So many people in the replies here are totally missing the point. Yes, there is an infinite number of valid possible answers for i^i. But EVERY SINGLE ONE OF THEM IS A REAL NUMBER.

So many people in the replies here are totally missing the point. Yes, there is an infinite number of valid possible answers for i^i. But EVERY SINGLE ONE OF THEM IS A REAL NUMBER.

https://x.com/mathladyhazel/status/1800905395961241811

I think about it like this:

i is a number in 2D, and some operations on it end up in one axis (the real axis in this case). If you visualize it, it is easier. Although you started with a number or a vector on 2D, this specific operation lands on 1D always

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idanlevin.xyz

i^x = rotation (counterclockwise)
^i = "take the operation you were doing, and rotate *that operation itself* by another 90 degrees counterclockwise

Hence i^i = move along the vector going counterclockwise along the circle, rotated another 90 degrees = move along the vector going direct toward the origin