Does $\frac{1}{x}=0$ have a solution?

Question

Does $\frac{1}{x}=0$ have a solution ?

since any number multiplied by 0 equals 0，this equation has no solution in elementary math. I wonder is there a solution in higher mathematics.

Answer 1

It depends on the number system you are working in. This equation has solutions in the extended real number line for example.

However, the extended real number line doesn't have very many nice properties. It's not a not a field or ring for example. This means that you lose the ability to do even simple things like cancelling. For this reason it's often avoided.

Answer 2

No, it does not. This is because $x\frac{1}{x}=1$. This would not be possible if $\frac{1}{x}$ was zero, as $x\frac{1}{x}$ would be $x\cdot0=0$

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You may also want to read on wheels, they are structures in which division is always defined, although I don't recommend you go in very deep just yet.