Rational Number

Definition

$\mathbb{Q}$

The quotient or fraction $\frac{p}{q}$ of two integers, a numerator $p$ and a non-zero denominator $q$.

The $\mathbb{Q}$ form an ordered Field which contains the Integer

Construction

$(m_1,n_1)\sim (m_2,n_2)\iff m_1{n}_2=m_2{n}_1$ Define an Equivalence Class $\sim$ on the set of ordered pairs $(m,n)\in (\mathbb{Z}\times (\mathbb{Z}\setminus \{0\}))$ of integers such $n\ne 0$,

The set of rational numbers, denoted by $\mathbb{Q}$, is defined as the set of equivalence classes under the relation $\sim$ A rational number is represented by the Equivalence Class $[(m,n)]$ of ordered pairs under the $\sim$

Operations

$[(m_1,n_1)]+[(m_2,n_2)]:=[(m_1{n}_2+n_1{m}_2,n_1{n}_2)]$ $[(m_1,n_1)]\cdot [(m_2,n_2)]:=[(m_1{m}_2,n_1{n}_2)]$ $[(m_1,n_1)]÷[(m_2,n_2)]=\frac{[(m_1,n_1)]}{[(m_2,n_2)]}:=[(m_1{n}_2,n_1{m}_2)]$ where $[(m_2,n_2)]\ne 0$ $[(m_1,m_1)]:=1$ $\frac{1}{[(m_1,n_1)]}:=[(n_1,m_1)]$ where $[(n_1,m_1)]\ne 0$