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If y is irrational, then f(y) = 0. Again, we can take ε = 1 ⁄ 2, and this time, because the rational numbers are dense in the reals, we can pick z to be a rational number as close to y as is required. Again, f(z) = 1 is more than 1 ⁄ 2 away from f(y) = 0.
Irrational behavior can be useful when used tactically in certain conflict, game and escape situations. The moves of an irrational opponent are not (or only very limitedly) predictable. An irrational negotiator cannot be put under rational pressure. [52] An indirect tactic is the rational use of the irrationalism of third parties.
Rational numbers have irrationality exponent 1, while (as a consequence of Dirichlet's approximation theorem) every irrational number has irrationality exponent at least 2. On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers, including all algebraic irrational numbers , have an irrationality exponent exactly ...
A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible. The set of discontinuities of any function must be an F σ set. If such a function existed, then the irrationals would be an F σ set.
Rational irrationality is not doublethink and does not state that the individual deliberately chooses to believe something he or she knows to be false. Rather, the theory is that when the costs of having erroneous beliefs are low, people relax their intellectual standards and allow themselves to be more easily influenced by fallacious reasoning, cognitive biases, and emotional appeals.
Rational choice theory uses a much more narrow definition of rationality. At its most basic level, behavior is rational if it is reflective and consistent (across time and different choice situations). More specifically, behavior is only considered irrational if it is logically incoherent, i.e. self-contradictory.
This shows that any irrational number has irrationality measure at least 2. The Thue–Siegel–Roth theorem says that, for algebraic irrational numbers, the exponent of 2 in the corollary to Dirichlet’s approximation theorem is the best we can do: such numbers cannot be approximated by any exponent greater than 2.
Farey sequences are very useful to find rational approximations of irrational numbers. [15] For example, the construction by Eliahou [ 16 ] of a lower bound on the length of non-trivial cycles in the 3 x +1 process uses Farey sequences to calculate a continued fraction expansion of the number log 2 (3) .