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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You have several things in that formula:
* cos and sin are trigonometric functions, related to the circle, related to angles, very useful for geometry and also for “repeating patterns” a.k.a waves, frequencies, etc.
* i indicates complex numbers, and complex numbers can be used to represent “things” that may have two dimensions, such as vectors or, again, waves or frequencies. Physics and engineering use vectors a lot.
* e and pi are “special” numbers. Pi obviously relates to the circle, but also (again) to frequencies and waves, and e is used a lot in probability and statistics, as well as in “things that repeat” a certain way like arithmetic and geometric series.
So the applications can be broad. For example, the formula is used quite a bit for AC / power electricity calculations.
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