Number Theory/Irrational, Rational, Algebraic, And Transcendental Numbers

      Rational numbers \mathbb{Q} \, can be expressed as the ratio of two integers p and q \ne \!\, 0 expressed as p/q. In set notation: { p/q: p,q \in \!\, \Z \,q \ne \!\, 0 }

      Irrational numbers are those real numbers contained in \R \, but not in \mathbb{Q} \,, where \R \, denotes the set of real numbers. In set notation: { x: x \in \!\, \R \,, x \notin \!\, \mathbb{Q} \, }

      Algebraic numbers, sometimes denoted by \mathbb{A}, are those numbers which are roots of an algebraic equation with integer coefficients (an equivalent formulation using rational coefficients exists). In math terms: { x: anxn + an-1xn-1 + an-2xn-2 + ... + a1x1 + a0 = 0, x \in \!\, \C \,, a0,...,an\in \!\, \Z \, }

      Transcendental numbers are those numbers which are Real ( \R \,) , but are not Algebraic (\mathbb{A}). In set notation: { x: x \in \!\, \R \,, x \notin \!\, \mathbb{A} \, }

      Last modified on 14 July 2009, at 11:15