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Intuitive Understanding of Euler’s Formula
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Re: Intuitive Understanding of Euler’s Formula
#52People that like this article might be interested in 'Visual Complex Analysis' by Tristan Needham. It's (IMHO) a rare book of collections of "concrete" analogies for complex analysis.
Yes, this page appears to be lifted directly from that book.
Re: Intuitive Understanding of Euler’s Formula
#53This is a very confusing way to explain a simple thing. In fact
"Eulers formula simply shows how one can parametrize a helix using the exponential function", see
https://math.stackexchange.com/questions/3510/how-to-prove-e...
And that's it.
Re: Intuitive Understanding of Euler’s Formula
#54This is a very confusing way to explain a simple thing. In fact "Eulers formula simply shows how one can parametrize a helix using the exponential function", see https://math.stackexchange.com/questions/3510/how-to-prove-e... And that's it.
There are many answers in that thread; I think different things work for different people.
My favorite angle on this is the following graphic/animation, also present in the thread:
https://upload.wikimedia.org/wikipedia/commons/0/0e/ExpIPi.g...
This shows how (1+i * Pi/N)^k, k=1..N traces out a semi-circle for large values of N.
Geometrically, all it says is:
* Draw a right triangle ABC with AB=1, BC=Pi/N, and ABC the right angle
* Make a copy of ABC, call it A'B'C', and scale it so that A'B'(the long leg) = AC (the hypotenuse)
* Put A'B'C' over ABC so that A'B' and AC coincide
* Let ABC=A'B'C'
* Repeat the process N times
* Look where you end up when N is large enough
The answer is: when N is large, Pi/N is small, and the right triangle ABC is almost isosceles, AB ~= AC. So you end up with N slices of a pie that make up a fraction of a circle.
Which fraction? Well, the perimeter is N/Pi * N = Pi - so half a circle. So if A=(0,0) and B=(1,0), you end up at (-1,0).
Now (1+x/n)^n approaches e^x, so it makes sense to define e^(i * Pi) to be the same limit - which we found out to be -1 + i * 0.
Re: Intuitive Understanding of Euler’s Formula
#55The 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…
Always replace "intuitive" with "familiar" and you get an insight into what the person is talking about. So the task is to explain Euler's equation in terms of what you're already familiar with. Hmmm.
[1] Theorem by Jeff Raskin - https://www.asktog.com/papers/raskinintuit.html
Re: Intuitive Understanding of Euler’s Formula
#56I 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/…
>> I tried to define the integration operator in terms of e^x. The 1 + x/N needed to be one "infinitesimal" iteration of integration, that adds an extra infinitesimal rectangle to the area. But it didn't seem to work out. You're close! This can indeed be done properly and is then called the Euler-Maclaurin formula. For this, you define the "shift to the left by n operator" e^(nD) where D is the differentiation operat…
Amazing, thanks.
Re: Intuitive Understanding of Euler’s Formula
#57my favourite intuition about Euler, which is not really an explanation, somewhat tautological, and may or may not be wildly incorrect, but I like it nonetheless: e^x is a function whose value is its rate of change. (De^x=e^x). Now imagine the unit circle by taking a point an unit away from O, and set "rate of change" perpendicular to that vector. You will end up with Df(x) = i f(x), which really only works when f(x)…
That is exactly what it is!
In the real plane you get exponential growth: the rate of change is equal to the current value.
In the Argand plane you get a curling action due to the quadrature effect of the imaginary unit. The rate of change at each point is the tangent, and the result is therefore a circle.
Lie infinitesimal displacements capture this nicely, and also render the generic case which is a similarity transformation, e.g. rotation through two half reflections, e^(-w/2) * x * e^(w/2) like those found in quaternions and Clifford algebras.
Re: Intuitive Understanding of Euler’s Formula
#58Earlier quoted context omitted.
Indeed. There were 18th century mathematicians like Gauss who realized the importance of the geometric nature of complex numbers, but it didn't become central to the subject until the 19th century. The appearance of imaginary numbers as formal square roots of negative numbers goes back to the mid 16th century. As for Cartesian coordinates, i^2 = -1 has an intrinsic, coordinate-free interpretation in terms that would…
> and thinking of geometric operators as generalized numbers would have seemed pretty alien for most of the 19th century as well. Do you know of any resource treating that subject explicitly? I assume you mean the same kind of operator as in 'differential operator'—is that right? I can kinda see it maybe... but would definitely be interested in hearing the idea expanded on :)
By an operator I just mean a transformation. Certainly linear operators like differential operators qualify. The fact that these have an algebra in their own right goes back to work in the 19th century by Felix Klein and Sophus Lie on transformation groups and to later 20th work on linear algebra and functional analysis. It's stuff pretty much everyone learns as an undergrad nowadays, but the fact that you can do algebra on operators hardly without thinking is a relatively modern perspective.
Re: Intuitive Understanding of Euler’s Formula
#59I don't understand why this is so profound. I knew Euler's formula was a trig formula simply by looking at it. I don't know why there needs to be this complex explanation of where it came from where its obvious that it comes from trig and the concept of a unit circle. The problem is that reliance on intuition doesn't prove anything and its very deceptive. You end up having to store several cases of explanations inste…
Intuition is not an absolute thing. Intuition changes as we learn. Sometimes we learn things without building an accompanying intuition. This post is about building a sound intuition for something learned.
That's nice. I didn't say there was no value in intuition. I just said it was unreliable and I demonstrated an example of why its unreliable.
Its better to focus on solutions that don't rely on intuition because intuition is based solely on experience and often can't be applied to everything. This is why there's a strive to find general solutions to things. Intuition can help you, but you should not base your understanding off it. Intuition can lead you astray easily as in my example. People just assumed it was correct for many years until Galileo actually proved it through experimentation.
In fact, I found this explanation way more complicated than just stating that you are performing trigonometry in two different domains. Your sine is scaled by an imaginary value and your cosine is scaled by a real value. The radius equates to an exponential. When you move it pi units you get sin = 0, cosine = -1 which you use algebra to get Euler's identity.
You don't need to introduce complex intuition in order to get it. Intuition is often a crutch people use to try and think they understand something and just blame others when they don't get it much like this post. If intuition was so important, why not submit it as a legitimate mathematical way of proving things?
Re: Intuitive Understanding of Euler’s Formula
#60The 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…
> The key idea of imaginary numbers is to consider multiplication -1 to be a rotation by half a turn rather than a reflection. That is a lot easier to do half of! As soon as you have multiplication by -1 as a rotation by half a turn, it is obvious to identify i as rotation by a quarter turn. It gets even more interesting when you add additional degrees of freedom so that such rotations can happen via more than one pa…
Even in the complex numbers, there is more than 1 path, i^2 and (-i)^2.
On a non-geometric note, i and -i are algebraicly equivelent over the reals.