Recently, a comment made by Matt Parker on a Numberphile video made me start to think about the implications of positive infinity equaling negative infinity.
First off, what would the number line look like? If , then we could represent the number line as a number "circle", with both ends wrapping around to meet at the single point of infinity, like this:
This already feels very intuitive, because an infinite distance from , whether you go in one direction or the other, seems as if it should be treated as just one distance: infinity.
But things get even more interesting!
How would a coordinate plane look if ? I propose that it would look like a torus:
At first this may seem confusing, but there is a reason I chose the torus over what might seem more natural: a sphere.
On a sphere, both the and axes would intersect at a single point of infinity. However, there are combinations of and which must be accounted for: , , and . A sphere would not be able to distinguish between these points, whereas a torus could. These points are all labelled in the picture above.
One of the exciting results of this model is that division by zero now becomes defined.
A common example of division by zero being undefined is the graph of the function . The graph is shown below:
As this function's input gets closer to from the right (the function getting closer to ), its output approaches , but as its input gets closer to from the left, its output approaches . Thus, on a coordinate plane, division by zero is undefined, since this graph doesn't approach one value for .
But on a coordinate torus, this would not be an issue. Excluding the torus shape for clarity, the graph of the function would look like this:
Not only does this graph define division by zero since it approaches infinity from both directions, but it just looks beautiful!
I should note that the reason infinity is not commonly treated in this way is because it violates the axioms which make number sets such as the real numbers what is called a field. This means that infinity breaks the normal rules of arithmetic such as addition and multiplication so that proper algebra cannot be done with it.
However, I would propose that we adopt a single infinity into our fields and treat it in a special way. For instance, define or even (where ) as equal to , but define as undefined. This way, expressions such as remain undefined (since we can't derive them from without using an undefined expression first).
I should also note that the concept of a number "circle" and a coordinate "torus" only exist in infinity. On a finite level, all we can see is a number line and a coordinate plane. Notwithstanding this, it is still fun and practical to contemplate properties which only exist in infinity.
Does anyone else have any thoughts on the subject? I would love to hear them!