It is an accepted knowledge in mathematics that division by zero is undefined. But this is not intuitive to me. I'm not a mathematician, but how about if there is an axiomatic definition that any division by zero is zero?
Going back to the most accepted answer that $\frac{x}{0}$ is undefined:
"the equation that defines it has no solutions: for something to be equal to $\frac{x}{0}$, with $x ≠ 0$, we would need $0z = x$. But $0z = 0$ for any z, so there are no solutions to the equation. Since there are no solutions to the equation, there is no such thing as "$\frac{x}{0}$". So $\frac{x}{0}$ does not represent any number."
I get that $ z = \frac{x}{y}$ is the same as $ x = yz $:
Step 1: $ z = \frac{x}{y}$
Step 2: $ y ⋅ z = y ⋅ \frac{x}{y}$
Step 3: $ yz = x $
Step 4: $ x = yz $
If $ y = 0 $ and $ x ≠ 0 $, then there are no solution to the equation $ x = 0z $ because:
Step 5: $ x = 0z $
Step 6: $ x = 0 $
Therefore $ x = 0 $ for for any value of $z$ and conflicts with the requirement that $ x ≠ 0 $.
But if we define $ \frac{x}{0} = 0 $ as an axiom, then we do not even arrive at the equation $x = 0z$.
Step 1: $ 0 = \frac{x}{0}$ ____________ as an axiom
This already reduces to $ 0 = 0 $, but if we go further:
Step 2: $ 0 ⋅ 0 = 0 ⋅ \frac{x}{0}$ ________multiply both sides by zero
Step 3: $ 0 = 0 $ ____________any number multiplied by zero is zero
Therefore the equation $ x = 0z $ (and consequently $ x = 0 $ since $ z = 0$) simply cannot be derived.
But if we continue with the re-arrangement and have the equation $ x = 0z $, then because it passed a step where $ y $ is a divisor, therefore, it implies that $ y ≠ 0 $ under this axiom, so we can't make further assumption that $ y = 0 $ after step 4, so the equation $ x = yz $ is only valid where $ y ≠ 0 $.
Also, there would be no problem also for $\frac{0}{0}$ as described here, for now, the result of $\frac{0}{0}$ could not be anything but zero.
What would be the problems if division by zero is defined as an axiom as defined above?
If $\frac x 0 = 0$, then it has to be the neutral element of the summation. But $$ 0 + 1 = \frac x 0 + \frac 1 1 = \frac {x \cdot 1 + 1 \cdot 0}{0 \cdot 1} = \frac x 0 = 0 $$ which never holds, unless in the zero ring.