Relativistic addition

Let c be a positive constant and define a new addition operation on numbers in the interval (-c, c) by

x \oplus y \equiv \frac{x + y}{1 + \dfrac{xy}{c^2}}

This addition has several interesting properties. If x and y are small relative to c, then xy is approximately x + y. But the closer x or y get to c the more xy differs from x + y. In fact, xy can never escape the interval (-c, c).

The number c acts as a sort of point at infinity: cx = 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 (-cc) with ⊕ for addition. This is the transformation I used yesterday to graph a function that had too large a range to plot directly.

Update: Now that we have relativistic addition, how would we define relativistic multiplication?