What is $(1/0)^{-1}$?

Question

What is $(\frac{1}{0})^{-1}$?

Would the fraction be reciprocated first to give you $\frac{0}{1}=0$.

Or would you not be able to evaluate this as $\frac{1}{0}$ is undefined?

Answer 1

You would not be able to evaluate this as 1/0 is undefined. Algebraic manipulation rules such as $(a/b)^{-1} = b/a$ (which you are using in your first option) are only valid if every operation in it is valid. In this case, only if $a, b\neq 0$.

Answer 2

Well, given a commutative ring $R$ with 1 and without zero divisors such as the ring of integers, one can define the quotient ring $$K = \{p/q\mid p,q\in R,q\ne 0\},$$ which is a field. In the case of integers, one obtains the field of rational numbers.

So the quotient $1/0$ is not defined and cannot be written down and so its inverse is also undefined.