Euler's identity is said to be the most beautiful theorem in mathematics.
"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."
- Benjamin Pierce
"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."
- Benjamin Pierce
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8 comentários:
Wow, I thihnk you might actually be on to something here.
jess
www.privacy.de.tc
eCalc is a free online calculator that supports complex numbers in both rectangular and polar formats. http://www.ecalc.com
All that equation (e**it = cos(t) + i sin(t)) says is that if your velocity is perpendicular-to and proportional-to a displacement from a central point (that is, i times the displacement vector in 2d), then you will move in a circle about the point at an angular velocity of 1 radian per time unit (say seconds). After pi radians you are 180 degrees from where you started, or -1 times your initial position. Obvious :)
My head hurts...
Meester-burns:
thats a physical result of the property of these numbers. The idea that 2 transcendental numbers behaving with such a simple relationship to form back to unity is incredible, from a number theory point of view. Physics just takes advantage of this fact by representing waves (cf circular motion in your case) with complex phases
Sorry guys, but the condition for the Maclaurin Series of e^x says that x must be a real number, and therefore, we cannot replace x by i∏ since i∏ is an imaginary number...
Anonymous: These numbers are intimately related to the 2-dimensional plane and circular motion. It is not a mere physical convenience and not just a formality of Taylor expansions. And e is not random transcendental. Ask yourself what e means. :) (Hint, in the complex plane, it comes up as the solution to the evolution equation df/dt = kf. Consider what that means for k positive, negative and imaginary - i.e. rate of change is +/- proportional to or perpendicular to displacement from 0, e is by any sane definition related to motion and continuous evolution.)
Alternately, you might think of the limit, so try to imagine what multiplying a number z by (1+ it/n) looks like for large n. This is gives you z plus a little component perpendicular to z. It is not shocking that doing that n-times might give you a rotation of the initial z (of course that's not a proof!)
My main point is that this still comes as such a shock to folks now mainly because complex numbers and e are taught way too formalistically.
and people don't understand what they are looking at.
J-P, study laurent series and you will understand
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