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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]], 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> or <math>\pi</math>.
  
 
===Number of algebraic numbers===
 
===Number of algebraic numbers===
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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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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]].
  
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===Properties===
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* All rational numbers are algebraic, but not all algebraic numbers are rational. For example, consider <math>\sqrt{2}</math>, which is a root of <math>x^{2}-2</math>.
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* The if a and b are algebriac than <math>\log_a{b}</math> is either rational or transendental
 
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Revision as of 20:56, 22 April 2022

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 $e$ or $\pi$.

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 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 $\sqrt{2}$, which is a root of $x^{2}-2$.
  • The if a and b are algebriac than $\log_a{b}$ is either rational or transendental

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