answer and explanation
([11, -1], 122)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([11, -1], 122).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
50. Exact Modulus Squared without an Unrequested Root
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[12, 0]. Give (value,modulus squared).
View answer and explanation
([12, 0], 144)
Add the two integer squares. This is the squared modulus, not the modulus itself. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([12, 0], 144).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-modulus-square · candidate
51. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-12, -3]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([-12, -3], [-12, 3], [153, 0], 153)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -3], [-12, 3], [153, 0], 153).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
52. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-11, -2]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([-11, -2], [-11, 2], [125, 0], 125)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -2], [-11, 2], [125, 0], 125).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
53. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-10, -1]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([-10, -1], [-10, 1], [101, 0], 101)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, -1], [-10, 1], [101, 0], 101).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
54. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-9, 0]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([-9, 0], [-9, 0], [81, 0], 81)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 0], [-9, 0], [81, 0], 81).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
55. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-8, 1]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([-8, 1], [-8, -1], [65, 0], 65)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 1], [-8, -1], [65, 0], 65).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
56. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-7, 2]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([-7, 2], [-7, -2], [53, 0], 53)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 2], [-7, -2], [53, 0], 53).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
57. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-6, 3]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([-6, 3], [-6, -3], [45, 0], 45)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 3], [-6, -3], [45, 0], 45).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
58. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-5, -3]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([-5, -3], [-5, 3], [34, 0], 34)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, -3], [-5, 3], [34, 0], 34).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
59. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-4, -2]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([-4, -2], [-4, 2], [20, 0], 20)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, -2], [-4, 2], [20, 0], 20).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
60. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-3, -1]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([-3, -1], [-3, 1], [10, 0], 10)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, -1], [-3, 1], [10, 0], 10).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
61. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-2, 0]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([-2, 0], [-2, 0], [4, 0], 4)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 0], [-2, 0], [4, 0], 4).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
62. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[-1, 1]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([-1, 1], [-1, -1], [2, 0], 2)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 1], [-1, -1], [2, 0], 2).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
63. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[0, 2]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([0, 2], [0, -2], [4, 0], 4)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2], [0, -2], [4, 0], 4).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
64. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[1, 3]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([1, 3], [1, -3], [10, 0], 10)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 3], [1, -3], [10, 0], 10).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
65. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[2, -3]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([2, -3], [2, 3], [13, 0], 13)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, -3], [2, 3], [13, 0], 13).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
66. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[3, -2]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([3, -2], [3, 2], [13, 0], 13)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2], [3, 2], [13, 0], 13).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
67. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[4, -1]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([4, -1], [4, 1], [17, 0], 17)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1], [4, 1], [17, 0], 17).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
68. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[5, 0]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([5, 0], [5, 0], [25, 0], 25)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0], [5, 0], [25, 0], 25).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
69. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[6, 1]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([6, 1], [6, -1], [37, 0], 37)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1], [6, -1], [37, 0], 37).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
70. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[7, 2]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([7, 2], [7, -2], [53, 0], 53)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2], [7, -2], [53, 0], 53).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
71. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[8, 3]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([8, 3], [8, -3], [73, 0], 73)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([8, 3], [8, -3], [73, 0], 73).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
72. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[9, -3]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([9, -3], [9, 3], [90, 0], 90)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([9, -3], [9, 3], [90, 0], 90).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
73. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[10, -2]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([10, -2], [10, 2], [104, 0], 104)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([10, -2], [10, 2], [104, 0], 104).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
74. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[11, -1]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([11, -1], [11, 1], [122, 0], 122)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([11, -1], [11, 1], [122, 0], 122).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
75. The Product with the Conjugate Is a Real Embedding
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. value=[12, 0]. Give (value,conjugate,product as a complex pair,modulus squared).
