Integer
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. First construct the set of natural numbers according to the Peano axioms, call this P {\displaystyle P}. Then construct a set P − {\displaystyle P^{-}} which is disjoint from P {\displaystyle P} and in one-to-one correspondence with P {\displaystyle P} via a function ψ {\displaystyle \psi }. For example, take P − {\displaystyle P^{-}} to be the ordered pairs (1, n) {\displaystyle (1,n)} with the mapping ψ = n ↦ (1, n) {\displaystyle \psi =n\mapsto (1,n)}. Finally let 0 be some object not in P {\displaystyle P} or P − {\displaystyle P^{-}}, for example the ordered pair (0, 0) {\displaystyle (0,0)}. Then the integers are defined to be the union P ∪ P − ∪ { 0 } {\displaystyle P\cup P^{-}\cup \{0\}}.

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: − x = { ψ (x), if x ∈ P ψ − 1 (x), if x ∈ P − 0, if x = 0 {\displaystyle -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}}}

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.
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