For those who don’t know, and correct me if I’m wrong, Euler’s Formula is e^(xi) = cos(x) + isin(x) and Euler’s Identity is where “x” is equal to pi, therefore e^(πi) = -1. It’s fascinating to me how something so complex can be equal to something so simple (talking about Euler’s Identity). But what I don’t get is how they’re practical. What are they used for, and why are they important?
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To give a deep mathematical application as well:
A number a is called _algebraic_ if there are integers (or rationals, by clearing denominators) c0, …, cn such that
cn·α^n + c(n-1)·α^(n-1) + … + c1·α + c0 = 0.
For example α=sqrt(2) is algebraic because α^2 – 2 = 0. As are the golden ratio, the silver ratio, 0, 1, 65537/257 or any other rational number. A number that is not algebraic is called _transcendental_. Very sloppily, those are the numbers that are the most complictaed from the point of view of Algebra.
**Lindemann’s Theorem** now states that **if α is algebraic and not 0, then e^α is transcendental!**
Now, we know that e^(i·π) = -1 is as algebraic as it gets, thus there is no way i·π is algebraic. But a short argument then shows that this means that π cannot be algebraic either.
tl;dr: Euler’s identity is the reason why π is not the root of any rational polynomial, i.e. **π is transcendental**.
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