Algebraic number – Definition & Meaning

Algebraic number is a concept that is used in mathematics to describe a number that is a solution to a polynomial equation with integer coefficients. This concept has been widely used in various fields of mathematics, including number theory, algebra, and geometry. In this article, we will explore the definition and meaning of algebraic numbers, their origin, and their significance in different dictionaries.

Definitions

An algebraic number is a complex number that satisfies a polynomial equation with integer coefficients. In other words, it is a number that can be expressed as a root of a polynomial equation with integer coefficients. For example, the number √2 is an algebraic number because it is a solution to the equation x^2 – 2 = 0.

Origin

The concept of algebraic numbers was first introduced by mathematicians in the 19th century. The German mathematician Ernst Eduard Kummer was the first to develop a theory of algebraic numbers, which he used to solve Fermat’s Last Theorem for certain cases.

Meaning in different dictionaries

The meaning of algebraic number in different dictionaries is consistent with the definition given above. The Oxford English Dictionary defines an algebraic number as “a complex number that is a root of a polynomial equation with integer coefficients.” Similarly, the Merriam-Webster dictionary defines an algebraic number as “a complex number that is a root of a polynomial equation with rational coefficients.”

Associations

Algebraic numbers are associated with various branches of mathematics, including number theory, algebra, and geometry. They are used in the study of Diophantine equations, which are polynomial equations with integer coefficients that have solutions in integers. They are also used in algebraic geometry, which is the study of geometric objects defined by polynomial equations.

Synonyms

Some synonyms of algebraic number include algebraic integer and polynomial root.

Antonyms

There are no direct antonyms of algebraic number, as it is a specific mathematical concept.

The same root words

The root words of algebraic number are algebra and number.

Example Sentences

  1. The number √5 is an algebraic number because it is a solution to the equation x^2 – 5 = 0.
  2. The concept of algebraic numbers is used in the study of Diophantine equations.
  3. Algebraic numbers are used in algebraic geometry to define geometric objects.
  4. An algebraic integer is a special case of an algebraic number where the polynomial equation has integer coefficients.
  5. The study of algebraic numbers has led to important breakthroughs in number theory.
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