Algebraic number

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In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational (or equivalently, integer) coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental. (Here "almost all" has the sense "all but a set of Lebesgue measure zero"; see Properties below.)

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Examples

  • The rational numbers, expressed as the quotient of two integers a and b, b not equal to zero, satisfy the above definition because x = a / b is the root of bxa.[1]
  • The numbers \scriptstyle\sqrt{2} and \scriptstyle\sqrt[3]{3}/2 are algebraic since they are the roots of polynomials x2 − 2 and 8x3 − 3, respectively.
  • The golden ratio φ is algebraic since it is a root of the polynomial x2x − 1.
  • The constructible numbers (those that, starting with a unit, can be constructed with straightedge and compass, e.g. the square root of 2).
  • The quadratic surds (roots of a quadratic polynomial ax2 + bx + c with integer coefficients a, b, and c) are algebraic numbers. If the quadratic polynomial is monic (a = 1) then the roots are quadratic integers.

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