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 \[q^{-1}=a^{-1}b=\frac 1ab=\frac ba\in \mathbb Q.\]

  1. Proposition: Definition of Rational Numbers
  2. Proposition: Algebraic Structure of Rational Numbers Together with Addition
  3. Proposition: Algebraic Structure of Non-Zero Rational Numbers Together with Multiplication
  4. Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication
  5. Definition: Order Relation for Rational Numbers - Positive and Negative Rational Numbers
  6. Definition: Absolute Value of Rational Numbers

Chapters: 1
Parts: 2

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