Which rule has more hierarchy?

Question

There are these 3 rules:

$1)$ Any number divided by $0$ it is indeterminate

$2)$ $0$ divided by any number is $0$

$3)$ Any number multiplied by its reciprocal is $1$

These rules are contradictory, when they are together in a fraction.

$\frac{0}{0}$,

If i follow the first rule and it will be indeterminate.

If i follow the second rule and it is $0$

If i follow the third rule, is $1$

So, which rule has more hierarchy?

Answer 1

The usual rule for forming a fraction is that it is $\frac ab$ with $b \neq 0$. Neither $\frac a0$ nor $\frac 00$ has any official meaning at all, so your first rule is not a rule and the second and third are correct. We can speak informally of things like $\frac 00$ and often do when we are talking about a limit. We can have an expression like $\lim_{x \to 1}\frac {a(x)}{b(x)}$ where $a$ and $b$ both go to zero as $x \to 1$. This is where we call it indeterminate. Now your first rule should be any nonzero number divided by zero is infinite. Your second should be indeterminate, and the third does not apply because $0$ does not have a reciprocal.

Answer 2

In the usual system, the first is enforced first. Hence, $0/0$ is indeterminate, and so is $x/0$ for any real $x$. Thus, $0$ has no multiplicative inverse.

Then, (2) and (3) don't contradict each other anymore.