# Relative numbers representation (graphic)

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

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 $$n_2=-1\ n_1-1$$.