Definition: A number α is algebraic if α satisfies an equation of the form
xn + a1xn−1 + · · · + an−1x + an = 0
in which the coefficients ai are rational numbers. Notice that all rational numbers are algebraic, as well as numbers like √2.
An algebraic number is a fundamental concept of mathematics that comes from the understanding of algebraic equations and number theory. An algebraic number confines a vast range of numbers, containing rational numbers, specific types of irrational numbers, and integers.
These numbers built a connection between the algebraic and analytic properties of numbers. Algebraic numbers also have different types of properties. By using the degree of algebraic numbers, we can classify these numbers. [1]
References
[1] “Algebraic Numbers – Algebra”. 2023. allmath.com. https://www.allmath.com/algebra/algebraic-numbers.
Additional Reading
“Algebraic Number”. 2023. mathsisfun.com. https://www.mathsisfun.com/numbers/algebraic-numbers.html.
“Algebraic Number – Encyclopedia Of Mathematics”. 2023. encyclopediaofmath.org. https://encyclopediaofmath.org/wiki/Algebraic_number.
“Algebraic Number — From Wolfram MathWorld”. 2023. mathworld.wolfram.com. https://mathworld.wolfram.com/AlgebraicNumber.html.
“Art Of Problem Solving”. 2023. artofproblemsolving.com. https://artofproblemsolving.com/wiki/index.php/Algebraic_number.
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.
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The featured image on this page is from the Algebraic Numbers: Definition, Properties, Applications, and Examples page of the AllMath website.
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