Relative numbers representation (graphic)

Question

Could you please explain this picture to me well?

1. I don't understand why every oblique line is an integer

2. I don't understand for example why $(0,1), (1,2)$ and $(4,5)$ are in the same equivalence class, which is that of $−1$

3. I see that the difference between the first coordinate and the second coordinate is constant, but I don't understand why it is an equivalence relation on the set $N × N$ of the pairs of naturals.

The integers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers $(a,b)$

Algebraically I think I understand, but graphically I can't visualize

Answer 1

You understand algebraically that a linear function is given by $y=f(x)=m\ x+b$ where $m$ is the slope and $b$ is the y-intercept $f(0)$? In your figure the diagonal lines are $n_2=-1\ n_1+b$ corresponding to a slope of $m=-1$. If $n_1$ and $b$ are both integers, then obviously $n_2$ is also an integer. In the case related to your question 2 the relationship becomes $n_2=-1\ n_1-1$, and the number pairs $(0,1)$, $(1,2)$, and $(4,5)$ are integer pair examples of $(n1,-n2)$ where $n_2=-1\ n_1-1$.