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Euler's formula states e^(iθ) = cos θ + i sin θ, connecting exponentials to rotation. Setting θ = π gives Euler's identity e^(iπ) + 1 = 0, tying together e, i, π, 1 and 0.
Euler's formula
e^(iθ) = cos θ + i·sin θ
Euler's identity
e^(iπ) + 1 = 0
cos 180° + i sin 180° = −1 exactly, which is Euler's identity.
The power series for e^(iθ) splits into the cosine and sine series, so multiplying by i repeatedly cycles through the four directions.