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

Rational numbers ${\displaystyle \mathbb {Q} \,}$ can be expressed as the ratio of two integers p and q ${\displaystyle \neq \!\,}$ 0 expressed as p/q. In set notation: { p/q: p,q ${\displaystyle \in \!\,}$ ${\displaystyle \mathbb {Z} \,}$ q ${\displaystyle \neq \!\,}$ 0 }
Irrational numbers are those real numbers contained in ${\displaystyle \mathbb {R} \,}$ but not in ${\displaystyle \mathbb {Q} \,}$, where ${\displaystyle \mathbb {R} \,}$ denotes the set of real numbers. In set notation: { x: x ${\displaystyle \in \!\,}$ ${\displaystyle \mathbb {R} \,}$, x ${\displaystyle \notin \!\,}$ ${\displaystyle \mathbb {Q} \,}$ }
Algebraic numbers, sometimes denoted by ${\displaystyle \mathbb {A} }$, 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 ${\displaystyle \in \!\,}$ ${\displaystyle \mathbb {C} \,}$, a0,...,an ${\displaystyle \in \!\,}$ ${\displaystyle \mathbb {Z} \,}$ }
Transcendental numbers are those numbers which are Real (${\displaystyle \mathbb {R} \,}$) , but are not Algebraic (${\displaystyle \mathbb {A} }$). In set notation: { x: x ${\displaystyle \in \!\,}$ ${\displaystyle \mathbb {R} \,}$, x ${\displaystyle \notin \!\,}$ ${\displaystyle \mathbb {A} \,}$ }