Difference between revisions of "Algebraic number"

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An '''algebraic number''' is a [[root]] of a [[polynomial]] with [[integer]] [[coefficient]]s. Examples include <math>\frac{1}{3}</math>, <math>\sqrt{2}+\sqrt{3}</math>, <math>i</math>, and <math>\frac{4+\sqrt[27]{19}}{\sqrt[3]{4}+\sqrt[7]{97}}</math>. A number that is not algebraic is called a [[transcendental number]], such as <math>e</math> and <math>\pi</math>.
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A [[complex number]] is said to be '''algebraic''' if it is a [[root]] of a [[polynomial]] with [[rational]] [[coefficient]]s. Examples include <math>\frac{1}{3}</math>, <math>\sqrt{2}+\sqrt{3}</math>, <math>i</math>, and <math>\frac{4+\sqrt[27]{19}}{\sqrt[3]{4}+\sqrt[7]{97}}</math>. A number that is not algebraic is called a [[transcendental number]], such as <math>e</math> and <math>\pi</math>.
  
 
===Number of algebraic numbers===
 
===Number of algebraic numbers===

Revision as of 08:51, 25 April 2008

A complex number is said to be algebraic if it is a root of a polynomial with rational coefficients. Examples include $\frac{1}{3}$, $\sqrt{2}+\sqrt{3}$, $i$, and $\frac{4+\sqrt[27]{19}}{\sqrt[3]{4}+\sqrt[7]{97}}$. A number that is not algebraic is called a transcendental number, such as $e$ and $\pi$.

Number of algebraic numbers

Although it seems that the number of algebraic numbers is large, there are only countably many of them. That is, the algebraic numbers have the same cardinality as the integers.

Algebraic numbers are studied extensively in algebraic number theory.

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