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.