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1.2.2 Completeness of R and measurement
Even in Euclid’s geometry, a real number was, implicitly, understood in terms of all rationals
which exceeded it and all rationals below it. However, the traditional axioms of Euclidean
geometry, with requirements on intersections of lines and circles, can work with a field which is
larger than the rationals but smaller than R, and completeness is not essential.
The simplest measurement problem is to devise a measure of sets of points in the plane which
are made up of a finite collection of segments constructed by Euclidean geometry. Two such sets
should have the same measure if they can be decomposed into a finite collection of congruent
segments. For this we would not need the full complete system R of real numbers. Moving up a
dimension, with the task of measuring areas of polygonal regions constructed by Euclidean
geometry, one could still get away with a less-than-complete system of numbers. However, it
was shown by Max Dehn in 1900, in resolving Hilbert’s Third Problem, that there are polyedra in
three dimensions which have equal volumes (as defined by requirements of ‘upper’ and ‘lower’
approximations) which cannot be decomposed into congruent pieces. This, along with, of course,
the utility of measuring areas of curved regions even in two dimensions, makes it absolutely
essential to work with a notion of measure that goes beyond simply decomposing into
geometrically congruent figures. For a truly useful theory of measure, the completeness of the
number system is essential.
Capturing a real number between upper approximations and lower approximations proves
to be very useful. Archimedes and others computed areas of curved regions by such upper and
lower approximations. In modern calculus, this method lives on in the Riemann integral, as we
shall see later.

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1.2.2 Completeness of R and measurement Even in Euclid’s geometry, a real number was, implicitly, understood in terms of all rationals which exceeded it and all rationals below it. However, the traditional axioms of Euclidean geometry, with requirements on intersections of lines and circles, can work with a field which is larger than the rationals but smaller than R, and completeness is not essential. The simplest measurement problem is to devise a measure of sets of points in the plane which are made up of a finite collection of segments constructed by Euclidean geometry. Two such sets should have the same measure if they can be decomposed into a finite collection of congruent segments. For this we would not need the full complete system R of real numbers. Moving up a dimension, with the task of measuring areas of polygonal regions constructed by Euclidean geometry, one could still get away with a less-than-complete system of numbers. However, it was shown by Max Dehn in 1900, in resolving Hilbert’s Third Problem, that there are polyedra in three dimensions which have equal volumes (as defined by requirements of ‘upper’ and ‘lower’ approximations) which cannot be decomposed into congruent pieces. This, along with, of course, the utility of measuring areas of curved regions even in two dimensions, makes it absolutely essential to work with a notion of measure that goes beyond simply decomposing into geometrically congruent figures. For a truly useful theory of measure, the completeness of the number system is essential. Capturing a real number between upper approximations and lower approximations proves to be very useful. Archimedes and others computed areas of curved regions by such upper and lower approximations. In modern calculus, this method lives on in the Riemann integral, as we shall see later. Name: Description: ...
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