The integers: construction, signs and order

The construction of the integers

In the lesson on number systems the integers were presented as the set that gathers the natural numbers, zero and the opposites of the natural numbers, and their properties were stated. Now, stating these properties is not proving them: as long as it has not been said what a negative number is, the rule of signs or the compatibility of order with multiplication are conventions, not theorems. This lesson constructs the integers with no other resources than the Peano axioms and the operations on natural numbers, and it deduces all those rules from the construction. As in the rest of the course, the natural numbers begin at $1$; zero appears for the first time in this construction.

The idea is as follows. The equation $x + b = a$ has a natural solution only when $b < a$, and that solution is the difference $a - b$. If this is so, a negative number can be conceived as a pending subtraction, described by the pair of natural numbers $(a, b)$ that would produce it. Many pairs describe the same subtraction; for example, $(5, 2)$, $(4, 1)$ and $(9, 6)$ describe $3$, and $(2, 5)$ and $(1, 4)$ describe what does not yet exist and will be $-3$. Two pairs $(a, b)$ and $(c, d)$ describe the same subtraction when $a - b = c - d$; this condition uses subtractions that may not exist, but adding $b + d$ to both sides turns it into an equality between natural numbers that always makes sense:

Definition (equivalent pairs). Two pairs of natural numbers $(a, b)$ and $(c, d)$ are equivalent, and one writes $(a, b) \sim (c, d)$, when $a + d = b + c$.

On a grid whose points are the pairs of natural numbers, with $a$ on the horizontal axis and $b$ on the vertical one, the pair $(a + 1, b + 1)$ is equivalent to $(a, b)$, because $a + (b + 1) = b + (a + 1)$; hence the pairs equivalent to a given one lie on a line parallel to the diagonal, along which one moves one place to the right and one place up. For this relation to serve to identify pairs, it must behave like an equality: every pair is equivalent to itself, the relation does not depend on the order in which the pairs are named, and two pairs equivalent to a third are equivalent to each other. That is, it must be reflexive, symmetric and transitive: an equivalence relation, in the sense of the lesson on sets, relations and functions, whose example on the relation that will construct the integers anticipated this construction.

The integers as classes of pairs

Since the equivalence of pairs is an equivalence relation, the general results of the lesson on sets, relations and functions hold for it, and there is no need to repeat them. The class of the pair $(a, b)$ is the set of all the pairs of natural numbers equivalent to $(a, b)$; in the notation of that lesson it would be $C((a, b))$, and here it is written, more briefly, $C(a, b)$. Each pair of the class is a representative of it. On the grid, a class is the set of the points of a line parallel to the diagonal. By the lemma on equivalence classes proved there, two pairs are equivalent if and only if their classes are equal, and two classes with some pair in common coincide; and, by the theorem on equivalences and partitions, the classes form a partition of the pairs of natural numbers. From this follows the rule by which two classes are compared:

Definition (integers). An integer is a class of pairs of natural numbers. The set of the integers is denoted by $\mathbb{Z}$; it is the quotient set of the pairs of natural numbers by the equivalence of pairs.

The definition may seem strange, because a number turns out to be an infinite set of pairs. But this is what already happens with fractions: $\frac{1}{2}$ and $\frac{2}{4}$ are different ways of writing one and the same number, and what is defined is what all of them have in common. The class $C(5, 2)$ will be $3$; the class $C(2, 5)$, $-3$; the class $C(1, 1)$, $0$. Before using these names, the operations must be defined and it must be checked that the natural numbers are indeed inside the integers.

The sum and the product of classes

The operations are defined by imitating what would happen if the subtractions existed. If $(a, b)$ represents $a - b$ and $(c, d)$ represents $c - d$, the sum of the two subtractions would be $(a + c) - (b + d)$, and their product, once expanded, $(ac + bd) - (ad + bc)$. This computation is not a proof, because it uses subtractions that do not yet exist; it serves to choose the definitions, in which only sums and products of natural numbers occur:

The definition computes with representatives, and one and the same class has infinitely many of them. If changing the representative changed the result, the definition would define nothing: it would give several values for one and the same operation. It must therefore be proved that the result depends only on the classes; the operation is then said to be well defined.

The properties of addition and multiplication

With the operations well defined, each property of the integers is proved by computing with representatives, and each one inherits the property of the natural numbers that supports it. For all integers $x$, $y$ and $z$ these properties hold:

1. Commutativity and associativity of addition, inherited component by component from those of the natural numbers.
2. Commutativity of multiplication, inherited from the commutativity of addition and of multiplication of the natural numbers, as in step 5 of the preceding proof.
3. Associativity of multiplication and distributivity, inherited from the associativity, the commutativity and the distributivity of the natural numbers.
4. Additive identity: the class $C(1, 1)$, which contains all the pairs $(m, m)$.
5. Opposite: for each class $C(a, b)$, the class $C(b, a)$, which is obtained by reflecting the pairs in the diagonal.
6. Multiplicative identity: the class $C(2, 1)$.

The last four are brief computations. For distributivity, both sides are expanded with the definitions and the same pair is obtained:

For the associativity of multiplication, the two ways of grouping $C(a, b) \cdot C(c, d) \cdot C(e, f)$ give, after distributing, the same pair of sums of four terms:

For the identities and the opposite it suffices to apply the rule for comparing classes: $C(a + 1, b + 1) = C(a, b)$, because $(a + 1) + b = (b + 1) + a$; the pair $(2a + b, a + 2b)$, which defines $C(a, b) \cdot C(2, 1)$, is equivalent to $(a, b)$, because both sides of the condition equal $2a + 2b$; and $C(a, b) + C(b, a) = C(a + b, a + b)$, which is the class of the pairs with two equal terms.