A similar trick (point at infinity or ideal point) is used in projective geometry to distinguish between directions (vectors) and places (points) by using coordinates only: https://en.wikipedia.org/wiki/Projective_geometry
But if you actually want to do calculations with infinities and infinitesimals the surreal numbers might be better suited for that: https://en.wikipedia.org/wiki/Surreal_number
They could have been more precise, but they probably shouldn't have to in the space of a comment. The Riemann Sphere defines a value for the expression x/0, and it's often useful, but it fails to uphold the most important property division should have -- that it undoes multiplication. Division by 0 (with some assumptions about not being in a trivially small space and how those operations behave with respect to addition) does lead to contradictions in that latter sense.
"but it fails to uphold the most important property division should have -- that it undoes multiplication"
I'm not sure I follow that as it's most important property. I'm not sure if division could even be defined as an operation that undoes multiplication.
Number theory, fields, and rings I believe make it clear while subtraction and addition can be viewed as the same function; multiplication and division cannot.
Apologize if that's not clear as to why that is; it's been a while since I read up on those being defined.
However I recommend One, Two, Three: Absolutely Elementary Mathematics by David Berlinski that gives in my opinion pretty good layman understanding of these nuances and number theory.
Take a look into division rings as a concept. The usual definition for division in rings and fields is via multiplicative inverses for some subset of the nonzero elements. Not all algebraic spaces have division, but that doesn't change what it is, especially from the "number theory, fields, and rings" point of view.
Unless you're talking about some higher-order concept?
Edit: For a bit of completeness, what's happening with the Riemann Sphere is that the algebraic definition is being extended in a way that has some useful analytic, topological, and quality-of-life properties, but which is no longer wholly compatible with the underlying algebra. The algebraic issues are isolated to the extra point at infinity, so they're not terrible to work around, but the operation in question is a proper extension of the underlying algebraic definitions -- much how the gamma function in no way can be defined as multiplication of integers but is a useful extension of the factorials nonetheless.
Division is multiplication by the multiplicative inverse. Subtraction is addition by the additive inverse. Both division and subtraction undo their corresponding operation. Multiplying by a (provided it’s not zero) is undone by dividing by a. Adding a is undone by subtracting a.
In a ring the elements form a group under addition and thus every element has an additive inverse. The additive identity element, let’s call it e, has the property that ea = e and ae = e. For this reason we use 0 instead of e. In a nontrivial ring 0 can’t have a multiplicative inverse because if it did then every element would be equal to the multiplicative identity (which is unique).
Okay so if you can get to a ring without a multiplicative inverse and then applying that operation to the ring forms it into a field then wouldn't it be fair to say that division is not really the opposite of multiplication the same way that subtraction absolutely is for addition?
The definition of division is multiplication by the multiplicative inverse. It may be the case that some elements don’t have such an inverse but the definition is analogous to that of subtraction. The analogy is not perfect because every element has an additive inverse while not every element had a multiplicative inverse.
What you're saying is that the analogy between subtraction and division is good as far as it goes. So why should "as far as it goes" end at zero not having an inverse, rather than division by zero producing something other than the multiplicative inverse of zero? The two choices end up having different structure, and so they end up being applicable to different things, but there is nothing wrong with either choice.
That's a good example of where defining division by zero leads to interesting maths, but it ends up sacrificing some of the usual rules of arithmetic, so it comes down to a choice of which is more useful in the relevant circumstance.
This just goes to show that you really have to be careful when slinging out math facts. I've done some under grad maths and the only line on that page that I understand is
"The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1 / 0 = ∞ 1/0=\infty well-behaved."
It clearly does not satisfy a primitive understanding of 1/0.
1/0 is the limit of x/y as x approaches 1 and y approaches 0. It works fine if you choose to put a a point there called \infinity, with an appropriate notion of nearness.
A similar trick (point at infinity or ideal point) is used in projective geometry to distinguish between directions (vectors) and places (points) by using coordinates only: https://en.wikipedia.org/wiki/Projective_geometry
But if you actually want to do calculations with infinities and infinitesimals the surreal numbers might be better suited for that: https://en.wikipedia.org/wiki/Surreal_number