Math 2.0

definition

Transcendental number

Definition

A complex number is transcendental if it is not an algebraic number. That is, it is not a root of any nonzero polynomial with rational coefficients.

Remarks

The numbers ee and π\pi are transcendental, by the Lindemann–Weierstrass theorem. Each Liouville number is transcendental, by Liouville's theorem. The algebraic numbers are countable, so almost all real numbers are transcendental. Joseph Liouville first proved that transcendental numbers exist, in 1844.

Explanations

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A transcendental number solves no polynomial equation with integer coefficients. It is usually hard to prove that a given number is transcendental, even though most numbers are.

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Algebraic number, definition, no evidence levelAlgebraic numberdefinitionTranscendental number, definition, no evidence levelTranscendental numberdefinition
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Sources

  1. Web https://en.wikipedia.org/wiki/Transcendental_number, lead section

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