Breaking up a sum

Question

Imagine we are adding two numbers: $$ 5+3 $$ But, we decide to break up and re-arrange the sum as follows: $$ \underbrace{(1+1+3)}_{\text{First Part}}+\underbrace{(1+1+1)}_{\text{Second part}} $$ where we have broken up the $5$ into $(1+1)$ which is added to $3$ which is then added to the remainder of the $5$ $((1+1+1)).$ My question is - without any more information- which $1's$ do the $(1+1+3)$ in the first part represent? In other words,could they represent \emph{any }of the $1's$ from the original $5?$ Is there any way of distinguishing them? In other words, I find it confusing that the same 1 from the original 5 can represent any of the underlying 5 objects, say.

Answer 1

Imagine you are watching some people, say,

$\tag 1 \{Joe,Paul,Dorothy,Mike,Cathy,Katie,Tamara,Andrew\}$

and they are all standing $\text{Outside}$.

If asked for the total number of people you might begin by writing

$\quad 1_J + 1_P + 1_D + 1_M + 1_C + 1_K + 1_T +1_A$

As you study the associative and commutative laws for addition you realize that it doesn't matter how you add them up - brackets for organizing your work do not matter. So you can just work abstractly on the math problem $1+1+1+1+1+1+1+1$.

Suppose that $\{Joe,Paul,Dorothy,Mike,Cathy\}$ move inside into $\text{Room A}$ with $\{Joe,Paul\}$ standing while $\{Dorothy,Mike,Cathy\}$ ($3$ people) sit down at a table. OK, to find the total number of people in the starting $\text{(1)}$ collection you can write

$\quad [1_J +1_P + 3_{\text{Table}} ]_{\text{Room A}} + (1_K+1_T+1_A)_{\text{Outside}}$

Again, due to the associativity and commutativity laws for addition (a binary operation), the labels can be dropped, leaving you with arithmetic (no labels),

$\quad (1 + 1 + 3) + (1+1+1)$