Multiplying by i is a 90-degree rotation, nothing more. Every complex multiplication fits in one sentence: moduli multiply while arguments add, so multiplying is scaling and turning at once. Euler’s formula e^(iθ) = cos θ + i sin θ says exactly why this happens, since adding exponents is adding angles. From there the powers z, z², z³… thread themselves into a logarithmic spiral, while the n solutions of zⁿ = 1 space themselves evenly around the unit circle as a regular n-gon.
Multiplying by i is a 90-degree rotation, nothing more. Every complex multiplication fits in one sentence: moduli multiply while arguments add, so multiplying is scaling and turning at once. Euler’s formula e^(iθ) = cos θ + i sin θ says exactly why this happens, since adding exponents is adding angles. From there the powers z, z², z³… thread themselves into a logarithmic spiral, while the n solutions of zⁿ = 1 space themselves evenly around the unit circle as a regular n-gon.
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