Nate
Reputation
604
Top tag
Next privilege 1,000 Rep.
Create tags
4 11
Impact
~10k people reached

# 89 Actions

 May21 comment How come the Euclidian distance for n-space involves only squares? Cool! Are there any uses to such alternative definitions? Also, what do you mean by saying that one allows rotation while the other does not? How can a distance metric prevent rotation? May21 asked How come the Euclidian distance for n-space involves only squares? Mar14 awarded Yearling Mar6 revised What's so special about sine? (Concerning $y'' = -y$) corrected spelling, fixed formatting Mar6 suggested approved edit on What's so special about sine? (Concerning $y'' = -y$) Mar3 comment What's so special about sine? (Concerning $y'' = -y$) I understand that, but I don't completely buy this. Wherever there is a circle in nature, looking closer we find it not a circle. Orbits are actually ellipses, the earth isn't quite spherical, and so on. Mar3 comment What's so special about sine? (Concerning $y'' = -y$) "Nearly every"? Which ones can't? Also, given the taylor series approximation for sine, can't anything approximated by a harmonic oscillator be modeled by even simpler functions? Feb29 comment What's so special about sine? (Concerning $y'' = -y$) Thanks! I suppose I probably should take this question over to physics, I'm also curious as to why physics seems to care so much about the second derivative (in everything from F=ma to the position/momentum interaction). Feb29 accepted What's so special about sine? (Concerning $y'' = -y$) Feb29 awarded Nice Question Feb28 awarded Editor Feb28 revised What's so special about sine? (Concerning $y'' = -y$) added 448 characters in body Feb28 asked What's so special about sine? (Concerning $y'' = -y$) Jun6 comment Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos? That helped a lot, thanks. In my own terms, the key point that I was missing was that taking $\lim_{n\to\infty}$ negates the drift off the unit circle that you see in $1+iX/n$, but does not negate the rotation caused by $1+iX/n$. It still astounds me that by taking the limit in this situation you negate the drift but not the rotation - that's an incredibly beautiful and powerful idea. :) Jun6 accepted Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos? Jun6 comment Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos? I get it! Multiplying by $1+ix$ gives us rotation by x but doesn't keep us on the unit circle. However, if we divide by n and raise to the power n, we manage to quell the drift off the circle but maintain the rotation. The drift is lost if we divide by and raise to the power n, but the rotation is not. Looking deeper, is there any field of math concerned with which operations are susceptible to elimination by such limits and which operations are unaffected? Jun6 comment Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos? Thanks! I don't quite understand the image above: I understand the first frame represents $1 + i*\pi$, but I don't understand the following frames. I know I'm being a bit dense, but could you break frame 3 down for me into what each segment represents? Jun5 asked Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos? May23 comment Complex Exponents Thanks, that clears a lot of things up. I still don't quite understand why multiplication by (1+iX/N) is equivalent to rotation by X/N - do you think you could clear that up a bit for me? May18 comment Complex Exponents See, that's exactly the part that I'm trying to understand by understanding complex exponents. Everybody just waves their hands and says "by definition" or pulls out a Taylor series, which is a proof, not a reason. I'm trying to figure out why $e^{ix}=cos(x) + i*sin(x)$. Lubos Motl's answer below helps, but if you have any extra pointers they'd be more than welcome.