Integer Construction

The integers may be defined as the set of all natural pairs $ℕ × ℕ$, and may be denoted as $ℤ$.

Equality

The proposition of whether two integers $(a_1, b_1), (a_2, b_2)$ are equal can be rewritten as an equality statement on two naturals.

Order

Addition

Let $(a_1, b_1), (a_2, b_2)$ be natural pairs. Then integer addition is defined as entry-wise natural addition.

Negation

Multiplication

Let $(a_1, b_1), (a_2, b_2)$ be natural pairs. Then integer multiplication is defined:

Integers as a Group

The integers qualifies as a group under addition and also as a commutative ring under both addition and multiplication.

Integer Addition is Well Defined

A function is well defined when expressions which are considered the same object are treated identically. For the integers under addition, we would want to know that $[0, 0]$ is treated as the same object as $[1, 1]$.

Let $a, a', b, b', x, y ∈ ℕ$.

But the truth of our original assumption $[a, b] = [a', b']$ rests on whether $a + b' = a' + b$, and thus addition is well defined. This proof can be repeated for the commutative case.

Integer Multiplication is Well Defined

By the equality of the integers.

But we have already assumed that $[a, b] = [a', b']$. This proof can be repeated for the commutative case.

Subset of Integers isomorphic to Naturals

Integers of the form $[n, 0]$ are isomorphic to naturals of the form $n$. Consider that integers in this form are closed under addition and multiplication:

And that two integers in this form are only equal when:

Trichotomy of the Integers

If $x$ is an integer, then only one of the three possibilities may be true:

1. $x = 0$.
2. $x$ is equivalent to a positive natural.
3. $x$ is equivalent to a negative natural.

First we will describe what we consider to be a “negative” natural. We have accepted that integers of the form $[n, 0]$ are isomorphic to the naturals under addition. Then by the definition of integer negation, a negative natural will take the form of:

Secondly, since we already accept that $x$ is an integer, then $x$ may take on the form of a natural pair $[a, b]$. Then by the trichotomy of the naturals we know that only one of three possibilities may occur at any time:

1. $a = b$
2. $a < b$
3. $a > b$

In the event of $a = b$ we can show that such integers are always equivalent to the natural $0$.

In the event of $a > b$, then by the definition of strict natural order:

In the event of $a < b$ we have by definition of strict natural order:

Integers cannot reciprocate $0$

We want to see if the integers have any zero divisors.