Conceptually, subtraction and addition of negatives are two very different processes. Subtraction involves an undoing of addition: It is an inverse function. Addition of negatives involves an extension of the number system to a mirror world. The Exploding Dots model, for instance, relies on this extension. That is, here are two ways we can see 5 – 4 = 1:
- There is a binary function, “Subtraction”. It takes two ordered inputs and finds their difference. In this case, the inputs are 5 and 4 (in that order) and the output is 1. This is the inverse of the binary “Addition” function, in that it takes the output of that function and uses one of its inputs to find the other one.
- There is a binary function, “Addition” and a set of numbers called “Negatives”. The binary function takes two unordered inputs and finds their sum. When adding a positive number to a negative one, we start by pairing up units and balancing them out. When only positive or only negative units remain, that is our output. In this case, the inputs are 5 and -4: +++++ and —-. We pair them: +- +- +- +- +. One positive unit remains.
We can also see addition as movement on a number line: We start at 0, move five units to the right, then four units to the left, and determine where we are. This can serve as a model for either of the conceptions above.
Assume there are two unary processes, “Addition” and “Subtraction”. Addition consists of moving in one direction (usually the right) on the number line. Subtraction consists of moving in the other direction.
We can see the process of “Subtraction” either as always facing forward but stepping backward (our first conception), or as turning around and moving forward (our second conception).
There is another way of describing this process, a hybrid of the two conceptions which I think matches how mathematics at its core is really practiced:
- There is a binary function, “Addition”, and a unary function, “Negation”. Addition functions as described above. Negation converts the units between + and -. So 5 – 4 is really Add(5, Neg(4)). In this model, negative numbers aren’t things in their own right; there are only positive numbers, with the function Neg applied when needed.
That does lead to a fundamental philosophical question: Do negative numbers really not “exist” in the mind of the mathematician, or do we just not have a way of representing them unambiguously?
An obvious comparison here is to complex numbers, with a similar fundamental question: Is \(i\)a number unto itself, or is it a unit times the number 1? We call \(i\) the “imaginary unit”, which suggests the latter. So is -4 the value of 4 times the “negative unit”, or is -4 a number in its own right?
I don’t have a definitive answer for that: I’m not sure there is one. My point here is that the two models above are conceptually quite different, and yet it feels like we often teach mathematics while conflating them.