PlanetMath: transcendental number

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transcendental number

(Definition)

A transcendental number is a complex number that is not an algebraic number. That is, it is a complex number that is transcendental over $\mathbb{Q}$ (or, equivalently, over $\mathbb{Z}$ ).

Well known transcendental numbers include $\pi$ and (the base of natural logarithms).

Cantor showed that, in a sense, “almost all” complex numbers are transcendental: there are uncountably many complex numbers, but only countably many algebraic numbers.

"transcendental number" is owned by yark. [ full author list (2) | owner history (1) ]

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See Also: pi, irrational

Attachments:: proof of the existence of transcendental numbers (Proof) by kidburla2003
example of transcendental number (Example) by alozano
e is transcendental (Theorem) by pahio

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Cross-references: natural logarithms, base, transcendental, algebraic number, complex number
There are 18 references to this entry.

This is version 8 of transcendental number, born on 2001-11-04, modified 2008-01-17.
Object id is 656, canonical name is TranscedentalNumber.
Accessed 6910 times total.

Classification:

AMS MSC:	11J81 (Number theory :: Diophantine approximation, transcendental number theory :: Transcendence )
	11J82 (Number theory :: Diophantine approximation, transcendental number theory :: Measures of irrationality and of transcendence)

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