Algebraic Number - Properties

Properties

  • The set of algebraic numbers is countable (enumerable).
  • Hence, the set of algebraic numbers has Lebesgue measure zero (as a subset of the complex numbers), i.e. "almost all" complex numbers are not algebraic.
  • Given an algebraic number, there is a unique monic polynomial (with rational coefficients) of least degree that has the number as a root. This polynomial is called its minimal polynomial. If its minimal polynomial has degree, then the algebraic number is said to be of degree . An algebraic number of degree 1 is a rational number.
  • All algebraic numbers are computable and therefore definable and arithmetical.
  • The set of real algebraic numbers is linearly ordered, countable, densely ordered, and without first or last element, so is order-isomorphic to the set of rational numbers.

Read more about this topic:  Algebraic Number

Famous quotes containing the word properties:

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    —John Locke (1632–1704)

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