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

Can any giga brain explain me what the math is? 🥲

IRL: Creative Director who makes movies and ads.

Are you looking for an explanation that shows the algebraic steps to prove it? Or just, the idea in general, without math stuff shown?

Mathematics/Design linktr.ee/sinusoidalsnail
⟢⟢ᚖ⟡⟡ᚖ⟡⟡ᚖ⟡⟡ᚖ⟡⟡ᚖ⟡⟡ᚖ⟡⟡ᚖ⟣⟣

Okay! I'll try lol. Will get back to you in a bit.

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

Hmmm I tried for quite awhile, but couldn’t come up with a clear way of describing it conceptually. I kept falling back on showing it algebraically lol, sorry.

But now I see that Vitalik has explained it in the comments in a very nice, clear, concise way (quoted below).

We can think of the number 𝑖 as an action: a rotation by 90°.
And we can think of raising something to the power of 𝑖 as a rotating action upon that action: a 90° rotation of the act of rotating 90°.

I think the reason it’s not intuitive, is because most of us learned that “exponentiation is repeated multiplication.” When that’s an oversimplification that doesn’t work once we get to complex numbers.

Sorry I couldn’t come up with a better explanation! But I think Vitalik’s nails it perfectly anyway!

Hmmm I tried for quite awhile, but couldn’t come up with a clear way of describing it conceptually. I kept falling back on showing it algebraically lol, sorry.

But now I see that Vitalik has explained it in the comments in a very nice, clear, concise way (quoted below).

We can think of the number 𝑖 as an action: a rotation by 90°.
And we can think of raising something to the power of 𝑖 as a rotating action upon that action: a 90° rotation of the act of rotating 90°.

I think the reason it’s not intuitive, is because most of us learned that “exponentiation is repeated multiplication.” When that’s an oversimplification that doesn’t work once we get to complex numbers.

Sorry I couldn’t come up with a better explanation! But I think Vitalik’s nails it perfectly anyway!

https://warpcast.com/vitalik.eth/0xd941680b