Most definitions of division that I have seen use q * d + r = n if q is unique and abs(r)<abs(d), which doesn't require the definition of an inverse. Rather, d

Most definitions of division that I have seen use q * d + r = n if q is unique and abs(r)<abs(d), which doesn't require the definition of an inverse. Rather, d that exist for n = 1 and r = 0 can be labelled q's inverse but it doesn't require a new definition. Additionally, if inverses are defined as separate objects then what is 2 plus the inverse of 2? It doesn't simplify to 2.5 because there's no addition axiom for numbers and multiplicative inverses, or for that matter any rules for inverses with inverses. So you might have 1/2 and 5/10 but they're not equal and can't be multiplied together.