I was hoping to pick a term that was a quick and easy one to dash off. I learned better.
This is a simple one. It’s about notation. Notation is never simple. But it’s important. Good symbols organize our thoughts. They tell us what are the common ordinary bits of our problem, and what are the unique bits we need to pay attention to here. We like them to be easy to write. Easy to type is nice, too, but in my experience mathematicians work by hand first. Typing is tidying-up, and we accept that being sluggish. Unique would be nice, so that anyone knows what kind of work we’re doing just by looking at the symbols. I don’t think anything manages that. But at least some notation has alternate uses rare enough we don’t have to worry about it.
“Hat” has two major uses I know of. And we call it “hat”, although our friends in the languages department would point out this is a caret. The little pointy corner that goes above a letter, like so: . . . It’s not something we see on its own. It’s always above some variable.
The first use of the hat like this comes up in statistics. It’s a way of marking that something is an estimate. By “estimate” here we mean what anyone might mean by “estimate”. Statistics is full of uses for this sort of thing. For example, we often want to know what the arithmetic mean of some quantity is. The average height of people. The average temperature for the 18th of November. The average weight of a loaf of bread. We have some letter that we use to mean “the value this has for any one example”. By some letter we mean ‘x’, maybe sometimes ‘y’. We can use any and maybe the problem begs for something. But it’s ‘x’, maybe sometimes ‘y’.
For the arithmetic mean of ‘x’ for the whole population we write the letter with a horizontal bar over it. (The arithmetic mean is the thing everybody in the world except mathematicians calls the average. Also, it’s what mathematicians mean when they say the average. We just get fussy because we know if we don’t say “arithmetic mean” someone will come along and point out there are other averages.) That arithmetic mean is . Maybe if we must. Must be some number. But what is it? If we can’t measure whatever it is for every single example of our group — the whole population — then we have to make an estimate. We do that by taking a sample, ideally one that isn’t biased in some way. (This is so hard to do, or at least be sure you’ve done.) We can find the mean for this sample, though, because that’s how we picked it. The mean of this sample is probably close to the mean of the whole population. It’s an estimate. So we can write and understand. This is not but it does give us a good idea what should be.
(We don’t always use the caret ^ for this. Sometimes we use a tilde ~ instead. ~ has the advantage that it’s often used for “approximately equal to”. So it will carry that suggestion over to its new context.)
The other major use of the hat comes in vectors. Mathematics types do a lot of work with vectors. It turns out a lot of mathematical structures work the way that pointing and moving in directions in ordinary space do. That’s why back when I talked about what vectors were I didn’t say “they’re like arrows pointing some length in some direction”. Arrows pointing some length in some direction are vectors, yes, but there are many more things that are vectors. Thinking of moving in particular directions gives us good intuition for how to work with vectors, and for stuff that turns out to be vectors. But they’re not everything.
If we need to highlight that something is a vector we put a little arrow over its name. . . That sort of thing. (Or if we’re typing, we might put the letter in boldface: x. This was good back before computers let us put in mathematics without giving the typesetters hazard pay.) We don’t always do that. By the time we do a lot of stuff with vectors we don’t always need the reminder. But we will include it if we need a warning. Like if we want to have both telling us where something is and to use a plain old to tell us how big the vector is. That turns up a lot in physics problems.
Every vector has some length. Even vectors that don’t seem to have anything to do with distances do. We can make a perfectly good vector out of “polynomials defined for the domain of numbers between -2 and +2”. Those polynomials are vectors, and they have lengths.
There’s a special class of vectors, ones that we really like in mathematics. They’re the “unit vectors”. Those are vectors with a length of 1. And we are always glad to see them. They’re usually good choices for a basis. Basis vectors are useful things. They give us, in a way, a representative slate of cases to solve. Then we can use that representative slate to give us whatever our specific problem’s solution is. So mathematicians learn to look instinctively to them. We want basis vectors, and we really like them to have a length of 1. Even if we aren’t putting the arrow over our variables we’ll put the caret over the unit vectors.
There are some unit vectors we use all the time. One is just the directions in space. That’s and and for that matter and I bet you have an idea what the next one in the set might be. You might be right. These are basis vectors for normal, Euclidean space, which is why they’re labelled “e”. We have as many of them as we have dimensions of space. We have as many dimensions of space as we need for whatever problem we’re working on. If we need a basis vector and aren’t sure which one, we summon one of the letters used as indices all the time. , say, or . If we have an n-dimensional space, then we have unit vectors all the way up to .
We also use the hat a lot if we’re writing quaternions. You remember quaternions, vaguely. They’re complex-valued numbers for people who’re bored with complex-valued numbers and want some thrills again. We build them as a quartet of numbers, each added together. Three of them are multiplied by the mysterious numbers ‘i’, ‘j’, and ‘k’. Each ‘i’, ‘j’, or ‘k’ multiplied by itself is equal to -1. But ‘i’ doesn’t equal ‘j’. Nor does ‘j’ equal ‘k’. Nor does ‘k’ equal ‘i’. And ‘i’ times ‘j’ is ‘k’, while ‘j’ times ‘i’ is minus ‘k’. That sort of thing. Easy to look up. You don’t need to know all the rules just now.
But we often end up writing a quaternion as a number like . OK, that’s just the one number. But we will write numbers like . Here a, b, c, and d are all real numbers. This is kind of sloppy; the pieces of a quaternion aren’t in fact vectors added together. But it is hard not to look at a quaternion and see something pointing in some direction, like the first vectors we ever learn about. And there are some problems in pointing-in-a-direction vectors that quaternions handle so well. (Mostly how to rotate one direction around another axis.) So a bit of vector notation seeps in where it isn’t appropriate.
I suppose there’s some value in pointing out that the ‘i’ and ‘j’ and ‘k’ in a quaternion are fixed and set numbers. They’re unlike an ‘a’ or an ‘x’ we might see in the expression. I’m not sure anyone was thinking they were, though. Notation is a tricky thing. It’s as hard to get sensible and consistent and clear as it is to make words and grammar sensible. But the hat is a simple one. It’s good to have something like that to rely on.