# The integers

## The integers

### 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:

$$-1= \{ \{ 0 \},\{0,1\}\}$$

$$-1= \{ \{ 5 \},\{5,6\}\}$$

$$(a,b)=a-b$$

#### Converting natural numbers to integers

Natural numbers can be shown as integers by using:

$$(n,0)$$

Natural numbers can be converted to integers:

$$\{\{a\},\{a,0\}\}$$

### Ordering of the integers

#### Ordering integers

Integers are an ordered pair of naturals.

$$\{\{x\},\{x,y\}\}$$

For example $$-4$$ can be:

$$\{\{4\},\{4,8\}\}$$

$$\{\{0\},\{0,8\}\}$$

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:

$$\{\{a\},\{a,b\}\}=\{\{s(a)\},\{s(a),s(b)\}\}$$

And either of:

$$\{\{a\},\{a,b\}\}=\{\{0\},\{0,A\}\}$$

$$\{\{a\},\{a,b\}\}=\{\{B\},\{B,0\}\}$$

$$\{\{a\},\{a,b\}\}=\{\{0\},\{0,0\}\}$$

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:

$$(a,b)+(c,d)=(a+c,b+d)$$

Integer addition can then be defined:

$$a+b=\{\{a_1\},\{a_1,a_2\}\}+\{\{b_1\},\{b_1,b_2\}\}$$

$$a+b=\{\{a_1+b_1\},\{a_1+b_1,a_2+b_2\}\}$$

Or:

$$a+b=c$$

$$c_1=a_1+b_1$$

$$c_2=a_2+b_2$$

#### Multiplication

Similarly, multiplication can be defined as:

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

$$ab=c$$

$$c_1=a_1b_1+a_2b_2$$

$$c_2=a_2b_1+a_1b_2$$

#### Subtraction

$$a-b=c$$

$$c_1=a_1+b_2$$

$$c_2=a_2+b_1$$