As a layperson who is interested in math, imaginary numbers always fascinated me. Like in the real world you taking the square of a negative makes no sense whatso ever, but in theoretical math you can just invent new imaginary numbers, make it so that *i*^2 = -1 and suddenly you have just revolutionized math. If this is useful, why can’t you break other rules and account for them with new imaginary symbols?
So let’s pretend that we call them made up numbers and use *m* to represent them. Why is *m*=1/0 impossible when something like *i*^2 = -1 is not?
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Others have given you a lot of examples of why **m** cannot be a real number – if we assume that division is the inverse of multiplication, this results in an inconsistency.
We can append **m** to the real numbers, similar to how we appended **i**. The rules will be different, but this is what is done to form the [Riemann sphere](https://en.wikipedia.org/wiki/Riemann_sphere) or the less useful [real/complex wheel](https://en.wikipedia.org/wiki/Wheel_theory).
We can also give up on division being the inverse of multiplication. This was done by a programming language called Pony, and [this post by Hillel Wayne](https://www.hillelwayne.com/post/divide-by-zero/) is a soft defense of allowing 1/0 = 0. It’s no longer generally true that x * (1 / x) = 1, but only true if x != 0. However, that might just be ok sometimes.
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