| This does not imply that tan(pi/8) is 0.5. There’s some overloaded language here that I’d like to unpack: The original issue at hand is to talk sensibly about “half way” between 0 and infinity. But in the standard way we think about distance between numbers, there’s obviously no way to do that — you can’t add and subtract real numbers to infinity! So, implicitly what’s happening here is that by talking about a map between a finite interval and (0,inf), we are equipping (0,inf) with a new, special definition of distance between numbers. This is called a metric. Usually, when we talk about the distance between x and y, we mean `d(x,y) = |x-y|` — this is called the Euclidean metric (in 1 dimension). Here, we’ve introduced a new metric on (0,inf): `d2(x,y) = |tan^-1(x) - tan^-1(y)|/(pi/8)` The half way point between 0 and 1 under the Euclidean metric is 0.5. The half way point between 0 and 1 under our fancy new metric d2 is ~0.414. TL;DR: You can generalize the notion of “distance between two numbers”, using distance functions called metrics. Under the typical metric, the half way point between 0 and 1 is 0.5. Under our cool new infinite-tan metric, the half way point between 0 and 1 is ~0.414. |