How the Associative and Commutative Properties
There are clients, especially during tax season, who will come in with a large paper bag, or maybe several large paper bags, full of receipts, invoices, cancelled checks, deposit slips, statements, and many other things. To make sense of this material, the first step is to sort it by category. Then each category is sorted by whether the amount is to be added to or subtracted from the previous balance. Finally, an adding machine tape is run on and stapled to each of these final piles. Only the difference between the adds and the subtracts is entered in the adjustment column.
The final separation is done because it is faster when dealing with significant numbers of entries to hit the + key every time instead of having to decide whether to hit the + key or the - key on each and every item. The hand movement alone will slow the process, and the decision factor serves to slow it even more.
Okay. That is mechanically what is happening. This works because the account will end with the same balance regardless of the order in which the entries are made. This last statement is true because of the Associative and Commutative Properties of Addition after we take into account the formal definition of Subtraction. If we put in all of the mathematical steps, the process looks a lot like a physical representation of what programmers call a bubble sort. An example follows.
3 + 7 5 2 + 1 6 + 9 8 4 + 3 ; The starting point
3 + 7 + ( 5) + ( 2) + 1 + ( 6) + 9 + ( 8) + ( 4) + 3 ; Def. of Subtraction
3 + 7 + ( 5) + [( 2) + 1] + ( 6) + 9 + ( 8) + ( 4) + 3 ; Assoc. Prop. of Add.
3 + 7 + ( 5) + [1 + ( 2)] + ( 6) + 9 + ( 8) + ( 4) + 3 ; Comm. Prop. of Add.
3 + 7 + [( 5) + 1] + ( 2) + ( 6) + 9 + ( 8) + ( 4) + 3 ; Assoc. Prop. of Add.
3 + 7 + [1 + ( 5)] + ( 2) + ( 6) + 9 + ( 8) + ( 4) + 3 ; Comm. Prop. of Add.
3 + 7 + 1 + ( 5) + ( 2) + [( 6) + 9] + ( 8) + ( 4) + 3 ; Assoc. Prop. of Add.
3 + 7 + 1 + ( 5) + ( 2) + [9 + ( 6)] + ( 8) + ( 4) + 3 ; Comm. Prop. of Add.
3 + 7 + 1 + ( 5) + [( 2) + 9] + ( 6) + ( 8) + ( 4) + 3 ; Assoc. Prop. of Add.
3 + 7 + 1 + ( 5) + [9 + ( 2)] + ( 6) + ( 8) + ( 4) + 3 ; Comm. Prop. of Add.
3 + 7 + 1 + [( 5) + 9] + ( 2) + ( 6) + ( 8) + ( 4) + 3 ; Assoc. Prop. of Add.
3 + 7 + 1 + [9 + ( 5)] + ( 2) + ( 6) + ( 8) + ( 4) + 3 ; Comm. Prop. of Add.
3 + 7 + 1 + 9 + ( 5) + ( 2) + ( 6) + ( 8) + [( 4) + 3] ; Assoc. Prop. of Add.
3 + 7 + 1 + 9 + ( 5) + ( 2) + ( 6) + ( 8) + [3 + ( 4)] ; Comm. Prop. of Add.
3 + 7 + 1 + 9 + ( 5) + ( 2) + ( 6) + [( 8) + 3] + ( 4) ; Assoc. Prop. of Add.
3 + 7 + 1 + 9 + ( 5) + ( 2) + ( 6) + [3 + ( 8)] + ( 4) ; Comm. Prop. of Add.
3 + 7 + 1 + 9 + ( 5) + ( 2) + [( 6) + 3] + ( 8) + ( 4) ; Assoc. Prop. of Add.
3 + 7 + 1 + 9 + ( 5) + ( 2) + [3 + ( 6)] + ( 8) + ( 4) ; Comm. Prop. of Add.
3 + 7 + 1 + 9 + ( 5) + [( 2) + 3] + ( 6) + ( 8) + ( 4) ; Assoc. Prop. of Add.
3 + 7 + 1 + 9 + ( 5) + [3 + ( 2)] + ( 6) + ( 8) + ( 4) ; Comm. Prop. of Add.
3 + 7 + 1 + 9 + [( 5) + 3] + ( 2) + ( 6) + ( 8) + ( 4) ; Assoc. Prop. of Add.
3 + 7 + 1 + 9 + [3 + ( 5)] + ( 2) + ( 6) + ( 8) + ( 4) ; Comm. Prop. of Add.
[3 + 7 + 1 + 9 + 3] + [( 5) + ( 2) + ( 6) + ( 8) + ( 4)] ; Assoc. Prop. of Add.
[3 + 7 + 1 + 9 + 3] + ( [5 + 2 + 6 + 8 + 4]) ; Distributive Prop.
[3 + 7 + 1 + 9 + 3] [5 + 2 + 6 + 8 + 4] ; Def. Of Subtraction
[23] [25] ; Add (Order of Operations)
2 ; Subtract
Okay. So actually doing it takes a lot less time and effort than justifying it. However, if it cannot be justified in some manner, we cannot trust the result to be accurate.