An intuitive understanding of Euler's Formula

[pokerfan]

From the Better Explained blog...

Euler's identity seems baffling. {...} Could this ever be intuitive? Not according to 1800s mathematician Benjamin Peirce: "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."

Argh, this attitude makes my blood boil! Formulas are not magical spells to be memorized: we must, must, must find an insight.

http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula">Intuitive Understanding Of Euler’s Formula

[tridim]

1. Anyone interested in seeing Euler rotations in action..

Start here:
http://www.skytopia.com/project/fractal/mandelbulb.html
http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/540/

Nerds beware, this stuff is a major time sucker.

[pokerfan]

2. We electrical types practically worship Euler

but it's really not that big of a deal. For our purposes, complex numbers are just a way to express a number in two dimensions. Euler's Identity can be viewed graphically as:



e^i*pi is the curved arrow leading to -1 on the horixontal axis.

+1 is the vector leading you from -1 back to 0.

Thus e^i*pi +1 = 0.

[Occulus]

7. You can also use this other program:

http://sourceforge.net/projects/mandelbulber/

Both apps- the one at skytopia, and this one- will work, but I prefer this one; it has more options. Be aware that you better have a multicore processor to generate these, or you could be waiting for hours... or days.

[Igel]

3. This is a problem?

After 15 weeks of complex analysis, it made sense. Actually, it didn't take nearly that long.

A trite equation masked by a notational convention.

[pokerfan]

4. Not for someone as smart as you!

[Dr. Strange]

6. That's the beauty of complex analysis.

I suppose Benjamin Peirce was still viewing the exponential function and the trig functions as real functions and being surprised at how closely related they are. Which is, to a large extent, surprising: the exponential function comes from analysis and the trig functions come from geometry. Why should they be related at all? But in the context of complex variables it's easy to see just how intimately related they are.

[rug]

5. It makes more sense after you take a day off.