3 ms·

Well, the tangent and the secant will converge to the same thing as h -> 0. I'm not sure what you mean by "the equivalence of curved and straight lines" though

by rcfox 16y ago

Well, the tangent and the secant will converge to the same thing as h -> 0. I'm not sure what you mean by "the equivalence of curved and straight lines" though.

zeynel1 16y ago
Tangent and secant converge to the same [point] but as far as I understand this is not a statement about the equivalence of the curved and straight lines. By this I mean something like Archimedes's method of exhaustion http://en.wikipedia.org/wiki/Method_of_exhaustion#Use_by_Archimedes http://en.wikipedia.org/wiki/Method_of_exhaustion#Use_by_Arc... when he approximated a circle (curved line) with a polygon (straight line). I would think that there would be a theorem in mathematics generalizing this fact; something like; "curved line is equivalent to straight line." It depends on how we divide (or scale) a line that gives it its "curvature." Or "there is no curve but curvature." Something like that?
- rcfox 16y ago
  Around a given operating point, you can approximate any function with a polynomial (http://en.wikipedia.org/wiki/Taylor_series http://en.wikipedia.org/wiki/Taylor_series). This includes approximating a polynomial with a lower-order polynomial. A line is first-order polynomial. Therefore, around a given operating point, a line can approximate a function. The key phrase there is "around a given operating point". If you zoom in close enough, a line and an exponential will appear to be the same. Outside of this area, they will clearly diverge. So you can't really say that a curved line is equivalent to a straight line, but you can say that sin(x) = x, for small values of x (and tolerable levels of error in sin(x)). In fact, that comes up quite often in engineering.
  - zeynel1 16y ago
    "you can say that sin(x) = x" Yes, this is the small angle approximation http://en.wikipedia.org/wiki/Small-angle_approximation http://en.wikipedia.org/wiki/Small-angle_approximation and also the example I gave above: 3 points on a sin curve may be considered a straight line or a curved line; 1 meter on earth's orbit is a straight line. The same idea comes up in the string theory too as a pun between point and line. I am surprised that there is no mathematical theorem to express this fundamental property of nature. Or maybe I am not expressing myself clearly.

by rcfox 16y ago

Well, the tangent and the secant will converge to the same thing as h -> 0. I'm not sure what you mean by "the equivalence of curved and straight lines" though.

zeynel1 16y ago

Tangent and secant converge to the same [point] but as far as I understand this is not a statement about the equivalence of the curved and straight lines. By this I mean something like Archimedes's method of exhaustion http://en.wikipedia.org/wiki/Method_of_exhaustion#Use_by_Archimedes http://en.wikipedia.org/wiki/Method_of_exhaustion#Use_by_Arc... when he approximated a circle (curved line) with a polygon (straight line). I would think that there would be a theorem in mathematics generalizing this fact; something like; "curved line is equivalent to straight line." It depends on how we divide (or scale) a line that gives it its "curvature." Or "there is no curve but curvature." Something like that?

rcfox 16y ago
Around a given operating point, you can approximate any function with a polynomial (http://en.wikipedia.org/wiki/Taylor_series http://en.wikipedia.org/wiki/Taylor_series). This includes approximating a polynomial with a lower-order polynomial. A line is first-order polynomial. Therefore, around a given operating point, a line can approximate a function. The key phrase there is "around a given operating point". If you zoom in close enough, a line and an exponential will appear to be the same. Outside of this area, they will clearly diverge. So you can't really say that a curved line is equivalent to a straight line, but you can say that sin(x) = x, for small values of x (and tolerable levels of error in sin(x)). In fact, that comes up quite often in engineering.
- zeynel1 16y ago
  "you can say that sin(x) = x" Yes, this is the small angle approximation http://en.wikipedia.org/wiki/Small-angle_approximation http://en.wikipedia.org/wiki/Small-angle_approximation and also the example I gave above: 3 points on a sin curve may be considered a straight line or a curved line; 1 meter on earth's orbit is a straight line. The same idea comes up in the string theory too as a pun between point and line. I am surprised that there is no mathematical theorem to express this fundamental property of nature. Or maybe I am not expressing myself clearly.