Writing $0$ as $0^2\div 0^1$ gives $0\div 0$. Does that make $0$ undefined?

Question

$0$ is a unique real number which means that it is a definite number, right?

$$0^2 = 0\times0 = 0$$

$0\div 0$ is considered as not defined.

So, if I write $0$ as $0^2\div0^1$ then that will become $0\div0$. Does that mean that now $0$ is undefined?

Answer 1

Hint: Divison can be viewed as an operation derived from multiplication.

Then the quotient $a/b$ is defined as $a\cdot b^{-1}$ where $b^{-1}$ is the inverse of $b$, i.e., $b\cdot b^{-1}=1$. One can show that the inverse of a number, if existent, is uniquely determined. Therefore the writing $b^{-1}$ as the inverse of $b$.

If we talk about real numbers (any field), each number different from zero has an inverse. E.g., the inverse of $2$ is $1/2$.

Zero has no inverse, since zero is absorbing: $0\cdot a = 0$ for all $a$.