In classical physics, velocities add like real numbers. For instance, if you are traveling on an open-air platform moving at 10 mph, and throw a baseball at 50 mph in the direction of motion, then an observer on the ground sees the ball move at 10+50 = 60 mph. However, according to the (special) theory of relativity, no particle travels faster than the speed of light, c. If your platform is moving at (3/4)c and you throw the ball at (1/2)c in the direction of motion, then it appears to an observer on the ground that the ball's velocity is not (3/4)c + (1/2)c = (5/4)c, which exceeds the speed of light, but instead (10/11)c.

We can think about these two situations from a mathematical point of view, without worrying about the physics involved. They describe two different types of addition, one on R using ordinary addition, and the other on the interval (-c,c) using relativistic addition. It turns out that there is a "relativistic logarithm," that converts relativistic addition on (-c,c) into ordinary addition on R. This means relativistic addition is just ordinary addition in disguise, even though they may seem quite different.

We will explain what it means for any interval (a,b) to have an "addition law," and then prove that all such addition laws are just a disguised form of ordinary addition on R. The proof involves an interesting mix of calculus and algebra. No physics will be assumed or used.