Intuitive Understanding of Euler’s Formula

Post by vnorilo » Tue, Oct 30, 2018, 2:50 PM UTC

pjbk wrote: post #57

vnorilo wrote: post #15

my 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 tran…

Thank you for this! The wikipedia rabbit hole beckons :D

Post by dstipp » Wed, Oct 31, 2018, 1:18 AM UTC

sweetheart wrote: post #21

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 romantic…

Thanks for your kind words about my book. I never quite knew when writing it whether it would find its way to the people I mainly wrote it for -- those interested in math who don't know a whole lot about it. They aren't thick on the ground. So it's a real, sort of rare blast for me to hear about it arriving where I'd hoped.

Re: Intuitive Understanding of Euler’s Formula

Re: Intuitive Understanding of Euler’s Formula