Objective:
There are two types of numbers. One type is the rational numbers that we can write in fraction I/J of integer numbers I and J; for instance, ³/₇ for I =3 and J =7. The other type is the irrational numbers that cannot be expressed in a fraction form. In principle, we can count all the rational numbers, though there are infinitely many. On the other hand, we do not know the full membership of irrational numbers, an archetypical example of which is . Our main interest here is the decimal expression of rational and irrational numbers. Take, for instance, a rational fraction ³/₇, with six digits (4, 2, 8, 5, 7, 1) repeating in the decimal expression. Then, by visualizing the decimal digits as the path of a bouncing ball, we say ³/₇ is period 6.

All rational numbers have a finite period, though may be very long. On the other hand, the irrational numbers have never ending or repeating decimal digits, so we say their periods are infinitely long. In any event, when period becomes long the decimal digits appear to have been chosen randomly out of (0, 1, ...,9), as demonstrated here by the first 21 decimal digits of = 3.141592653589793238462.

Scary words:
Rational and irrational numbers, unit fraction, finite and infinite periods, trajectory, Farey tree, countable and uncountable, stability and instability.

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