# Part: Rational Numbers

### A motivation for the set of rational numbers $$\mathbb Q$$

The set of integers $$\mathbb Z=\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}$$ can be extended to the set of all rational numbers by introducing the ratios $\mathbb Q=\left\{q:=\frac xy, x\in \mathbb Z, y\in \mathbb Z\setminus\{0\}\right\}.$ The major motivation for this extension is the solvability of the equation $ax=b,~~~~~~~(b\in \mathbb Z, a\in \mathbb Z\setminus\{0\})~~~~~~~~~~~~~~~~~~( * )$ which cannot be always solved by an integer for any two given initial integers $$b\in \mathbb Z, a\in \mathbb Z\setminus\{0\}$$. For instance, while $$-3x=15$$ has the integer solution $$-5\in\mathbb Z$$, there is no integer solution of the equation $$-2x=15$$. The main reason for the (in general) missing solutions to the equation $$( * )$$ is the algebraic structure of integers together with addition and multiplication $$(\mathbb Z,+,\cdot)$$, which turns out to be a ring. Since, in general, the elements of a every ring do not have a multiplicative inverse, the equation $$( * )$$ cannot be solved by $ax=b\Longleftrightarrow a^{-1}ax=a^{-1}b\Longleftrightarrow x=a^{-1}b,$ since the multiplicative inverse $$a^{-1}$$ of $$a\in\mathbb Z$$ simply does not exist in $$\mathbb Z$$.

However, $$(\mathbb Z, + ,\cdot)$$ is a special kind of a ring, called an integral domain. Following the construction of fields from integral domains, the integral domain $$(\mathbb Z,+ ,\cdot)$$ is extended to the field $$(\mathbb Q, +, \cdot)$$, which turns out to be the smallest "extended" algebraic structure, in which the equation $$( * )$$ becomes solvable, regardless which initial integers $$b\in \mathbb Z, a\in \mathbb Z\setminus\{0\}$$ we choose. The solvability of $$( * )$$ is then ensured by the existence of uniquely defined multiplicative inverse elements $q^{-1}=a^{-1}b=\frac 1ab=\frac ba\in \mathbb Q.$

Chapters: 1
Parts: 2

Github: ### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013