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 ${\frac {p}{q}}$. In set notation:

$\left\{{\frac {p}{q}}:p,q\in \mathbb {Z} ,q\neq 0\right\}$

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:

$\{r\in \mathbb {R} :r\notin \mathbb {Q} \}$

Algebraic numbers ${\overline {\mathbb {Q} }}$ (sometimes denoted by ${\mathbb {A}}$) are those numbers which are roots of polynomial equations with rational coefficients (an equivalent formulation using integer coefficients exists). In math terms:

${\bigl \{}c\in \mathbb {C} :\exists \,P\in \mathbb {Q} [z],P(c)=0{\bigr \}}$

Transcendental numbers are those numbers which are Real ($\mathbb {R}$), but are not Algebraic (${\overline {\mathbb {Q} }}$). In set notation:

$\{r\in \mathbb {R} :r\notin {\overline {\mathbb {Q} }}\}$