Algebraic number

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.


  • 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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