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Redbelly98
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#12
Nov7-10, 09:06 AM
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Quote Quote by D H View Post
There are four solutions if A and B are restricted to the reals. If A and B can be complex numbers or vectors in RN, N>1, the number of solutions is infinite.
Agreed. But we haven't found all of the complex-valued solutions. Jimmy's solution did not include skeptic2's or my answers.

Spoiler
Geometric argument:

In the complex plane, the numbers 0, A, and A+B form the vertices of a triangle. The vertex at A+B must be a distance √2 from the origin, while the two shorter sides' lengths must add up to 2.

For a fixed value of A+B, say √2 + 0i, the possible values of A forms an ellipse with foci at 0 and A+B, and major axis of length 2.

Furthermore, any solution pair can be multiplied by e to obtain all possible solutions.
Uh, and θ must be real.