Disclaimer: this is more of a disscussion, rather than a question. if there's a better place to put this, let me know.
Say we have an equation $f(x)=mx$
Where m is the slope.
If $m=\infty$
Then $f(x)=\infty x$
Which appears to be $x=0$
Also, If $f(x)=-\infty$
Then $f(x)=-\infty x$
Which also is $x=0$
This means that $\infty x=-\infty x$
If we divide by x, then we get this:
$\infty=-\infty$
Now at first, this appears to be wrong. However, go follow this link to see what i mean. move m close to infinity, and compare this to when m is close to negative infinity. They appear to be the same (if they don't, then set m more closer to infinity/negative infinity).
I want to see your opinions about this. I'm sure that when people see this, they will think i'm wrong, but who knows.
The answer, basically, is that infinity is weird.
There are a couple distinct ways to extend the real line to include infinity. One way is to add two elements, one at either "end" of the number line; often they're denoted $+\infty$ and $-\infty$. This gives a structure called the extended real line.
Another approach, though, is to view the real line as a sort of horseshoe-shape, where both ends curve around towards each other. In this view, the natural thing to do is to add a single point (frustratingly also usually called "$\infty$") which "glues" the ends together. This results in a structure called the projective real line, and is also the one-point compactification of the real line.
What you've discovered is that - essentially - "slope" is a map from $\{$lines$\}$ to the projective real line, rather than from $\{$lines$\}$ to the extended real line. There are other contexts where the extended real line is the "right" object to be looking at instead of the projective real line.
The crucial methodological takeaway, in my opinion, is: