if 0*0=0, why can’t you divide 0 by 0 (the square root of 0) and get 0 (the square root of 0)?

Question

ditto for cube root, nth root, etc.

In: 5

Answer 1

You can… in the trivial ring.

In other division rings, unfortunately, it’s not possible, because there is no multiplication inverse of 0. I’ll mark the multiplication inverse of a number n as n^(T)

0 0^(T) = 1 (definition of multiplication inverse)

0 = 1 (definition of 0)

That statement is only compatible with a trivial ring. To show that, we pick any element of a ring R. Then

0r = 1r (multiplying both sides with r)

0 = r (definition of 0 and 1)

This shows that all elements of that ring are equal to zero. This means that the ring is just {0}

There is a way to work around that, by working with a wheel instead of a division ring, Wheel is similar to a ring, but with a slightly different axiom: [https://en.wikipedia.org/wiki/Wheel_theory](https://en.wikipedia.org/wiki/Wheel_theory)

A common way to obtain a wheel is by adjoining a bottom element to a projective line. The bottom would be the solution of 0/0