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 $$\in \!\,$$ $$ \Complex \,$$, 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 $$\notin \!\,$$ $$ \mathbb{A} \,$$ }