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A stronger result is the following: [31] Every rational number in the interval ((/) /,) can be written either as a a for some irrational number a or as n n for some natural number n. Similarly, [ 31 ] every positive rational number can be written either as a a a {\displaystyle a^{a^{a}}} for some irrational number a or as n n n {\displaystyle n ...
17 (seventeen) is the natural number ... it is believed that Theodorus had proved all the square roots of non-square integers from 3 to 17 are irrational by ...
Irrational numbers ... 11, 13, 17, 19, 23, 29, 31, ... Composite number: A positive integer that can be factored into a product of smaller positive integers. Every ...
In mathematics, an irrational number is any real number that is not a rational number, i.e., ... This page was last edited on 11 December 2022, at 04:17 (UTC).
Hippasus is sometimes credited with the discovery of the existence of irrational numbers, following which he was drowned at sea. Pythagoreans preached that all numbers could be expressed as the ratio of integers, and the discovery of irrational numbers is said to have shocked them. However, the evidence linking the discovery to Hippasus is unclear.
It is assumed that Theodorus had proved that all of the square roots of non-square integers from 3 to 17 are irrational by means of the Spiral of Theodorus. [2] Plato does not attribute the irrationality of the square root of 2 to Theodorus, because it was well known before him. Theodorus and Theaetetus split the rational numbers and irrational ...
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 ...
[17] [better source needed] By 130 AD, Ptolemy, ... A real number that is not rational is called irrational. A famous irrational real number is the ...