algebraic number
Noun:
An algebraic number is a number that is a root of a non-zero polynomial equation in one variable with integer (or, equivalently, rational) coefficients. In simpler terms, it is a number that can be a solution to an equation like a_n*x^n + ... + a_1*x + a_0 = 0, where all the a coefficients are whole numbers (or fractions).
This term is used in the field of mathematics, specifically in number theory and abstract algebra, to classify numbers.
* All rational numbers (like 2, -1/3, 0.75) are algebraic numbers because they are roots of simple linear equations (e.g., x - 0.75 = 0).
* Many irrational numbers are also algebraic numbers, such as √2 (which is a root of x² - 2 = 0) and the golden ratio φ (which is a root of x² - x - 1 = 0).
- In a textbook: "The set of algebraic numbers is countable, unlike the set of real numbers."
- In a proof: "To demonstrate that √5 is an algebraic number, consider the polynomial equation ."
- In a lecture: "Pi (π) and Euler's number (e) are famous examples of numbers that are algebraic numbers; they are transcendental."
- Algebraic Number Field: An extension field of the rational numbers that contains only algebraic numbers.
- Algebraic Integer: A special type of algebraic number where the leading polynomial coefficient is 1 (a monic polynomial with integer coefficients).
- Algebraic (adjective): Pertaining to algebra or satisfying a polynomial equation. (e.g., an expression, an solution).
- Transcendental Number (noun): The conceptual opposite; a real or complex number that is an algebraic number (e.g., π, e).
- There is no perfect single-word synonym. The term is defined precisely by its mathematical property. Descriptive phrases include:
- A root of a polynomial with rational coefficients.
- A number that is algebraic over the field of rational numbers (ℚ).
Given the technical nature of the term, it is not associated with phrasal verbs or idioms. Key related mathematical concepts include: * Polynomial * Root / Zero (of a polynomial) * Rational Number * Field Extension
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root of an algebraic equation with rational coefficients