View answer and explanation
([12, 0], [12, 0], [144, 0], 144)
Multiply the ordered pairs and verify the imaginary coefficient cancels while the real part is the sum of squares. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([12, 0], [12, 0], [144, 0], 144).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-product-conjugate · candidate
76. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-12, -3], right=[-2, -9]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([-12, -3], [-2, -9], [-3, 114], [-3, -114], [-3, -114], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -3], [-2, -9], [-3, 114], [-3, -114], [-3, -114], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
77. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-11, -2], right=[-1, -7]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([-11, -2], [-1, -7], [-3, 79], [-3, -79], [-3, -79], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -2], [-1, -7], [-3, 79], [-3, -79], [-3, -79], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
78. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-10, -1], right=[0, -5]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([-10, -1], [0, -5], [-5, 50], [-5, -50], [-5, -50], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-10, -1], [0, -5], [-5, 50], [-5, -50], [-5, -50], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
79. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-9, 0], right=[1, -3]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([-9, 0], [1, -3], [-9, 27], [-9, -27], [-9, -27], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 0], [1, -3], [-9, 27], [-9, -27], [-9, -27], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
80. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-8, 1], right=[2, -1]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([-8, 1], [2, -1], [-15, 10], [-15, -10], [-15, -10], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 1], [2, -1], [-15, 10], [-15, -10], [-15, -10], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
81. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-7, 2], right=[-2, 1]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([-7, 2], [-2, 1], [12, -11], [12, 11], [12, 11], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 2], [-2, 1], [12, -11], [12, 11], [12, 11], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
82. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-6, 3], right=[-1, 3]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([-6, 3], [-1, 3], [-3, -21], [-3, 21], [-3, 21], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 3], [-1, 3], [-3, -21], [-3, 21], [-3, 21], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
83. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-5, -3], right=[0, 5]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([-5, -3], [0, 5], [15, -25], [15, 25], [15, 25], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-5, -3], [0, 5], [15, -25], [15, 25], [15, 25], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
84. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-4, -2], right=[1, 7]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([-4, -2], [1, 7], [10, -30], [10, 30], [10, 30], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-4, -2], [1, 7], [10, -30], [10, 30], [10, 30], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
85. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-3, -1], right=[2, 9]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([-3, -1], [2, 9], [3, -29], [3, 29], [3, 29], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-3, -1], [2, 9], [3, -29], [3, 29], [3, 29], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
86. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-2, 0], right=[-2, 11]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([-2, 0], [-2, 11], [4, -22], [4, 22], [4, 22], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 0], [-2, 11], [4, -22], [4, 22], [4, 22], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
87. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[-1, 1], right=[-1, 13]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([-1, 1], [-1, 13], [-12, -14], [-12, 14], [-12, 14], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 1], [-1, 13], [-12, -14], [-12, 14], [-12, 14], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
88. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[0, 2], right=[0, 15]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([0, 2], [0, 15], [-30, 0], [-30, 0], [-30, 0], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2], [0, 15], [-30, 0], [-30, 0], [-30, 0], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
89. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[1, 3], right=[1, 17]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([1, 3], [1, 17], [-50, 20], [-50, -20], [-50, -20], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([1, 3], [1, 17], [-50, 20], [-50, -20], [-50, -20], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
90. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[2, -3], right=[2, 19]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([2, -3], [2, 19], [61, 32], [61, -32], [61, -32], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([2, -3], [2, 19], [61, 32], [61, -32], [61, -32], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
91. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[3, -2], right=[-2, 21]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([3, -2], [-2, 21], [36, 67], [36, -67], [36, -67], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2], [-2, 21], [36, 67], [36, -67], [36, -67], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
92. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[4, -1], right=[-1, 23]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([4, -1], [-1, 23], [19, 93], [19, -93], [19, -93], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1], [-1, 23], [19, 93], [19, -93], [19, -93], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
93. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[5, 0], right=[0, 25]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([5, 0], [0, 25], [0, 125], [0, -125], [0, -125], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0], [0, 25], [0, 125], [0, -125], [0, -125], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
94. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[6, 1], right=[1, 27]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([6, 1], [1, 27], [-21, 163], [-21, -163], [-21, -163], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1], [1, 27], [-21, 163], [-21, -163], [-21, -163], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
95. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[7, 2], right=[2, 29]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([7, 2], [2, 29], [-44, 207], [-44, -207], [-44, -207], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2], [2, 29], [-44, 207], [-44, -207], [-44, -207], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
96. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[8, 3], right=[-2, 31]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([8, 3], [-2, 31], [-109, 242], [-109, -242], [-109, -242], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([8, 3], [-2, 31], [-109, 242], [-109, -242], [-109, -242], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
97. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[9, -3], right=[-1, 33]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([9, -3], [-1, 33], [90, 300], [90, -300], [90, -300], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([9, -3], [-1, 33], [90, 300], [90, -300], [90, -300], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
98. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[10, -2], right=[0, 35]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([10, -2], [0, 35], [70, 350], [70, -350], [70, -350], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([10, -2], [0, 35], [70, 350], [70, -350], [70, -350], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
99. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[11, -1], right=[1, 37]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([11, -1], [1, 37], [48, 406], [48, -406], [48, -406], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([11, -1], [1, 37], [48, 406], [48, -406], [48, -406], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
100. Conjugating a Product versus Multiplying the Conjugates
Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[12, 0], right=[2, 39]. Give (left,right,product,conjugate of product,product of conjugates,equal?).
View answer and explanation
([12, 0], [2, 39], [24, 468], [24, -468], [24, -468], Yes)
Compute both expressions in the declared algebra; retain every intermediate pair and its order. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is ([12, 0], [2, 39], [24, 468], [24, -468], [24, -468], Yes).
Reviewed step-by-step derivation
Complex modulus and conjugation · qf-mod23-complexmod-conjugation-product · candidate
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