Arithmetic

Arithmetic deals the 4 basic operations: add, sub, mul, div.

Add and sub on the one hand, and mul and div on the other are each other's opposite operations. However, sub and div are somewhat of the second-class operations not only because they are easily defined in terms of add/mul with appropriate invertability axioms:

• sub: 9 - 5 = 9 + (-5)

• div: 5 ÷ 3 = 5 ⅓ = 5 3⁻¹

After all, add/mul could be defined in terms of sub/div as well, but the bigger reasons for their "lesser" treatment is because sub and div do not obey all the axioms that they inverses do; namely, unlike add and mul, sub and div do not uphold the axioms of associativity, commutativity and closure (totallity).