What is wrong with my proof regarding division?

Question

I wrote down $$12 \times 0 = 0$$ Then, I divided both sides by $0$ like so: $$12 = \frac {0}{0}$$ I know that $$ \frac {0}{x} = 0, x \in R$$ Therefore, $$12=0$$ which is a false statement. Where did I go wrong?

Answer 1

There are several things wrong with your proof. The first thing is when you divide by $0$ in your second line of work. Division by zero is an invalid operation. Then again you claim that $\frac{0}{x} =0$ when $x$ is a real number is also wrong. This is because in the real numbers $x$ can be zero and division by zero is an invalid operation. Overall both your second and third steps are incorrect because of division by zero.

Answer 2

You cannot divide anything by 0. That is just not possible unless you want an undefined answer.