I would like to know the logic behind adding two intervals $$[a, b]$$ $$[c, d]$$ together and for the sake of simplicity let's say that the numbers in the interval are just integers and not real numbers. I know that the idea is to just add the lowest and highest of the two $$[a+c, b+d]$$ and this is the result of the addition.
My question is whether this is just defined to be like this mathematically or is there a real world example that can shed some light on why it is calculated so?
Here's a proof that if $r$ is in $[a+c,b+d]$ then $r=x+y$ for some $x$ in $[a,b]$ and some $y$ in $[c,d]$:
First, note that $a+c\le r\le a+d$ or $a+d\le r\le b+d$.
In the former case, there must be $y$ in $[c,d]$ such that $r=a+y$. Let $x=a$.
In the latter case, there must be $x$ in $[a,b]$ such that $r=x+d$. Let $y=d$.
Note: I'm not sure that this engages with OP's concerns. And I suspect that what I've written here has been written before on this site, probably multiple times. But it does seem to speak to some of the comments on the question.