The Integers

Why Extend the Natural Numbers?

The natural numbers begin at zero and continue by taking successors. They are suited to counting, but subtraction does not always stay within them: \(2-5\) is not a natural number. The integers extend the number system by including negative numbers, so that every natural number has an additive opposite.

We write \(\mathbb Z\) for the integers. Informally, they are the numbers \[ \ldots,-3,-2,-1,0,1,2,3,\ldots. \] A precise way to build them from \(\mathbb N_0\) is to represent a possible difference by an ordered pair \((a,b)\) of natural numbers. The pair stands for “\(a\) minus \(b\).” For instance, \((7,4)\) and \((5,2)\) should represent the same integer, since both differences are \(3\). We must therefore specify when pairs represent the same integer before treating the pairs as numbers.

We use the elementary arithmetic laws for the natural numbers, including associativity and commutativity of addition, distributivity, and cancellation for addition. We also use the usual order on \(\mathbb N_0\): any two natural numbers can be compared, and if \(a\geq b\), then \(a=b+k\) for some \(k\in\mathbb N_0\).

The equality test uses only addition in \(\mathbb N_0\). It avoids presupposing that subtraction is already available there. For example, \((7,4)\sim(5,2)\) because \(7+2=4+5\). Thus the two pairs determine one integer, not two different integers.

Why the Equality Rule Works

An equivalence relation must be reflexive, symmetric, and transitive. These properties ensure that representing the same integer behaves consistently: every pair represents the same value as itself, equality can be reversed, and equality can be passed along a chain.

Proof. For reflexivity, let \((a,b)\) be any pair. Since \(a+b=b+a\), we have \((a,b)\sim(a,b)\).

For symmetry, suppose \((a,b)\sim(c,d)\). Then \(a+d=b+c\). Reversing the two sides gives \(c+b=d+a\), so \((c,d)\sim(a,b)\).

For transitivity, suppose \((a,b)\sim(c,d)\) and \((c,d)\sim(e,f)\). The definitions give \[ a+d=b+c\qquad\text{and}\qquad c+f=d+e. \] Adding \(f\) to the first equality and using the second yields \[ a+d+f=b+c+f=b+d+e. \] By associativity and commutativity, this says \(a+f+d=b+e+d\). Cancellation gives \(a+f=b+e\), so \((a,b)\sim(e,f)\). The relation is therefore transitive, and hence an equivalence relation. \(\square\)

The integer represented by \((a,b)\) is often informally written \(a-b\), but its precise meaning here is the entire equivalence class \([(a,b)]\). In particular, two different pairs may name the same integer. The equality criterion is

Zero, Positive Integers, and Negative Integers

The pair \((n,0)\) represents the nonnegative integer \(n\). The pair \((0,n)\) represents its negative. In particular, both \((0,0)\) and \((n,n)\) represent zero: for the latter, \((n,n)\sim(0,0)\) because \(n+0=n+0\). More generally, every pair can be put into one of these two forms.

Proof. Let an integer be represented by \((a,b)\). Since natural numbers are comparable, either \(a\geq b\) or \(b\geq a\). If \(a\geq b\), write \(a=b+k\) for some \(k\in\mathbb N_0\). Then \[ a+0=b+k, \] so \((a,b)\sim(k,0)\). If \(b\geq a\), write \(b=a+k\). Then \(a+k=b+0\), so \((a,b)\sim(0,k)\). Thus every class has at least one of the stated forms.

Now suppose \((k,0)\sim(0,m)\), where \(k,m\in\mathbb N_0\). The equality criterion gives \(k+m=0+0=0\). A sum of natural numbers is zero only when both terms are zero, so \(k=m=0\). Therefore a nonnegative form and a negative form can represent the same integer only when both represent zero. In particular, if \(k>0\), \((k,0)\) and \((0,k)\) represent different integers. \(\square\)

This description agrees with the familiar notation: \([(k,0)]\) is written \(k\), and \([(0,k)]\) is written \(-k\). When \(k=0\), both notations refer to zero, not to two distinct integers. For positive \(k\), \(k\) and \(-k\) are different.

Arithmetic with Integer Classes

The operations on integers must not depend on which pair is chosen to represent an integer. Addition is defined by adding the corresponding entries, and multiplication is defined by expanding a product of differences.

The multiplication formula reflects \[ (a-b)(c-d)=(ac+bd)-(ad+bc). \] The definitions are independent of the representatives chosen: if either pair is replaced by an equivalent pair, the resulting pair for the sum or product is equivalent to the original result. This follows by substituting the defining equalities \(a+d=b+c\) for equivalent pairs and using the natural-number laws stated above. Thus the formulas define operations on equivalence classes, not merely on individual pairs.

The additive opposite works because \[ [(a,b)]+[(b,a)]=[(a+b,b+a)]=[(a+b,a+b)]=[(0,0)]. \] The middle pair represents zero by the equality criterion. Therefore the extension supplies an additive opposite for every integer. Subtraction can now be defined using addition and this opposite:

The natural numbers embed into the integers by sending \(n\) to \([(n,0)]\). This embedding preserves addition and multiplication: the definitions give \([(m,0)]+[(n,0)]=[(m+n,0)]\) and \([(m,0)]\cdot[(n,0)]=[(mn,0)]\). It also preserves equality, since \((m,0)\sim(n,0)\) exactly when \(m+0=0+n\), which is equivalent to \(m=n\). We can therefore regard \(\mathbb N_0\) as a subset of \(\mathbb Z\), with its familiar arithmetic unchanged.

Ordering the Integers

The ordering also comes from the natural numbers. An integer \(x\) is less than an integer \(y\) when the difference \(y-x\) is positive. In terms of representatives, this gives a direct test that uses only the natural-number order.

For example, to compare \(-2\) and \(3\), use \((0,2)\) and \((3,0)\). The test gives \(0+0<2+3\), or \(0<5\), so \(-2<3\). To compare \(-5\) and \(-2\), use \((0,5)\) and \((0,2)\); the test becomes \(0+2<5+0\), so \(-5<-2\). The order extends the usual order on the nonnegative integers: comparing \((m,0)\) and \((n,0)\) reduces to comparing \(m\) and \(n\).

What the Integers Make Possible

The central gain is closure under subtraction: if \(x,y\in\mathbb Z\), then \(x-y\in\mathbb Z\). The construction achieves this because each integer has an additive opposite and the integers are closed under addition. This is why a calculation such as \(3-8\) has a value in the number system rather than being declared impossible.

There is an important limitation. The integers do not make every division possible. For example, there is no integer \(z\) such that \(2z=1\): if \(z\geq0\), then \(2z\) is either zero or at least two; if \(z<0\), then \(2z<0\). The number \(1/2\) therefore requires a further extension of the number system. That extension, from integers to rational numbers, is the next step.