Measurement of length begins with a unit of measurement and units have
'serial numbers' in geometry. This 'serial number' is Ordinal number.
The unit of length lies between 'serial numbers' or Ordinal numbers.
Ordinal numbers are dimensionless, they do not occupy 'length'. The
unit of length between ordinal numbers are equal and they can be as
as small as we please (say 10^-100 mm). In applied mathematics unit
of length cannot be zero but in geometry when ordinal number directy
gives length (without unit of length)and unit of length between
ordinal
numbers becomes 0; X and Y coordinates become contiguous but cadinal
and ordinal numbers become same at every point. (We have no entities
or
units to count!) In this condition coordinates can only give a series
of points that lie on a straight line [distance x1,x2 is minimum,
distance y1,y2 is minimum therefore distance between x1,y1 and x2,y2
is minimum and it must be a straight line - {(x2-x1)^2+(y2-y1)^2}^1/2.
No line in plane coordinate geometry is a continuous line. Geometry of
continuous lines has to be a different and a new concept. Integral of
dX/X gives a STRAIGHT line that is continuous.
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