Let c be a positive constant and define a new addition operation on numbers in the interval (-c, c) by
This addition has several interesting properties. If x and y are small relative to c, then x ⊕ y is approximately x + y. But the closer x or y get to c the more x ⊕ y differs from x + y. In fact, x ⊕ y can never escape the interval (-c, c).
The number c acts as a sort of point at infinity: c ⊕ x = c for any x in (-c, c), i.e. nothing you add to c can make it any larger.
The constant is called c after the speed of light. The addition above is simply adding x and y as velocities in special relativity.
You can show that mapping z to c tanh z gives an isomorphism from the real numbers with ordinary addition to the interval (-c, c) with ⊕ for addition. This is the transformation I used yesterday to graph a function that had too large a range to plot directly.