Difference between revisions of "Algebraic number"
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Revision as of 11:29, 27 September 2024
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 rational numbers are algebraic, but not all algebraic numbers are rational. For example, consider
, which is a root of 
. - The if a and b are algebriac than
is either rational or transendental
, which is a root of 
.
is either rational or transendentalThis article is a stub. Help us out by expanding it.
