- #76

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Do you mean this?Is what I wrote in the first paragraph of my last post (#73) correct at all?

I think the above still has issues. The main one is that none of what you say depends on the graphical representation; you're just restating what a "metric" or "distance function" is. A "unit" is just the result of applying the function ##d(x, y)## to the two members ##x = 1## and ##y = 0## of the set of real numbers. The statement ##d(x, 0) = |x|## is just a special case of the statement ##d(x, y) = |x - y|##, with ##y = 0##. All of this is true independently of any one-to-one correspondence between real numbers and points on a line.With this graphical representation, upon defining the distance between two real numbers as ##d(x,y)=\lvert x-y\rvert##, we see from this graphical representation that any real number ##x## can be viewed as being a distance ##d(x,0)=\lvert x\rvert## from ##0##. If one then one considers the unit length ##d(1,0)=1## as a "unitof length" along the real number line, then in this graphical representation, one can view a given real number ##x## as being a distance of ##\lvert x\rvert## units from the origin, and in general, any two real numbers ##x## and ##y##, as being separated by a distance of ##\lvert x-y\rvert## units.