Euler's identity, mathematical beauty and applications of it

[audire]

Learned this identity a year ago randomly studying for adv biomechanics and was wondering if there were real-world applications for this outside of mathematicians appreciating the formula.

 

[axmls]

Well, it's not that Euler's identity in and of itself is particularly useful (though it is quite beautiful). What's useful is where it comes from--Euler's formula:
$$e^{ix} = \cos x + i \sin x.$$

The exponential is sort of a way of representing polar coordinates (or at least, that's how a former professor of mine taught it).

But it does allow us to prove a couple of cool things. For instance, I don't like to memorize some of the trig identities, because it's easy to derive them from this formula. For instance, if you want to know what $$\cos(a + b)$$ is, you can simply look at $$e^{i(a + b)}$$ so we have
$$e^{i(a + b)} = \cos(a + b) + i \sin(a + b) = e^{i a} e^{i b} = (\cos(a) + i \sin(a))(\cos(b) + i \sin(b))$$ $$= \cos(a) \cos(b) - \sin(a) \sin(b) + i(\cos(a) \sin(b) + \sin(a) \cos(b))$$
Now take the real part to get

$$\cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b).$$

So it's very useful to this sort of bridge between trigonometry and analysis. In my electrical engineering courses, we've used Euler's formula to derive a number of important equations. Euler's identity is just a wonderful side note to Euler's formula.

 

[RaulTheUCSCSlug]

So it's very useful to this sort of bridge between trigonometry and analysis. In my electrical engineering courses, we've used Euler's formula to derive a number of important equations. Euler's identity is just a wonderful side note to Euler's formula.

Definitely agree with axmls on this one, it is also very important in physics as it will be used to explain some of the quantum nature of light and other things.

 

[audire]

Thanks for the responses, just on a personal curious level, how often does the formula or identity come up in nature? The simplified Euler's identity version and not the formula just seems a very elegant and simple way to describe something that must have taken a great amount of rigor to reach.

I understand concepts of expontential growth, pie and obviously 1 and 0 since they appear a lot in nature but to link all 4 to i, a concept which is truly foreign to me outside of solving equations in algebra 3 and trig seems to just be a piece of brilliant work.

I've formally only taken university math level classes up to trig as I believe that in kinesiology and in macro-level momentum, velocity, speed, movement, etc you would only need to trigulate a position but i now believe this to be a naive concept and was wondering how math "beyond calc" could be applied to such scenarios.

 

[axmls]

The identity I don't believe comes up in a large way anywhere. It's just a special case of Euler's formula where $x = \pi$.

The remarkableness of the formula is that it contains addition, exponentiation, multiplication, two irrational numbers (and two very important ones at that), the imaginary unit, 0, and 1. And it's mighty nice to look at. That's about it, though. There really aren't many applications for an equality like this. The applications come from the general case, which actually helps us a lot in various areas of engineering and physics.

The usefulness doesn't come from the fancy numbers in it, but from the way it connects two important areas of mathematics.

Knowing that $\sin^2 ((17.8392)\pi^{e + \int_{1} ^7 5 - \log(\sqrt{14}) \ dx}+8/\phi) + \cos^2 ((17.8392)\pi^{e + \int_{1} ^7 5 - \log(\sqrt{14}) \ dx}+8/\phi) = 1$ (which is true) doesn't mean anything to us. What's applicable is the generalized case: $\sin^2 x + \cos^2 x = 1$ for all $x$. That's what's useful to us. The former example is just a special case of the latter.