8.2: The Integers

We construct the integers out of the natural numbers. The algebraic purpose of the integers is to include additive inverses for natural numbers. Of course this naturally gives rise to the operation of subtraction.

Let \(Z=\mathbb{N} \times \mathbb{N}\). Define an equivalence relation, \(\sim\) on \(Z\) by \[\left\langle m_{1}, n_{1}\right\rangle \sim\left\langle m_{2}, n_{2}\right\rangle \quad \Longleftrightarrow \quad m_{1}+n_{2}=m_{2}+n_{1} .\] Then the integers are \[\mathbf{Z}:=Z / \sim .\] We think of the ordered pair \(\langle m, n\rangle \in \mathbf{Z}\) as being a representative of the integer \(m-n\). We say that an integer is positive if \(m>n\) and negative if \(m<n\). It should be clear that the set of non-negative integers (that is \(\mathbb{N}\) ) is \[\{[\langle m, n\rangle] \mid m \geq n\}=\{[\langle m, 0\rangle] \mid m \in \mathbb{N}\} .\] Let \(\mathbb{Z}\) be the (intuitive) integers and let \(i: \mathbf{Z} \rightarrow \mathbb{Z}\) be defined by \[i([\langle m, n\rangle])=m-n .\] Then \(i\) is a bijection. As we did with the natural numbers, we shall construct operations and order on \(\mathbf{Z}\) that agree with the usual operations and an order on \(\mathbb{Z}\). Of course, we could use \(i\) and the usual definitions in \(\mathbb{Z}\) to define operations and relations on \(\mathbf{Z}\), but that would miss the spirit of the construction, and would neglect the desire for set-theoretic definitions. Analogous to the construction of the previous section, we define \(\mathbb{Z}\) as \(\mathbf{Z}\). Let \(x_{1}, x_{2} \in \mathbb{Z}\) where \(x_{1}=\left[\left\langle m_{1}, n_{1}\right\rangle\right]\) and \(x_{2}=\left[\left\langle m_{2}, n_{2}\right\rangle\right]\). Addition is defined by \[x_{1}+x_{2}=\left[\left\langle m_{1}+m_{2}, n_{1}+n_{2}\right\rangle\right] .\] The additive inverse of \([\langle m, n\rangle]\) is \([\langle n, m\rangle]\) (i.e. the sum of these integers is \([\langle 0,0\rangle]\) - the additive identity in \(\mathbb{Z})\).

Multiplication is defined by \[x_{1} \cdot x_{2}=\left[\left\langle m_{1} \cdot m_{2}+n_{1} \cdot n_{2}, n_{1} \cdot m_{2}+m_{1} \cdot n_{2}\right\rangle\right] .\] The linear ordering on \(\mathbb{Z}\) is defined by \[x_{1} \leq x_{2} \Longleftrightarrow m_{1}+n_{2} \leq n_{1}+m_{2} .\] Addition and multiplication have been defined for the natural numbers, and the operations and linear ordering on \(\mathbb{Z}\) are defined with respect to operations and the linear ordering that were previously defined for \(\mathbb{N}\). Note that all our definitions were given in terms of representatives of equivalence classes. To show that \(+, \cdot\) and \(\leq\) are well-defined, we must show that the definitions are independent of the choice of representative - see Exercise 8.6.