3.3 Integers from equivalence classes

The natural numbers are not enough for every algebraic problem. A simple equation such as

$$1 = x + 2$$

has no solution in $N$. So if we want subtraction to become possible in a systematic way, we need a larger number system.

The source notes do not introduce negative numbers by intuition alone. Instead, they construct the integers from objects we already understand well, namely pairs of natural numbers.

The guiding idea

A pair (a,b) is meant to represent the formal difference

$$a-b.$$

From that viewpoint, different pairs can represent the same intended integer. For example,

$$(3,1), \quad (5,3), \quad (8,6)$$

all suggest the same difference, namely 2.

So the integer should not be the ordered pair itself. It should be the whole equivalence class of pairs that encode the same difference.

The relation on $N^2$

The equation $a+d=b+c$ is exactly what we would expect if (a,b) and (c,d) are both meant to represent the same formal difference:

$$a-b=c-d.$$

Indeed, rearranging this equality gives precisely $a+d=b+c$.

Why this relation is the right one

The proof is not difficult, but it is worth understanding because it explains why quotient constructions work.

What an integer now is

So an integer is not one pair, but an entire class of equivalent pairs.

This construction explains how the familiar numbers reappear:

• [(0,0)] behaves like 0;
• [(1,0)] behaves like 1;
• [(0,1)] behaves like $-1$;
• more generally, [(n,0)] gives the usual natural number n.

Embedding the natural numbers

The natural numbers are still present inside the integers. They are not lost; they are reinterpreted.

This is why quotient constructions do not destroy previous number systems. They usually enlarge them while preserving a recognizable copy inside the new one.

Positive, negative, and zero

The source notes emphasize that sign is not attached to one chosen pair, but to the entire class.

• an integer is positive when it has representatives (a,b) with $a>b$;
• it is negative when it has representatives with $a<b$;
• it is zero when it has representatives with $a=b$.

Because these properties must not depend on the chosen representative, one also has to check that sign is well defined on equivalence classes.

Arithmetic on equivalence classes

To turn the quotient set into a number system, we still need operations. Addition is defined by

$$[(a,b)] + [(c,d)] := [(a+c,b+d)].$$

This definition matches the formal-difference intuition:

$$(a-b)+(c-d)=(a+c)-(b+d).$$

The next step, developed further in the source notes, is to check that such definitions are well defined, meaning they do not depend on the representatives chosen.

A concrete class calculation

Common mistakes

Quick checks

Exercises

Read this first

This note depends on the language of equivalence relations from 2.2 Functions and relations and continues into 3.4 Rationals and well-defined operations.