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: a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + ... + a₁x¹ + a₀ = 0, x $\in \!\,$ $\C \,$, a₀,...,a_n $\in \!\,$ $\Z \,$ }

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