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| ▲ | itishappy 5 days ago | parent | next [-] |
| Euler's formula is a specific case of the exponential map from Lie theory. This means e^x can be used with all sorts of interesting x types, and it often has surprisingly intuitive behavior! When x is a real number you get continuous growth. When x is a purely imaginary number you get continuous rotation. When x is complex you get continuous growth and rotation. When x is a matrix you get a continuous linear transformation (growth, rotation, and shear). What's the similarity here? Euler's formula treats it's argument as a transformation which gets continuously applied in infinitesimal amounts. This also explains the formula for calculating the value of e: e = lim (1 + 1 / n) ^ n where (x -> infinity)
https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)https://en.wikipedia.org/wiki/Matrix_exponential https://www.youtube.com/watch?v=O85OWBJ2ayo |
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| ▲ | ColinHayhurst 5 days ago | parent | prev | next [-] |
| Complex numbers and (Pauli/Dirac) matrices not required if you use Geometric Algebra. I highly recommend the book by Doran and Lasenby [0], or you can get the details from their papers, notably [1]. [0] Geometric Algebra for Physicists, CUP, 2003 [1] https://arxiv.org/abs/quant-ph/0509178 |
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| ▲ | sparky_z 5 days ago | parent | prev | next [-] |
| My favorite intuitive explanation was actually written by science fiction author, Greg Egan. It takes the exact approach you're asking for, reinterpreting compound interest in a 2d rotational context on the complex plane, and doesn't use more than high school math: https://gregegan.net/FOUNDATIONS/04/found04.html#s2 Fig. 7 is the money shot. |
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| ▲ | dboreham 5 days ago | parent | prev | next [-] |
| IANAM but I'd go with "it's implicit in how complex numbers are defined". Complex numbers are a thing made up by humans (as are negative numbers), and we got to define i as "up the y-axis". Once you do that, and note that a rotation is therefore cos angle plus i sin angle, add in that e^something is an eigenfuncion of differentiation, and you're pretty much there. Fwiw I think it's Maclaurin series for this. Edit: obviously should be j not i. |
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| ▲ | Ono-Sendai 4 days ago | parent | prev | next [-] |
| The first thing to understand is that multiplying a complex number by i rotates the complex number by 90 degrees counter-clockwise around the origin.
For example, 1 * i = i (e.g. 1 + 0i is mapped to 0 + 1i).
And i*i = -1 (e.g. 0 + 1i is mapped to (-1 + 0i)
and so on.
e^ix is a continuous generalisation of this discrete rotation, as I understand it. |
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| ▲ | empiricus 5 days ago | parent | prev [-] |
| One possibility: take the unit circle, and a vertical line tangent to the circle at (1,0). Then e^ix takes that line and wraps it around the circle.
This |
