## When is commutator the right word?

Project: Looking back on the prior posts, when is commutator the right word?

Project: Does you answer depend on unary operators v binary?

Project: What if we have a unary and binary operator we want to permute jointly in some sense?

Project: Is multiplication assumed in commutator? [m,’]n and so on we assume the operator
is applied in a multiplicative way. What happens if try to make it apply in an additive way? Are there cases where we can define a multiplication commutator and addition commutator for the same data? Can we define a commutator on whether multiplication or addition is used? What does this tell us?

Project: Do commutators help us distinguish unary and binary operators? Their interaction? Student errors?

Project: Compare unary operators like successor, binary operators like addition and multiplication and the combination of two binary operators such as two or more from addition, subtraction, multiplication and division. Where are students most likely to make a systematic or common error? What type of commutator best illustrates the error?

Project: Student common errors can be thought of as a type of numerical error that propagates. I.e. it is like a type of error in the machine that then carries to the next operation on the data. Can commutators help us to understand how common errors propagate in standard calculations? In arithmetic? In algebra?

Project: Can we develop a sort of error propagator? Is it a function? An operator? Data driven or operator driven?