Idly passing the time this morning I thought of a – b = a + (-b).
Fair enough, it is the interpretation of subtraction in the extended positive/negative number system.
I then thought of a – (b + c)
Sticking to the rules I got a + (-(b + c))
To proceed further I had to guess that -(b + c) = (-b) + (-c)
and then, quite ok, a – (b + c) = a – b – c
But -(b + c) = (-b) + (-c) is guesswork.
I cannot see a rule to apply to this situation.
The only way forward is to use -1 as a multiplier:
So a – b = a + (-1)b = a + (-b),
and then -(b + c) = (-1)(b + c) = (-1)b + (-1)c = (-b) + (-c)
by the distributive law.
It’s not surprising that kids have trouble with negative numbers!
Do we just assert that the distributive law applies everywhere, even when it is only defined with ++’s ?