Note 6

This is because we can only work with what are called rational numbers, which are precisely defined by the ratio of two integers, while mathematics reveals an abundance of irrational numbers, much more unpleasant quantities described by a strictly infinite sequence of randomly occurring digits. Although there is an infinite number of rational numbers between zero and one (and in any interval for that matter), there are infinitely many more irrational numbers. Thus the rationals themselves, which are all we are able to handle ( The irrationals must always be approximated by rational numbers), form a highly abnormal selection. It is infinitely more probable that the values of the variables of the initial condition will be given by (i.e. actually are) irrational numbers. And even many rational numbers contain an infinite number of digits, such as 1 / 3 , which is 0.33333333333333333333333333333333 . . . . . , and must be rounded off to, say, 0.333333, which as such is an error, that can have dramatic consequences for the evolution of a chaotic system, i.e. of an ergodic system which is, for its long-term behavior, exquisitely sensitive to differences in initial condition (See COVENEY, P. & HIGHFIELD, R., The Arrow of Time, 1991, p.273).

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