Euler's equation

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This visual proof of the equation \( e^{i\pi} + 1 = 0 \) demonstrates how the most remarkable formula in mathematics — Leonhard Euler’s identity — emerges from the relationship between exponentials, complex numbers, and trigonometric functions. Follow each step below to explore its elegance and meaning.

\[\begin{aligned} &\text { Let } f(x)=\frac{\cos x+i \sin x}{e^{i x}}\\\\ &\Rightarrow f^{\prime}(x)=\frac{\left(i \cos x-\sin x) e^{i x}-i e^{i x}(\cos x+i \sin x)\right.}{e^{2 i x}}\\\\ &=\frac{e^{i x}(i \cos x-\sin x-i \cos x+\sin x)}{e^{2 i x}}\\\\ &f^{\prime}(x)=0\\\\ &\Rightarrow f \text { is constant }\\\\ &\text { then } f(x)=f(0)\\\\ &\Leftrightarrow \frac{\cos x+i \sin x}{e^{i x}}=1\\\\ &\text { thus } e^{i x}=\cos x+i \sin x \end{aligned}\]

\[e^{i x}=\cos x+i \sin x \]

Euler’s equation isn’t just a coincidence — it’s a perfect example of how mathematics unifies different ideas into one harmonious truth. If you enjoyed exploring this proof, take a look at other solved exercises on our site to discover more elegant mathematical connections and visual explanations.

Finally, don’t forget to explore more solved exercises to discover similar tricks and their logical explanations.

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