Difference between revisions of "Algebraic number"
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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> | + | 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> or <math>\pi</math>. |
===Number of algebraic numbers=== | ===Number of algebraic numbers=== | ||
| − | Although it seems that the number of algebraic numbers is large, there are only [[countable|countably]] many of them. That is, the algebraic numbers have the same [[cardinality]] as the | + | Although it seems that the number of algebraic numbers is large, there are only [[countable|countably]] many of them. That is, the algebraic numbers have the same [[cardinality]] as the [[natural numbers]]. |
Algebraic numbers are studied extensively in [[algebraic number theory]]. | Algebraic numbers are studied extensively in [[algebraic number theory]]. | ||
| + | |||
| + | ===Properties=== | ||
| + | * All of the rational numbers are algebraic. | ||
| + | * It is NOT necessary that an algebraic number is rational. | ||
{{stub}} | {{stub}} | ||
Revision as of 21:25, 26 September 2008
A complex number is said to be algebraic if it is a root of a polynomial with rational 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 
or 
.
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 natural numbers.
Algebraic numbers are studied extensively in algebraic number theory.
Properties
- All of the rational numbers are algebraic.
- It is NOT necessary that an algebraic number is rational.
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