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Intuitive Understanding of Euler’s Formula

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Re: Intuitive Understanding of Euler’s Formula

#31

Post by RivieraKid »

man-and-laptop wrote: post #12

I 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/…

The way I see it:

e^1 is what you get if you repeteadly multiply 1 by (1 + 0.001)

e^i is also what you get if you repeatedly multiply 1 by (1 + 0.001i)

Re: Intuitive Understanding of Euler’s Formula

#32

Post by sannee »

sevensor wrote: post #27

Euler'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.

Wasn't some sort of engineering-grade complex analysis part of your EE (undergraduate?) curriculum? I am sort of surprised Euler's formula would be a "secret weapon" in EE classes.

Re: Intuitive Understanding of Euler’s Formula

#33

Post by sevensor »

sannee wrote: post #32
sevensor wrote: post #27

Euler'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.

Wasn't some sort of engineering-grade complex analysis part of your EE (undergraduate?) curriculum? I am sort of surprised Euler's formula would be a "secret weapon" in EE classes.

> Wasn't some sort of engineering-grade complex analysis part of your EE (undergraduate?) curriculum?

You'd think so, but it wasn't. Closest we got was probably the linear systems course, but that wasn't super close. I did a little more math than usual in that I took real analysis as an elective, but I wasn't able to squeeze complex analysis in.

Re: Intuitive Understanding of Euler’s Formula

#35

Post by jerf »

crazygringo wrote: post #28
empath75 wrote: post #17

Earlier 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?

I think one of the keys to understanding math, and to a lesser but still significant extent, physics history is to remember what you believed as a child, and how you struggled with the concepts taught to you. And that's even with a math curriculum designed to lead you to modern math. (One can debate how effective it is at that, but that's a separate topic.) Those misconceptions we had as children are pretty fundamental to the human wetware. Even today, with centuries of refinement and educational advancement, really only a small fraction of people come away from school with the ability to think truly mathematically.

For instance, even "just" negative numbers is a fairly counterintuitive concept. Despite how they may seem universal today, they actually didn't pop up in all that many cultures historically before their current line. And that story gets repeated over and over for all sorts of developments. It takes time for fields of study to process and abstract these things, because they weren't just handed it on a silver platter in school.

(To the extent that that doesn't seem to be the case today, I'd say that as we have become more and more mathematically sophisticated and the area of mathematical inquiry exponentially increases, the dominant factor 'holding back' math today is our inability to cover territory. Today nobody can completely cover a major discipline before the next generation is already coming in with fresh brains. That's a relatively recent development.)

Re: Intuitive Understanding of Euler’s Formula

#36

Post by jshowa3 »

I 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 instead of deriving one through a proof.

For example, one would intuitively think dropping a heavy object and a light object would conclude with the heavy object hitting the ground first which is not the case. This is just one example.

I'd much rather derive things the traditional way and not be duped by "intuitive" explanations. Because ones intuition is often different from natures.

Re: Intuitive Understanding of Euler’s Formula

#38

Post by jonsen »

jshowa3 wrote: post #36

I 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.

Re: Intuitive Understanding of Euler’s Formula

#39

Post by nabla9 »

sannee wrote: post #32
sevensor wrote: post #27

Euler'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.

Wasn't some sort of engineering-grade complex analysis part of your EE (undergraduate?) curriculum? I am sort of surprised Euler's formula would be a "secret weapon" in EE classes.

Analog electronics and signal processing fundamentally tied into complex numbers but it's completely possible to do the math to pass exams etc. and become EE engineer without deep understanding complex math. You just use formulas and do arithmetic with them.

Most practically oriented engineering students bitch about the math heavy parts because they don't have any use for them. They are happy with just the formulas and arithmetic. They can calculate electronic circuits, do Laplace transforms and Fourier transforms mechanically. As long as they understand what goes in and what comes out, it works just fine.

Re: Intuitive Understanding of Euler’s Formula

#40

Post by jacobolus »

crazygringo wrote: post #28
empath75 wrote: post #17

Earlier 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?

Apollonius used what basically amounts to Descartes’s coordinate method in the 1st century for studying conic sections, and European mathematicians of the 16th–17th centuries were all quite familiar with his work. But the formulation was somewhat cumbersome and context-specific. https://en.wikipedia.org/wiki/Apollonius_of_Perga#The_coordi...

The world also had a long cartographic tradition based on coordinates.

Moreover, Descartes’s book only used one coordinate axis at a time, only used positive numbers, and used it as a tool for setting up geometry problems to be solved algebraically, not as a general tool for what we now think of as graphing functions/equations. The way we think of the “Cartesian plane” is not the way Descartes thought about it.

The history of these conceptual developments is richer and more complicated than popularly imagined today.

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