---
title: Differences
---

<small>Part of [my units pages](/projects/units)</small>

*Differences* between values are distinct from those values themselves. For
example, *n-dimensional points* can be related using an *affine space*; whilst
the *difference between these points* is better represented as a *vector space*.
The crucial difference is that vectors have a special, distinguished "origin"
(the $0$ vector, representing "no difference"), whilst affine spaces don't need
to distinguish any points as being more "special" than others.

A good example of this is a 1-dimensional line, which we often model as a
"number line" with entirely artificial labels for both position and magnitude.
In fact, the "number line" is a more abstract space: of *differences between
1-dimensional points*. The relationship between these two spaces is captured
algebraically by a [torsor](torsors.html)