j/0=1
The idea to define i^2=-1 brought up a lot of interesting math. Could it be similarly useful to define a number j so that j/0=1?
There are a few ways to add infinite to the operations, and add some operations for them, but the extensions breaks the properties of the operations https://en.wikipedia.org/wiki/Cardinal_number
Physicist use something like what you are proposing, using Taylor and hand waving. I read a few tries to formalize that, but they are just too weird and no one use them. I think the siplest example is https://en.wikipedia.org/wiki/Graded_ring that you can use to define ϵ and a number is something like 7.9+3.1ϵ+27.4ϵ^2+... So you have that ϵ is almost like 0, and ϵ/ϵ=1, but 7/ϵ is undefined.
It look hard to make an extension where ?/0=1. I think it's better to be happy with something similar enough.
> There are mathematical structures in which a/0 is defined for some a such as in the Riemann sphere (a model of the extended complex plane) and the Projectively extended real line; however, such structures do not satisfy every ordinary rule of arithmetic (the field axioms). ...
> This definition leads to many interesting results. However, the resulting algebraic structure is not a field, and should not be expected to behave like one. For example, ∞ + ∞ is undefined in this extension of the real line. ...
> Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define it, or similar operations, in other mathematical structures.