Distance

A metric on a set calculates the distance between any two elements in the set.

The distance separating two points in a two dimensional vector space:

  • (x_1, y_1)
  • (x_2, y_2)

distance = \sqrt{(y_2 - y_1)^2+(x_2 - x_1)^2}

You are accustomed to there being a distance between 6 and 7 that is the same as the distance between 8 and 9. For most of the math that you do, this is true, but there are a few situations where you actually have to let go of the idea of distance. How can we fathom this?

On a turnpike, the distance between Exit 6 and Exit 7 is different from the distance between Exit 8 and Exit 9. We cannot use Turnpike Exit Integers to determine distance.

However, we still have order. Exit 7 comes after Exit 6, Exit 8 comes after Exit 7, Exit 9 comes after Exit 8.

We say that Turnpike Exit integers lack distance, but still have order.Appendix A

What we have above might be called “Euclidean” or some other words. That calculation works for the map of a city,,,

…but consider a map of the world that makes Greenland look so much larger than Mexico, but we know that actually Greenwood and Mexico are about the same size. Imagine a complicated distance function that works on that map.