The integers

The integers


Defining integers

To extend the number line to negative numbers, we define:

\(\forall ab \in \mathbb{N} \exists c (a+c=b)\)

For any pair of numbers there exists a terms which can be added to one to get the other.

For \(1+x=3\) this is another natural number, however for \(3+x=1\) there is no such number.

Integers are defined as the solutions for any pair of natural numbers.

There are an infinite number of ways to write any integer. \(-1\) can be written as \(0-1\), \(1-2\) etc.

The class of these terms form an equivalence class.

Integers as ordered pairs

Integers can be defined as an ordered pair of natural numbers, where the integer is valued at: \(a-b\).

For example \(-1\) could be shown as:

\(-1= \{ \{ 0 \},\{0,1\}\}\)

\(-1= \{ \{ 5 \},\{5,6\}\}\)


Converting natural numbers to integers

Natural numbers can be shown as integers by using:


Natural numbers can be converted to integers:


Cardinality of integers

Ordering of the integers

Ordering integers

Integers are an ordered pair of naturals.


For example \(-4\) can be:



We extend the ordering to say:

\(\{\{x\},\{x,y\}\}\le \{\{s(x)\},\{s(x),y\}\}\)

\(\{\{x\},\{x,s(y)\}\}\le \{\{x\},\{x,y\}\}\)

So can we define this on an arbitrary pair:

\(\{\{a\},\{a,b\}\}\le \{\{c\},\{c,d\}\}\)

We know that:


And either of:




As the latter is a case of either of the other \(2\), we consider only the first \(2\).

So we can define:

\(\{\{a\},\{a,b\}\}\le \{\{c\},\{c,d\}\}\)

As any of:

\(1: \{\{0\},\{0,A\}\}\le \{\{0\},\{0,C\}\}\)

\(2: \{\{0\},\{0,A\}\}\le \{\{D\},\{D,0\}\}\)

\(3: \{\{B\},\{B,0\}\}\le \{\{0\},\{0,C\}\}\)

\(4: \{\{B\},\{B,0\}\}\le \{\{D\},\{D,0\}\}\)

Case 1:

\(\{\{0\},\{0,A\}\}\le \{\{0\},\{0,C\}\}\)

Trivial, depends on relative size of \(A\) and \(C\).

Case 2:

\(\{\{0\},\{0,A\}\}\le \{\{D\},\{D,0\}\}\)

We can see that:

\(\{\{D\},\{D,A\}\}\le \{\{D\},\{D,0\}\}\)

And therefore this holds.

Case 3:

\(\{\{B\},\{B,0\}\}\le \{\{0\},\{0,C\}\}\)

We can see that:

\(\{\{B\},\{B,0\}\}\le \{\{B\},\{B,C\}\}\)

And therefore this does not hold.

Case 4:

\(\{\{B\},\{B,0\}\}\le \{\{D\},\{D,0\}\}\)

Trivial, like case 1.

Functions of integers


Then we can define addition as:


Integer addition can then be defined:








Similarly, multiplication can be defined as:

\((a,b).(c,d)=(ac+bd, ad+bc)\)








Cardinality of the integers

Cardinality of integers