Does 0/0 = a new branch of numbers? Have I made a mistake in the equation?

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So I thought...

If $\frac00 = x$... then

$0 = x\cdot0$... then

$0x = 0$... then its technically possible to divide by $0$ again

$\frac{0x}0 = \frac00$ ... since $\frac00 = x$ and $\frac{0x}0 = \frac00$.. then $x = \frac{0x}0$ ...

$0x = 0 ... x = \frac00 ...$

$\frac00 = \frac00=...$ $\frac00$ can only equal itself, meaning the value of it is an exact value, but it cannot equal anything else because the value it has is not from any branch-type of numbers that we know. It isnt imaginary nor anything else. Does this mean we need a new type of number that will make the mechanics of $0$ in maths work?

T.S SVK 1998 (17)

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As has been pointed out division by zero is undefined. But let's just think why?

Well division by two can be defined as $\times2^{-1}$ in other words multiply by the multiplicative inverse of $2$ . To be specific the element $a $ such that $2 \times a=1$ Now such an $a$ is called $2^{-1}$

Can we find an $a $ such that $0 \times a =1$ ?

No because multiplication by $0$ is always $0$. So division by zero is undefined.