## Proper.

So there’s this family of mathematical jokes. They run about like this:

A couple people are in a hot air balloon that’s drifted off course. They’re floating towards a hill, and they can barely make out a person on the hill. They cry out, “Where are we?” And the person stares at them, and thinks, and watches the balloon sail aimlessly on. Just as the balloon is about to leave shouting range, the person cries out, “You are in a balloon!” And one of the balloonists says, “Great, we would have to get a mathematician.” “How do you know that was a mathematician?” “The person gave us an answer that’s perfectly true, is completely useless, and took a long time to produce.”

(There are equivalent jokes told about lawyers and consultants and many other sorts of people.)

A lot of mathematical questions have multiple answers. Factoring is a nice familiar example. If I ask “what’s a factor of 5,280”, you can answer “1” or “2” or “55” or “1,320” or some 44 other answers, each of them right. But some answers are boring. For example, 1 is a factor of every whole number. And any number is a factor of itself; you can divide 5,280 by 5,280 and get 1. The answers are right, yes, but they don’t tell you anything interesting. You know these two answers before you’ve even heard the question. So a boring answer like that we often write off as *trivial*.

A *proper* solution, then, is one that isn’t boring. The word runs through mathematics, attaching to many concepts. What exactly it means depends on the concept, but the general idea is the same: it means “not one of the obvious, useless answers”. A proper factor, for example, excludes the original number. Sometimes it excludes “1”, sometimes not. Depends on who’s writing the textbook. For another example, consider sets, which are collections of things. A subset is a collection of things all of which are already in a set. Every set is therefore a subset of itself. To be a *proper* subset, there has to be at least one thing in the original set that isn’t in the proper subset.