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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e-to-the-pi-i-equals-minus-one is just a pretty little consequence of the real “power tool”.
The real power is in exp(xi) = cos(x) + i*sin(x). This let’s you replace anything to do with rotation or oscillation with a simple algebraic formula.
Everything from the injector timings of an Internal Combustion Engine controlled dynamically by computer to alternating electrical current modelling is way way simpler this way.
Here’s a simple test: imagine you threw a hammer (eg. Thor’s Mijonir) upwards at 45-degress at 10 m/s and want to know if the head of the hammer or the handle will hit the ground? There’s an angular rate of rotation to consider! Try doing that with just cos() and sin()!!!
With exp(rateOfRotation * i * time) + ThorHeight you can solve it very very quickly and easily.
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