What is the inverse of infinity in geometry?

[Pjpic]

I seem to recall reading a geometry method that showed zero to be the inverse of infinity. Can you give me a reference for that?

 

[HallsofIvy]

That depends upon what you mean. "Inverse" in what sense? You can, for example, use "inversion" in a circle. Given a circle of radius "R" and center "O" and a point P inside the circle, we define its "inverse" to be the point, Q, lying on the same extended radius of the circle as P, such that |OP||OQ|= R^2 where |OP| and |OQ| are the distances from O to the two points. If P is on the circle, Q= P. As we move P closer to the center of the circle, the corresponding Q moves farther and farther from the circle. As P approaches the center, in the limit, Q goes to infinity.

But if you are looking for a "proof", geometrical or otherwise, that, in our usual arithmetic 1/0 is equal to infinity, that just isn't going to happen. It simply isn't true. There is no number called "infinity" in our usual arithmetic and you cannot divide 1, or any other number, by 0. "Infinity" is just a "shorthand" for limits.

 

[ellipsis]

Asking what 1/0 is is like asking what the color of an electron is. Or, whether or not the king of France is bald!

 

[HallsofIvy]

Asking what 1/0 is is like asking what the color of an electron is. Or, whether or not the king of France is bald!

NO! I am not bald!

 

[CellCoree]

Yes, there is a concept in geometry known as the "projective line" where zero and infinity are considered to be inverses of each other. This is often represented by the symbol ∞^-1 = 0. This concept can be found in various mathematical texts, such as "Projective Geometry" by H.S.M. Coxeter and "Geometry: A Comprehensive Course" by Dan Pedoe. I hope this helps.