Why 0 does not have a multiplicative inverse?

Answer 1

As n gets very small, 1/n goes towards infinity. A multiplicative inverse of 0 would be some number x, such that 0x=1. This is impossible with the real numbers we use, since 0x=0 for any number x. One might be tempted to invent a new number (calling it "infinity", "nullity", or any other name) that would be the inverse of 0. Of course, then you're not dealing with real numbers anymore, you're dealing with real numbers plus this invented number. There are serious issues even with this approach. Again, let x be this "multiplicative inverse of 0". Then 0*1=0 and 0*2=0. So 0*1 = 0*2. Multiply both sides by x to get x*0*1 = x*0*2. Since x*0 is 1, this means 1*1 = 1*2. So 1=2, which is an absurd conclusion. As you can see, there are good reasons not to allow a multiplicative inverse for 0 - it breaks all the laws of multiplication we're accustomed to.