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

Template:Merge Rational numbers $\mathbb{Q}$ can be expressed as the ratio of two integers $p$ and $q\ne 0$ expressed as $\frac{p}{q}$. In set notation:

$\{\frac{p}{q}:p,q\in \mathbb{Z},q\ne 0\}$

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:

$\{r\in ℝ:r\notin \mathbb{Q}\}$

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

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

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

$\{r\in ℝ:r\notin \bar{\mathbb{Q}}\}$