In the previous post we looked at man hours. Man hours are strange beasts – but not too strange – that represent the result of multiplying men and hours. They make sense in the context of billing, and when sizing up a job. If job A can be done by a crew of 3 in 4 hours, then job A is a 12 man hour job. If job B is done in the same amount of time, but needs twice the crew, job B takes twice the number of man hours of job A. If job C is done with the same crew as job A, but needs twice the amount of time, job C also takes twice the number of man hours of job A. This is all stuff that most of us are familiar with, and it only looks strange on close inspection. On close inspection, the notion of multiplying men with hours can suddenly come to look mighty strange – and certainly much different from what happens when you add stuff.
Mostly, when we add, we add items of the same ilk. I can add two men and three men, resulting in five men. I can add three hours and nine hours, resulting in twelve hours. But I’m not sure I have much use for adding two men and seven hours. It’s not that we never add two of different things – I can take 6 feet and add 2 inches. Or I can take three dollars and add fifty cents. Or I can put 5 apples in my grocery cart and then add 6 oranges. The person at the checkout counter knows exactly what to do with that. Though the things I add together may not be the same, they are in some sense comparable – that is, expressible in a common denomination. For feet and inches, there is a common denomination of inches; for dollars and cents there is a common denomination of cents. For the grocery cart, there isn’t an obvious common denomination for apples and oranges, but at the checkout counter what I care about is that there is a common denomination with regard to price, so that what’s added is really the cost of the five apples and the cost of the six oranges.
Though it is simple to add 3 oranges and 2 oranges, we are rarely called upon to multiply them. The question “what is 3 oranges times 2 oranges?” is a strange one. If it was a parlor joke, you might respond “6 oranges squared!” and feel smug about your answer. But what in the world would an orange squared be? Is the picture an example of 6 oranges squared?
I see simply six oranges, and – I suspect – so do you. Though we multiplied two columns by three rows, we didn’t multiply oranges by oranges.
And yet – we talk about square inches all the time. A letter-size sheet of paper is and this amounts to of paper, and if we were to cut the sheet in squares of one inch by one inch, using some Scotch tape, we could confirm that we got 93.5 of those. If that’s too messy, we could cut off a strip of first and discard it, leaving us with a sheet. This sheet cuts neatly into squares of an inch by an inch, without any Scotch tape required. How many of those squares would we get? Indeed, 88. So:
And the idea that somehow one inch times one inch makes a square inch could be pictured like this:
Does this makes sense? It seems fairly clear even from this representation that the left inch and the right inch aren’t really fully symmetrical. Perhaps this representation is closer:
(There is a fancy word for this not-quite-symmetry: the two inches are adjoint. Those of us who throw around area models for multiplication often pretend that we know exactly what we’re doing. I’m clear that I’ve been lucky that my lack of understanding of the subtleties of it has never been called.)