Creating the Integers

Last time, we defined the integers $\mathbb Z$ as a collection of sets of the form $[(x, y)]$ , each being a specific subset of $\mathbb N \times \mathbb N$ satisfying a special property that encapsulates subtraction, but without invoking negative numbers to begin with. More specifically,

$[(x, y)] := \{(z, w) \in \mathbb N \times \mathbb N : x+w = z+y\} \subseteq \mathbb N \times \mathbb N.$

We write $(x, y) \equiv (z, w)$ whenever $x + w = z + y$ , and have seen that this is an equivalence relation—a generalisation of equality. We have also seen that any generic point $(x, y)$ belongs to exactly one of these sets, namely, $[(x, y)]$ . We then denoted this collection as $\mathbb Z := \mathbb N^2/{\equiv}$ .

But how do we know this definition agrees with our prior intuition of the integers?

Well, firstly, we had better have $\mathbb Z$ contain $\mathbb N$ . But this isn’t possible since $\mathbb Z = \mathbb N^2/{\equiv}$ is a completely different set from $\mathbb N$ . So our naïve notion of including meaning “subsets” is too weak to capture this idea.

Thankfully, that’s not the only way to discuss inclusion—we have injections on our side.

Theorem 1. The canonical embedding $\iota : \mathbb N \to \mathbb Z$ defined by $\iota(n) = [(n, 0)]$ is injective.

Proof. Fix $n_1, n_2 \in \mathbb N$ such that $\iota(n_1) = \iota(n_2)$ . By the definition of $\iota$ ,

$[(n_1,0)] = \iota(n_1) = \iota(n_2) = [(n_2, 0)].$

This means that

$(n_1,0) \equiv (n_2,0) \quad \Rightarrow \quad n_1 = n_1 + 0 = n_2 + 0 = n_2,$

as required.

However, does that really count? The whole goal of the natural numbers is to do addition, among other processes. This raises the question: does $\iota$ preserve addition?

In fact, we can reverse the question: can we define integer addition such that $\iota(n_1 + n_2) = \iota(n_1) + \iota(n_2)$ ? If so, then we can define integer addition even for numbers not created via $\iota$ . This is a principle in mathematics known as extension—we extend our ideas from a smaller set to a larger set.

Now, the condition $\iota(m + n) = \iota(m) + \iota(n)$ is equivalent to the condition $[(n_1+n_2, 0)] = [(n_1, 0)] + [(n_2, 0)]$ . This requires us to define a sensible notion of addition on the right-hand side. To that end, let’s observe that given $(x_1, y_1) \in [(n_1,0)]$ and $(x_2,y_2) \in [(n_2,0)]$ , we have

$x_1 + 0 = n_1 + y_1, \quad x_2 + 0= n_2 + y_2.$

Adding yields

$x_1 + x_2 + 0 + 0 = (n_1 + n_2) + (y_1 + y_2),$

which establishes $(x_1+x_2, y_1+y_2) \in [(n_1+n_2, 0)]$ . Hence, we can define integer addition in a relatively straightforward manner, with its proof a similar verification.

Theorem 2. For any $x_1, y_1, x_2, y_2 \in \mathbb N$ , integer addition defined by

$[(x_1, y_1)] + [(x_2, y_2)] := [(x_1 + x_2, y_1 + y_2)]$

is well-defined. Furthermore, $\iota(n_1+n_2) = \iota(n_1) + \iota(n_2)$ .

This allows us to embed $\mathbb N$ into $\mathbb Z$ with addition preserved. Henceforth, we will identify $\mathbb N$ with $\iota(\mathbb N) \subseteq \mathbb Z$ and regard $\mathbb N \subset \mathbb Z$ . For completeness, we state the definition of integer multiplication, which arises essentially from the distributivity property.

Theorem 3. For any $x_1, y_1, x_2, y_2 \in \mathbb N$ , integer multiplication defined by

$[(x_1, y_1)] \cdot [(x_2, y_2)] := [(x_1 x_2 + y_1 y_2, x_1 y_2 + x_2 y_1)]$

is well-defined. Furthermore, for any $n_1,n_2 \in \mathbb N$ , $\iota(n_1+n_2) = \iota(n_1) \cdot \iota(n_2)$ .

The definition of integer addition helps us prove many group-theoretic properties of $\mathbb Z$ . Coupling that with multiplication turns $\mathbb Z$ into a ring. The inverse property of integer addition, in particular, stands out.

Theorem 3. For any $n \in \mathbb Z$ , $n + 0 = 0 + n = n$ . Furthermore, there exists some unique additive inverse $-n \in \mathbb Z$ of $n$ such that $n+(-n) = (-n)+n = 0$ .

Proof. Let $n := [(x,y)]$ . The first result is a matter of bookkeeping via

$n + 0 = [(x,y)] + [(0,0)] = [(x+0,y+0)] = [(x,y)] = n.$

For the second result, define $-n := [(y,x)]$ . Then

$n+(-n) = [(x,y)] + [(y,x)] = [(x+y,x+y)] = [(0,0)] = 0,$

since $(x+y,x+y) \equiv (0,0)$ . Furthermore, suppose there exists $-n'$ such that $n + (-n') = (-n') + n = 0$ . Then

$-n = -n + 0 = -n + (n + (-n')) = (-n + n) + (-n') = 0 + (-n') = -n',$

establishing the uniqueness of the additive inverse.

In particular, we can now formally define the negative number.

Definition 1. For any natural number $n\in \mathbb N$ , define $-n := [(0,n)]$ . Thus, subtraction of two integers $m, n \in \mathbb Z$ is defined by $m - n:= m + (-n)$ .

Discussing division will take us into the rational numbers. We will briefly define it in a similar manner.

Definition 2. For any set $K$ , denote $K^* := K \backslash\{0\}$ . Define the equivalence relation $\equiv$ on $\mathbb Z \times \mathbb Z^*$ by

$(x_1,y_1) \equiv (x_2,y_2)\quad \iff \quad x_1 y_2 = x_2 y_1.$

We define the rational numbers by $\mathbb Q := \mathbb (\mathbb Z \times \mathbb Z^*)/{\equiv}$ , with addition and multiplication defined by

$\begin{aligned}[(x_1,y_1)] + [(x_2,y_2)] &:= [(x_1y_2 + x_2y_1, y_1y_2)], \\ [(x_1,y_1)] \cdot [(x_2,y_2)] &:= [(x_1 x_2, y_1 y_2)].\end{aligned}$

Finally, we denote $r/s := [(r,s)]$ .

One can verify the expected properties of $\mathbb Q$ .

—Joel Kindiak, 7 Dec 24, 0112H

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Creating the Integers

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