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A motivation for the set of integers \(\mathbb Z\)

It is broadly known and usually taught already in the elementary school that the set of natural numbers \(\mathbb N=\{0,1,2,3,\ldots\}\) can be extended to the set of all integers \(\mathbb Z=\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}\) by introducing the negative whole numbers:\(-1,-2,-3,\ldots\). The major motivation for this extension is the solvability of the equation
\[a+x=b,~~~~~~~(a,b\in\mathbb N)~~~~~~~~~~~~~~~~~~( * )\]
which cannot be solved by a natural number for any two given initial values \(a,b\in\mathbb N\). For instance, while \(2+x=5\) has the solution \(3\in\mathbb N\), there is no such solution (in the set \(\mathbb N\)) of the equation \(9+x=5\). The main reason for the (in general) missing solutions to the equation \(( * )\) is the algebraic structure of natural numbers together with addition, which turns out to be a cancellative and commutative semigroup (more precisely a special case of a semigroup called monoid).

The following questions arise:

  1. Which is “the smallest extended” algebraic structure, in which the equation \(( * )\) becomes solvable, regardless which initial natural numbers \(a,b\in\mathbb N\) we choose?
  2. Exactly which features of this extended algebraic structure make the equation \(( * )\) solvable?
  3. Which technical steps are necessary to find this extended algebraic structure?
  4. How can these steps be logically derived from axioms?

This part of bookofproofs.org is dedicated to the set of integers. In particular, it provides answers to questions raised above. It turns out that the answer to the 1st question is the algebraic structure \((\mathbb Z, + )\), which is the (additive) commutative group of integers. Anticipating the answer to the 2nd question, the solvability of \(( * )\) is then ensured by the existence of uniquely defined inverse elements \(a^{-1}\in \mathbb Z\) (which are usually written in the additive notation \( – a\in \mathbb Z\)), for which \(( * )\) can be solved by

\[a+x=b~~~\Longleftrightarrow~~~x=(-a + b).\]
The technical steps necessary to establish \((\mathbb Z, + )\) as a new (and well-defined) algebraic structure (3rd question), turn out to be a simple application of the theorem dealing with the construction of groups from commutative cancellative semigroups. By applying this theorem to the special case of the respective semigroup \((\mathbb N, + )\), we will be led to the structure of \((\mathbb Z, + )\). Finally, the answer to the 4th question is established in bookofproofs.org for each theorem separately, providing overviews of the proceeding mathematical results derived from basic axioms (learn more about the axiomatic method).

| | | | created: 2014-02-20 22:18:14 | modified: 2019-01-20 20:37:24 | by: bookofproofs | references: [696]

1.Proposition: Definition of Integers

2.Proposition: Algebraic Structure of Integers Together with Addition and Multiplication

3.Proposition: Algebraic Structure of Integers Together with Addition

4.Definition: Order Relation for Integers - Positive and Negative Integers

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Bibliography (further reading)

[696] Kramer Jürg, von Pippich, Anna-Maria: “Von den natürlichen Zahlen zu den Quaternionen”, Springer-Spektrum, 2013

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