# 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: anxn + an-1xn-1 + an-2xn-2 + ... + a1x1 + a0 = 0, x $\in \!\,$ $\mathbb {C} \,$ , a0,...,an $\in \!\,$ $\mathbb {Z} \,$ }
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 $\notin \!\,$ $\mathbb {A} \,$ }