An alternative to multiplicative inverse is to think of division x/y as allocating an amount x to y recipients; this is a very common operation in finance, eg a

An alternative to multiplicative inverse is to think of division x/y as allocating an amount x to y recipients; this is a very common operation in finance, eg allocating x shares to y accounts, or if you prefer, splitting a bill x among y diners. In this interpretation, you can't allocate anything to 0 recipients, so its reasonable and convenient to set 1/0 = 0