To anyone who finds this sort of explanation interesting or helpful, I recommend you check out "A Most Elegant Equation" by David Stipp, who covers Euler's Formula from step 0 for those with zero formal math knowledge. I'm definitely in that camp of people, and I was able to get a lot out of it. It's actually the book that helped several mathematical concepts "click" for me. Plus David Stipp just writes very romantically about math, which I thought was warming, and gave a lot of life to a field that I know nothing about.
Intuitive Understanding of Euler’s Formula
21–30 of 62 posts
Re: Intuitive Understanding of Euler’s Formula
#22The way I thought about it is that the trigonometric, hyperbolic, and exponential functions are all nontrivial solutions of the differential equation y'''' = y. Sharing differential properties is a very powerful kinship, which is also why you can substitute any of them for any of the other: the solution space of a linear differential equation is a vector space, and these three families of functions are just a change of basis in this space.
Well, I don't know, this is what makes sense to me.
Re: Intuitive Understanding of Euler’s Formula
#23(i^i)^i is pretty easy to do with Euler's formula: it's just i^(i*i)=i^(-1)=1/i=-i.
> it's just i^(i * i)=i^(-1)
That's not as obvious as the usage of the word "just" seems to imply. When we start with something like i^i, even before raising it to another power of i, we first need to understand i^i. The rules for exponentiation that hold good for real numbers cannot be apply here.
How exactly is i^i defined? What does raising a complex number to the power of another complex number even mean? We define it! We first define the following: For complex numbers w and z, w^z = e^(z log w).
Now we use this definition to see what i^i is. We get i^i = e^(i log i) = e^(i * (2i * pi * n + i * pi /2)) = e^(-2 * pi * n - pi/2) for n ∈ ℤ. Note that this is the result of log(i) being multivalued.
So far we have established an interesting result that i^i is always a real number regardless of which value of log(i) we choose from. If we choose the principal value of log(i), i.e., log(i) = i * pi / 2, then i^i = e^(-pi / 2). But let us move on with the multivalued i^i.
We use the result of (i^i) and the definition of w^z to see what (i^i)^i is. We get (i^i)^i = e^(i log(i^i)) = e^(i(2i * pi * m - 2 * pi * n - pi/2)) = e^(-2 * pi * m - 2i * pi * n - i * pi/2)) = e^(-2 * pi * m - i * pi/2)) for m ∈ ℤ.
So we can see that (i^i)^i = -i holds good for a single value of m, i.e., m = 0.
Re: Intuitive Understanding of Euler’s Formula
#24I once had an interesting thought about the function e^x. I think this is a key idea in the theory of Lie groups. If the x in e^x = (1+x/N)^N is understood as some transformation, then e^x is essentially repeating an infinitesimal transformation lots of times. So it's like a for-loop where the body of the loop is some infinitesimal transformation. I tried to define the integration operator in terms of e^x. The 1 + x/…
There is certainly a connection between Euler's formula and the matrix exponential, but I think you have confused some details about how e^x is defined. The connection is to consider C as a 2-dimensional real vector space with basis 1,i. Multiplication by i is a linear transformation of this vector space. In more detail: The exponential of a matrix X is an infinite sum just like that of the normal exponential functio…
OK, and where was I confused?
e^X could also be defined by \lim_{N \to \infty}(I + X/N)^N for X a linear map. For my intuition, I find that better than your definition.
Re: Intuitive Understanding of Euler’s Formula
#25Earlier quoted context omitted.
I think you have to be careful here when using rules that hold for real number for complex numbers. In fact, Wolfram|Alpha says, that -i is just one of multiple results: http://www.wolframalpha.com/input/?i=(i%5Ei)%5Ei
Indeed you can easily produce completely incorrect results if you aren't careful, like 1 = sqrt(-1)/sqrt(-1) = (-i)/(i) = -1.
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Re: Intuitive Understanding of Euler’s Formula
#26The article is quite right that multiplying by i gives a rotation. But it doesn't quite explain the reason for this: it's because that's the whole point of defining imaginary numbers in the first place! Remember you start off wanting to find a solution for the equation: i^2 = -1 This is actually easier to think about if you multiply it by a general real number r: r i^2 = -r In other words you want i such that if you…
> But if you stop this half way then everything has just settled on zero, and doing that twice is obviously not going to get to the whole reflection.
I just wanted to say, this is probably the best intuitive explanation I've ever heard, so thanks.
Complex numbers being associated with rotation -- well the math and geometry always worked out no problem, but it always felt sort of... random or arbitrary to me.
But framing it that there ultimately needs to be a total reflection in the real numbers when multiplying by -1, and that rotation is the simplest way to achieve a smooth path to that which preserves all the necessary... that just clicks.
So thanks again!
Re: Intuitive Understanding of Euler’s Formula
#27Euler's identity was my secret weapon in graduate-level EE classes. Out of laziness, I only ever memorized a couple of trig identities. (Trig functions are a huge part of EE.) Whenever I needed a trig identity on a test, I whipped out Euler's formula. From there, you have one step to definitions of sin and cos that you can manipulate any way you want.
Re: Intuitive Understanding of Euler’s Formula
#28The article is quite right that multiplying by i gives a rotation. But it doesn't quite explain the reason for this: it's because that's the whole point of defining imaginary numbers in the first place! Remember you start off wanting to find a solution for the equation: i^2 = -1 This is actually easier to think about if you multiply it by a general real number r: r i^2 = -r In other words you want i such that if you…
> it's because that's the whole point of defining imaginary numbers in the first place! People used imaginary numbers for a long time before Cartesian coordinates even existed.
I'll be honest, I've never understood how humanity didn't invent Cartesian coordinates until 1637, with all the other engineering we had.
Once we had linear equations, for example with the ancient Greeks, not one person ever thought to plot a line with it? Or to use it to calculate the necessary building materials for something like a pediment or cathedral?
Re: Intuitive Understanding of Euler’s Formula
#29I love reading posts like these. I was talking to my friend about how doing math proofs during HS is pretty similar to a math "lab" even though it's tedious and seemingly useless to some. I wish we were taught higher levels proofs like this. Even though we couldn't appreciate it at that time, it would of definitely helped us in the future.
Re: Intuitive Understanding of Euler’s Formula
#30Earlier quoted context omitted.
> it's because that's the whole point of defining imaginary numbers in the first place! People used imaginary numbers for a long time before Cartesian coordinates even existed.
I'll be honest, I've never understood how humanity didn't invent Cartesian coordinates until 1637, with all the other engineering we had. Once we had linear equations, for example with the ancient Greeks, not one person ever thought to plot a line with it? Or to use it to calculate the necessary building materials for something like a pediment or cathedral?
Nicholas Oresme essentially thought of them in the 14th century. I'd be surprised if as you suggest there weren't a lot of sporadic particular uses of the idea. But Descartes's big accomplishment was to abstract the coordinates away from any specific problem. No matter how many people prior to his work used coordinate ideas in problem solutions that's a major accomplishment.