page_title: Algebraic number
section_title: Properties
hierarchical_section_title: Algebraic number / Properties
caption_attribution_description: Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours indicate degree of the polynomial the number is a root of (red = linear, i.e. the rationals, green = quadratic, blue = cubic, yellow = quartic...). Points becomes smaller as the integer polynomial coefficients become larger. View shows integers 0,1 and 2 at bottom right, +i near top.
context_page_description: An algebraic number is any complex number that is a root of a non-zero polynomial in one variable with rational coefficients.
  All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as π and e, are called transcendental numbers.
  While the set of complex numbers is uncountable, the set of algebraic numbers is countable and has measure zero in the Lebesgue measure as a subset of the complex numbers, and in this sense almost all complex numbers are transcendental.
context_section_description: 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 n, then the algebraic number is said to be of degree n. For example, all rational numbers have degree 1, and an algebraic number of degree 2 is a quadratic irrational.
  The algebraic numbers are dense in the reals, linearly ordered, and without first or last element (and therefore order-isomorphic to the set of rational numbers).
  The set of algebraic numbers is countable (enumerable), and therefore its Lebesgue measure as a subset of the complex numbers is 0 (essentially, the algebraic numbers take up no space in the complex numbers). That is to say, "almost all" real and complex numbers are transcendental.
  All algebraic numbers are computable and therefore definable and arithmetical.
  For real numbers a and b, the complex number a + bi is algebraic if and only if both a and b are algebraic.
caption: Algebraic numbers on the complex plane colored by degree (red = 1, green = 2, blue = 3, yellow = 4)