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>\imath</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]] | + | 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>\imath</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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===Number of algebraic numbers=== | ===Number of algebraic numbers=== | ||
Although it initially seems that the number of algebraic numbers is large, it turns out that there are only [[countable|countably]] many of them. That is, the algebraic numbers have the same [[cardinality]] as the integers. | Although it initially seems that the number of algebraic numbers is large, it turns out that there are only [[countable|countably]] many of them. That is, the algebraic numbers have the same [[cardinality]] as the integers. | ||
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Algebraic numbers are studied extensively in [[algebraic number theory]]. | Algebraic numbers are studied extensively in [[algebraic number theory]]. | ||
{{stub}} | {{stub}} | ||
Revision as of 20:18, 24 April 2008
An algebraic number is a root of a polynomial with integer coefficients. Examples include 
, 
, 
, and ![$\frac{4+\sqrt[27]{19}}{\sqrt[3]{4}+\sqrt[7]{97}}$](http://latex.artofproblemsolving.com/f/b/2/fb23020f8042aef7ccbbf8d8585c78fcb421adef.png)
. A number that is not algebraic is called a transcendental number, such as 
and 
.
Number of algebraic numbers
Although it initially seems that the number of algebraic numbers is large, it turns out that 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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