Which fundamental relationship does e^{iπ} illustrate a special case of?

Which fundamental relationship does e^{iπ} illustrate a special case of?

• A. Pythagorean theorem
• B. Euler's formula
• C. De Moivre's theorem
• D. Cauchy-Riemann equations

Answer: (B)

At the heart is Euler's formula, which links complex exponentials to trigonometric functions. For any real angle θ, e^{iθ} equals cos θ plus i sin θ. Setting θ = π gives e^{iπ} = cos π + i sin π = -1, so the exponential at angle π lands exactly at -1 on the real axis. This shows how the exponential function encodes rotation on the complex plane, precisely what Euler's formula describes. De Moivre's theorem is related and follows from this idea when you raise the exponential to a power, but the direct relationship shown by e^{iπ} is Euler's formula. The other concepts don’t describe this specific bridge between exponential form and circular trigonometry.

At the heart is Euler's formula, which links complex exponentials to trigonometric functions. For any real angle θ, e^{iθ} equals cos θ plus i sin θ. Setting θ = π gives e^{iπ} = cos π + i sin π = -1, so the exponential at angle π lands exactly at -1 on the real axis. This shows how the exponential function encodes rotation on the complex plane, precisely what Euler's formula describes. De Moivre's theorem is related and follows from this idea when you raise the exponential to a power, but the direct relationship shown by e^{iπ} is Euler's formula. The other concepts don’t describe this specific bridge between exponential form and circular trigonometry.