Last modified on 14 July 2009, at 11:15

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} \, }