Construction of Integers

Question

I was wondering how Integers were constructed? I looked at the wikipedia article but the explanation about how they were constructed did not make much sense to me, would anyone care to simplify what is being said?

Answer 1

The OP asked in a comment,

Is there an intuitive, yet mathematical way getting at this?

Here we directly construct the integers under addition with an approach that might be helpful to some readers in understanding what makes $\mathbb Z$ tick.

Imagine a line with you standing on it at a spot that is called 'home base'. Your feet straddle the home base and you are gazing in a direction perpendicular to the line. You are an expert at taking sideways steps, so that you can go right or left one step with your feet marking off a new spot that is exactly one meter away from home base.

You can keep taking steps of course, and they will be determined by instructions,

$\quad \, R:$ take a step to the right
$\quad \, L:$ take a step to the left
$\quad [\,]:$ don't move

You eagerly await at home base to be given a string of instructions $S$, like

$\quad S = RRLRLLRRRR$

which you follow by going Right, Right, Left, Right, Left, and so on to the last Right step.

When you follow any string of instructions you'll wind up on a spot either to the right or left of home base by so many meters.

You can concatenate any two instruction strings $S$ and $T$ giving $ST$ by first performing the directions in $S$ and then doing $T$.

If two instruction strings $S$ and $S^{'}$ bring you to the same spot, we write $S \sim S^{'}$. The concatenation operation is actually well-defined 'under $\sim$ substitutions:

$\quad \text{If } S_\alpha \sim S_\beta \text{ and } {S_\alpha}^{'} \sim {S_\beta}^{'} \text{ then } S_\alpha {S_\alpha}^{'} \sim S_\beta {S_\beta}^{'}$.

We now have $\mathbb Z$ as the instruction lists partitioned by $\sim$.

So, if $S = RRLRLLRRRR$ then $S = RRRR$.

Also, $RRLRLLRRRR + LL = RR$ and $RR + LL = [\,]$.

Might as well call $[\, ]$ zero and denote it by $0$.

The instruction strings $RL$ and $LR$ are both equal to $0$, and any string can be '$\sim$contracted' by removing these two 'dumb movements'. So any string has a direct or pure $\sim$-expression containing only $R$ or $L$ moves.

We know that the additive inverse of say, $RRRRR$, is $LLLLL$. But you can also get the inverse of any 'dumb bloated' instruction string $S$ the 'lazy way':

$\quad \text{Inverse of } S = \text{Transpose the letters, } R\text{<->}L \text{, in reverse_string(}S\text{)}$

We called this the lazy way since it is totally obvious that the defined string $\text{Inverse of } S$ is both a left and right inverse. But, the concatenation of two $\sim$-instruction stings is commutative, so you don't even have to reverse the string.

Might as well use $+$ for this binary operation on $\mathbb Z$ and denote the additive inverse of an integer as $-S$. Also, since we bypassed $\mathbb N$ in this construction, we might not know about $\text{Base-}b$ systems, so we agree to name $R$ as "one", denoted by $1$, and to name $RR$ as "two", denoted by $2$.

Intuitively, we are generating the integers with the 'unit vectors'
$\quad R: \rightarrow$
$\quad L: \leftarrow$
using 'head-to-tail' addition on a line.

Think this out and see if it works for you. If you like it then proceed and define multiplication by considering the integers $R, RR, RRR, RRRR, \dots$ as 'positive' numbers.

Answer 2

So, if you look at 3-2 and 4-3, you expect those will be equal. Hence, you write (3,2)~(4,3). The problem is going past that, finding a rule that reduces pairs like (3,4) which should make 3-4, but without using the negative numbers to define them. So you look at a law that normally describes subtraction: 3-2=4-3 iff 3+3=4+2. Hence, you write (a,b)~(c,d) iff a+d=c+b, because when $b<a$ and $d<c$, a+d=c+b iff a-b=c-d.