Why “Bigger” Makes No Sense for πⁱ and i^π

Why can’t we say which is “bigger” — πⁱ or i^π?
In the real world, exponentiation is the engine of growth: more power means more size.
But once we step into the complex plane, that intuition completely breaks.

In this video, we explore why “bigger” makes no sense for the strange powers πⁱ and i^π — and how Euler’s formula turns the impossible into something beautifully geometric.

Using the identities

aᵇ = e^{b·ln(a)}
and
e^{iθ} = cosθ + i sinθ,
we convert each expression into a rotation on the unit circle, revealing that both πⁱ and i^π have magnitude 1, even though they travel very different rotational paths.

🔍 What you’ll learn
• Why exponentiation behaves differently in the complex plane
• How to compute the magnitude and angle of πⁱ
• Why i^π rotates by π²/2 radians
• How both results end up on the unit circle
• Why comparing them using “bigger/smaller” simply doesn’t apply

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