Sometimes, when you assign a number to a quantity, a single number really isn’t enough. Take blue jeans sizes, as a simple example. Jeans sizes aren’t expressed with a single number, from small to large, like shoes. Since at least the days of the dominance of Levi’s, blue jeans sizes are indicated giving two numbers, like 34 x 32, which we are supposed to interpret as a waist size of 34 inches and a length of 32 inches. It’s an interesting use of the multiplication symbol, by the way, and I assume it is supposed to be pronounced as “34 by 32”.
Leaving aside, for now, the question of whether two numbers are sufficient to fully capture the size of jeans, let’s take a closer look to see if a single number might in fact be sufficient. My guess is that in the case of jeans sizes, a single number might actually suffice, though such a number might be hard to interpret. My reason for saying this is that jeans appear to come in certain waist sizes and lengths only, jumping from 34 inches waist to 36 inches waist, with nothing in between. The consequence is that though there might be many different combinations, there aren’t so many that each can’t have its own UPC code – its own single big number. In other words, Levi’s could have chosen to treat the jeans sizes as a single key rather than a pair of scalars. The advantages of doing it the way they did are pretty clear: if you try on a pair, and it doesn’t fit, you have a good clue as to what pair to try next.
Location is a quantity that can be characterized by a key or by a pair of intervals. In the USA, ZIP codes are an established way of indicating a location. ZIP codes fit our notion of a key, though they aren’t pure keys by any means. Another way of indicating a location by a key is alphabetically: “San Francisco” indicates a location. Good at identifying a location, the name of a locale isn’t necessarily helpful in finding the location unless you are already very familiar with the name, or have an alphabetic index handy, or use other look-up mechanisms such as Google Maps. An entirely different way of characterizing a location is the one you find in use on a GPS device: a pair of numbers, one indicating latitude and the second indicating longitude. Latitude is measured relative to the equator, which has astronomic significance; longitude is measured relative to Greenwich, which has no astronomic significance. Regardless, both references are arbitrary – it isn’t hard to imagine that latitudes could also be measured relative to Greenwich, or that latitudes be measured relative to the south pole. Because of the relative arbitrariness of zero latitude and zero longitude, I would characterize both latitude and longitude as interval numbers.
The kind of quantity, like jeans size, that is expressed in more than one number, let’s call that a vector. Depending on your background, you may have encountered vectors before, and you may think of them in different ways: like coordinate pairs, or like arrows, or like certain quantities in physics such as velocity and force. All of those fit with what I say here, though this may shine some new light on those.
One way to picture vector quantities is as if we’re looking at a description of what’s in a shopping cart, but ignoring the price:
Russet potatoes 2 five-pound bag
Classic Coke 1 six-pack
Aspirin 81mg 2 bottles, 500 tablets each
With each number, we have a description as well as a unit. Picturing this explicit long-hand description is useful even if we often revert to a standard abbreviation. For example, the blue jeans size:
waist 34 inches
length 32 inches
is not a bad description for the “34×32” short-hand. Similarly, we can look at coordinate pairs like this:
x 5 meters east of origin
y 2 meters north of origin
which is not a bad description for the more common (5,2) short-hand.
The longer descriptive version, which I’ll call annotated vector to distinguish it from the short-hand versions we usually encounter in textbooks, is extraordinarily useful to visualize things like vector inner products as well as matrix multiplication. I expect to show this in a subsequent post.
For now, I simply want to emphasize that the notion of a vector allows us to look at several numbers and yet be clear that we’re talking about a single quantity. More precisely, with the idea of a vector we can, at will, think of it as a single quantity or think of it as several separate quantities. It is hard to overemphasize how fundamental that idea is, in mathematics as well as physics and other sciences, including computer science.