On 11/25/2013 1:59 PM, Robert Hansen wrote: > > On Nov 25, 2013, at 4:47 PM, Gary Tupper <email@example.com > <mailto:firstname.lastname@example.org>> wrote: > >> So - what would we make of a line that that blinked red & blue, with >> a single pixel blinking green? > > After my epileptic fit or during?:) > > I don't understand what you are suggesting by color. That sounds more > like a symbol than a model. For example, the open circle denoting a > missing point is a notational device or symbol. A model of a missing > point would a very tiny gap in the line, but that would be a poor > model and wouldn't work at the end of the line, thus the use of > symbols instead. > > Bob Hansen > Hi, Bob:
Actually, the gap would be smaller than 'tiny' since its width would be zero. We should remember that any physical representation of a mathematical object is a model. The objective of our model is essentially to impart information - the open circle on a curve is interpreted to mean that a single point is missing.
Take, for example, the graph of y=x^2 /(x-1) : Usually drawn as a parabola with a hollow dot at (1,1).
On a computer screen, a blinking pixel at (1,1) can impart the same information. If we "zoom in" to that pixel on the computer screen, the display will remain - a blinking pixel. (unless we exceed the computer's display resolution, which I imagine depends on the computer's numerical capabilities.)
BTW, although we are usually interested in the graph of the solution of y=x^2/(x-1) over a region of the plane, and therefore draw the hollow dot at (1,1), we should realize that the division by x-1 affects the whole plane - ie the whole line x=1 is 'removed'. Maybe the easiest way to visualize this is to ask about 'truth': So, for y=x^2/(x-1), what about (2,4)? It satisfies the equation - ie 'true', What about (2,3)? - it generates 'false'. What about (1,1)? And what about (2,1)?