Thread: Math paradox
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12-22-09, 11:59 PM   #15
Amenity
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Oh, and to help explain it here's a bit on it known as the "ghosts of departed quantities", which is a paradox involving infinitesimal calculus. Sorry it's a bit late...took me a while to find this. It's been a loooong time.

To consider an example, the function y = x^2 is differentiated in calculus by forming the quotient

Δy/Δx

of the y-increment, usually denoted Δy, over the x-increment, usually denoted Δx. The resulting expression simplifies algebraically to

2x+Δx

To obtain, instead, the familiar expression for the corresponding derivative, namely 2x, one needs to strip away the infinitely small quantity Δx. Thus, the infinitesimal quantity Δx seems to be assumed nonzero at the stage of calculating the quotient, and yet it is assumed zero in the last phase of the calculation when Δx is stripped away. In summary, we have departed quantities (Δx = 0) which are yet present in some ghostly fashion (Δx ≠ 0).

Last edited by Amenity : 12-23-09 at 12:01 AM. Reason: Blue text is a quoted portion. Quote tags not used in order to preserve formatting.
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