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E to the pi i equals negative one because of a string of properties. Namely logarithmic properties. When you take the natural log of both sides you are able to bring the exponent of pi and i down. And replace the ln of e with 1. When you keep the natural log of negative 1 you get 3.145....i or pi times i. To understand this natural log place you must go back to the original equation E^pi i. When you take the i root of both sides you get e to the pi alone, and have the i root of negative one. By doing this you are just raising negative 1 to the inverse of i, negative i. When you do this you end up with 23.14069263....... because of derivatives. Guess what e to the pi is. you guessed right 23.14069263. This is commonly known as Eulers's identity. If you want the more in depth understanding Google it, or bing whatever you prefer.

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Q: Why does E to the pi i equal negative one?
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