Liouville number

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As Liouville number , named after Joseph Liouville , is known in the theory of numbers a real number that satisfies the condition that for all positive integers integers and with there, so

Irrationality and transcendence

All Liouville numbers are irrational : For every rational number with an integer numerator and a positive integer denominator there is a positive integer with If now and integers with and are, then is

In 1844, Liouville showed that numbers with this property are not only irrational but also transcendent . This was the first proof of the transcendence of a number, Liouville's constant:

(Follow A012245 in OEIS )

All Liouville numbers are transcendent, but not all transcendent numbers are Liouville. For example, Euler's number is transcendent, but not Liouvillesch.


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