Editing Integer (section)

=== Traditional development ===
In elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers, [[zero]], and the negations of the natural numbers. This can be formalized as follows.<ref>{{cite book |last1=Mendelson |first1=Elliott |title=Number systems and the foundations of analysis |date=1985 |publisher=Malabar, Fla. : R.E. Krieger Pub. Co. |isbn=978-0-89874-818-5 |page=153 |url=https://archive.org/details/numbersystemsfou0000mend/page/152/mode/2up}}</ref> First construct the set of natural numbers according to the [[Peano axioms]], call this <math>P</math>. Then construct a set <math>P^-</math> which is [[Disjoint sets|disjoint]] from <math>P</math> and in one-to-one correspondence with <math>P</math> via a function <math>\psi</math>. For example, take <math>P^-</math> to be the [[ordered pair]]s <math>(1,n)</math> with the mapping <math>\psi = n \mapsto (1,n)</math>. Finally let 0 be some object not in <math>P</math> or <math>P^-</math>, for example the ordered pair (0,0). Then the integers are defined to be the union <math>P \cup P^- \cup \{0\}</math>. 

The traditional arithmetic operations can then be defined on the integers in a [[piecewise]] fashion, for each of positive numbers, negative numbers, and zero. For example [[negation]] is defined as follows:

<math display="block">
-x = \begin{cases}
  \psi(x), & \text{if } x \in P \\
  \psi^{-1}(x), & \text{if } x \in P^- \\
  0, & \text{if } x = 0
\end{cases}
</math>

The traditional style of definition leads to many different cases (each arithmetic operation needs to be defined on each combination of types of integer) and makes it tedious to prove that integers obey the various laws of arithmetic.<ref>{{cite book |title=Number Systems and the Foundations of Analysis |series=Dover Books on Mathematics |first=Elliott |last=Mendelson |publisher=Courier Dover Publications |year=2008 |isbn=978-0-486-45792-5 |page=86 |url=https://books.google.com/books?id=3domViIV7HMC&pg=PA86 |access-date=2016-02-15 |archive-url=https://web.archive.org/web/20161208233040/https://books.google.com/books?id=3domViIV7HMC&pg=PA86 |archive-date=2016-12-08|url-status=live}}.</ref>