# Formalizing an observation

I just rewatched some old math books for the fun of it and was able to observe the following which I want to formalize (could you help me with it):

Choose 2 numbers with distance 1 for each pair. Proceed with...

## First Example $$(i)$$

(3,4) and (6,7)

$$3\cdot 6+4\cdot 7=46$$ and
$$3\cdot 7 + 4\cdot 6 = 45$$

## EDIT: Second example $$(ii)$$

(distance between pairs don't need to be the same distance)

(5,6) and (9,10)

$$5 \cdot 9 + 6 \cdot 10 = 105$$ and
$$5\cdot 10 + 6\cdot 9 = 104$$

Therefore, the outcome of both equations in $$(i)$$ and $$(ii)$$ have a distance of $$1^2$$.

## Observation

If you try this for different examples and for distances $$n\in \mathbb{N}_0$$, you'll most likely notice, that the observation is satisfied for all pairs $$(a,b),(c,d)\mid a,b,c,d\in\mathbb{N}\setminus \{0\}$$ with a specific distance $$n\geq 0$$. I want to formalize this fact, but I'm not really sure, if my following thoughts are correct.

Informal, we have: For all pairs $$(a,b),(c,d) \in \mathbb{N}\setminus 0:$$ exists one $$n$$ in $$\mathbb{N}_0$$ with $$b$$ is the successor of $$a$$ with distance $$n$$ and $$d$$ is the successor of $$c$$ with distance $$n$$ for which we obtain equivalent that the outcome of $$(ac+bd)$$ has distance $$n^2$$ to the outcome $$(ad+bc)$$.

I would formalize this as the following: $$\forall a,b,c,d \in \mathbb{N}\setminus 0 : \exists n \in \mathbb{N}_0:(b=a+n) \land (d=c+n) \iff \underbrace{(ac+bd)}_{=x+n^2}- \underbrace{(ad+bc)}_{=x}=n^2$$

Do you think this is correct? If not, could you share your thoughts about it? Could we use group-theory to discribe these equations easier?

• "Formalization" does not mean "writing things with symbols". It means "fully explaining everything, without leaving gaps". This is a common misconception - using a bunch of symbols doesn't necessarily make your statement any more formal, useful, or understandable. – Deusovi Dec 14 '18 at 14:25
• I would be glad if you share your thoughts with me. – Doesbaddel Dec 14 '18 at 14:54

If $$b = a + n$$ and $$d = c + n$$ then $$(ac+bd)-(ad+bc) = (b-a)(d-c) = n^2.$$ For example, if the numbers are $$(0,n)$$ and $$(0,n)$$ then $$0\cdot 0 + n \cdot n = n^2$$ whereas $$0 \cdot n + n \cdot 0 = 0$$.
• Note that $1^2=1$. – Yuval Filmus Dec 14 '18 at 9:24