Prove that 0 has not a multiplicative inverseSolutionAssume

Prove that 0 has not a multiplicative inverse.

Solution

Assume that there exists a number z such that z is the reciprocal of zero. In symbols, z = 0^-1 = 1/0.

The inverse property of multiplication states that any number x has an inverse 1/x, x not equal to 0, such that x*1/x = x/x = 1.

So, applying the rule, we have z * 0 = 1. This statement doesn\'t make sense. If there exists a number z in the real numbers, then z * 0 = 0. Right? Because any number multiplied to zero is of course zero. So, zero has no multiplicative inverse.