Wait, what? What do you mean, "what is [math]\pi[/math]"? It's the ratio of the circumference of a circle to its diameter. Or it's the area of a circle of radius 1, or something like that. Ever...(more)
You may not be aware of this, but mathematicians are very imaginative and creative people. They love defining new things, they love analogies, they love challenging accepted assumptions and they love completeness and symmetry.
Any one of those reasons alone provides ample motivation to explore the idea of adding [math]1/0[/math] to our algebraic, geometric or analytic domains.