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

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

Irrational numbers are those real numbers contained in $\mathbb {R} \,$ but not in $\mathbb {Q} \,$, where $\mathbb {R} \,$ denotes the set of real numbers. In set notation: { x: x $\in \!\,$ $\mathbb {R} \,$, 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 $\in \!\,$ $\mathbb {C} \,$, a₀,...,a_n $\in \!\,$ $\mathbb {Z} \,$ }

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