You might not have much use for the real projective line when tallying up prices in the grocery store, but projective geometry is definitely very useful. https://en.wikipedia.org/wiki/Projective_geometry

Yeah, I don't really see what this gets you. With basic real number division you have to make the exception for zero in the definition:

    a/b = c if and only if a = c*b and b!=0

And with this infinity thing you just have to make essentially the same exception for multiplication and infinity:

    c*b = a if and only if a/b = c and b!=infinity and c!=infinity

What is the "issue"?

"/" means "* reciprocal of".

If "infinity" is defined as "reciprocal of 0", what is the problem?

Yes it is an exception to 0*n=0.

It won't work in every setting, but it works in some settings, like inversive geometry.

In normal math 1/0 is undefined but in a math where 1/0 is defined to be inf the 0*inf is still undefined.