Is There a Simple Explanation for Why Division by Zero is Undefined?

[Hippasos]

let u = undefined

Let n be any integer

n / 0 = u

n = 0 / u

n = 0

Can we really do arithmetic operations with undefined? I assume the operation is however made when we declare: n / 0 = undefined.

is there any simple explanation for the fact that other operations with zero are defined except division by it?

 

[jgens]

Can we really do arithmetic operations with undefined?

Of course we can't do arithmetic operations with undefined values! When we say that some quantity is undefined, we mean to say that there is no real number with that property.

is there any simple explanation for the fact that other operations with zero are defined except division by it?

Yes! If you accept the other properties of real numbers involving zero, you are forced to accept that $\alpha/0$ is undefined for all real $\alpha$.

 

[HallsofIvy]

If we were to "define" a/0= x for some x, that would be equivalent to saying that a= (0)x. But (0)x= 0 for any number x so, as long as $a\ne 0$, that makes no sense. On the other hand, if a= 0, then (0)x= a= 0 for any x so a/0 still cannot be any specific number.