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between continuous and discrete structures. Compare a segment, which is continu-
ous with a set of golf balls, which is discrete. A subset of balls can be partitioned
into smaller subsets, but if the initial set is finite, then after a finite number of parti-
tions you cannot go on because one half of a golf ball is not a golf ball. But if you
start from a segment, which has a finite size, the partition process can be infinitely
performed, because one half of a segment is still a segment. The difference is in the
composition of the two structures.
However, distinguishability of objects needs a further subtle analysis. In fact,
two balls can be distinguished from three balls, but this does not mean that they
can be individually distinguished. For example, if they have the same shape and
their positions may change when you are not observing them, are you able to tell
whether they exchanged their positions, when you observe them again? Surely, you
can tell whether their number increases or decreases, but in many cases their single
individuality cannot be determined.
Let us mention that arithmetical operations on numbers, when numbers are rep-
resented by segments, are realized by using compass and ruler. Sum and difference
may be performed by putting segments on the same line, in straightforward manners,
while product and division are performed by using similar triangles, at it is shown
in Fig. 5.3. An algorithm is a procedure solving a given problem. Many famous ge-
ometrical constructions are essentially algorithms manipulating geometrical entities
to solve specific problems.
Fig. 5.3
Segment multiplication by similarity: the lengths of two parallel segments are in the
ratio of their intercepts with the same incident lines
For our further discussion we recall the famous Pythagoras' theorem about right-
angled triangles. Figure 5.4 is essentially a proof of this theorem (due to Chinese
mathematicians).
Pythagoras' theorem is the basis of the well-known goniometric equation: sin
2
x
+
cos
2
x
=
1. In the goniometric unitary circle, a radius which forms an angle
α
with
the
x
-axis has projections, over the two axes, sin
(
x
-
projection), which together with the radius form a right triangle. From this circle
all the important goniometric properties can be derived. Figure 5.5 is essentially a
α
(
y
-projection) and cos
α
