Does the commutative property of addition holds for uncertainties?

Question

Box A has a height of $2.0±0.1$m, box B has a height of $1.5±0.1$m. If box B is placed on top of box A, what is the total height of both boxes?

I know that to find the total height of both boxes, I have to do like the following $$A_{height} = 2.0 ± 0.1m$$ $$B_{height} = 1.5 ± 0.1m$$ $$A_{height} + B_{height} = (2.0 ± 0.1m) + (1.5 ± 0.1m)-----①$$ I know that somehow, I will end up with something like $$(2.0+1.5)±(0.1m+0.1m)=3.5±0.2m-----②$$ But, I'm not really sure how to go from ① to ②. I thought that I could use the commutative property of addition and "rearrange" it. But then I'll get $$(2.0+1.5)±(0.1m±0.1m)$$ I'm not sure how to add both $0.1m$ since between them there's a ± sign instead of a + sign. Can I just add both $0.1m$ together? Does the commutative property of addition holds for uncertainties like this?

Answer 1

Work this out by thinking about the meaning, not by trying to manipulate formulas with rules.

The largest value the total height can have is when each measured height is at its maximum possible value, so the error in the sum in that direction is at most $0.2m$. Reason similarly for the lower estimate.

Answer 2

Here is a way to think about it. $A=2 \pm 0.1$ means $2-0.1 \leq A \leq 2 + 0.1$ and similarly for $B$. It then follows that $3.5 - 0.2 \leq A+B \leq 3.5+0.2$, i.e., $A+B = 3.5 \pm 0.2$.