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To change this template's initial visibility, the |state= parameter may be used: {{Irrational number | state = collapsed}} will show the template collapsed, i.e. hidden apart from its title bar. {{Irrational number | state = expanded}} will show the template expanded, i.e. fully visible.
To change this template's initial visibility, the |state= parameter may be used: {{Rational numbers | state = collapsed}} will show the template collapsed, i.e. hidden apart from its title bar. {{Rational numbers | state = expanded}} will show the template expanded, i.e. fully visible.
However, there is a second definition of an irrational number used in constructive mathematics, that a real number is an irrational number if it is apart from every rational number, or equivalently, if the distance | | between and every rational number is positive. This definition is stronger than the traditional definition of an irrational number.
Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between A and B. [3] In other words, A contains every rational number less than the cut, and B contains every rational number greater than or equal to the cut. An irrational cut is equated to an irrational number which is in neither set.
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 ...
Download as PDF; Printable version; ... Rational numbers have two continued fractions; the version in this list is the shorter one. ... is irrational. If true, this ...
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