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Transcendental number

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Pi (π) is a well known transcendental number

In mathematics, a transcendental number is a complex number that is not algebraic—that is, not a root (i.e., solution) of a nonzero polynomial equation with integer or equivalently rational coefficients. The most popular transcendental numbers are π and e.^[1] [2]

Though only a few classes of transcendental numbers are known, in part because it can be extremely difficult to show that a given number is transcendental, transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers compose a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set. All real transcendental numbers are irrational numbers, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental. For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x² − 2 = 0. The golden ratio (denoted ${\displaystyle \varphi }$ or ${\displaystyle \phi }$) is another irrational number that is not transcendental, as it is a root of the polynomial equation x² − x − 1 = 0.

HistoryEdit

PropertiesEdit

Numbers proven to be transcendentalEdit

Possible transcendental numbersEdit

Sketch of a proof that e is transcendentalEdit

Mahler's classificationEdit

See alsoEdit

NotesEdit

ReferencesEdit

External linksEdit

Last edited on 15 June 2020, at 18:00

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