Units and Ratios
Cross-multiplication is the bastard child of a proper theory of ratios. With what right do we treat meters and seconds as if they were numbers, so that a meter (or ought we to simply say "meter") divided by a meter (meter divided by meter) is one?
There are 12 inches in a foot. Or we could say, a foot is 12 inches. If we write that out, we get:
(1) 1 foot = 12 inches.
So now I can divide both sides by 12, and I get:
(2) 1/12 foot = 1 inch.
So far, so good. Now what if I divide both sides, not by 12, but by 12 inches? (It is awkward to say that when we divide an inch by inches, both the inches and the inch cancel.)
We would write this:
(3) 1 foot / 12 inches = 1
Or we could divide equation (2) by 1 inch, and then also we would have:
(4) 1 foot / 12 / 1 inch = 1.
So now is 12 multiplied by 1 inch equal to 12 inches? What does it mean to multiply a number by an inch? I know what it means to multiply a number by 1, but I am not sure of this. And why when we multiply 12 by 1 inch do we get 12 inches and not 12 inch? (Perhaps that's just a linguistic problem.)
It makes far more sense, as far as I can see, to present the matter this way. We take equation (2) seriously -- that is to say, we hold that 1 foot *is* 12 inches. But then we have to divide both sides of this equation (which is really an identity) by 12 to get that 1/12 of a foot is an inch. Now how can we divide something which does not seem to be a number by 12?
So my confusion comes down to these two points:
(1) What is the difference between 12 and 12 inches?
(2) Why is it that we can divide a measure (12 inches) by a number (12)?
It seems we can go back and forth from numbers to measures and from measures to numbers.
There are 12 inches in a foot. Or we could say, a foot is 12 inches. If we write that out, we get:
(1) 1 foot = 12 inches.
So now I can divide both sides by 12, and I get:
(2) 1/12 foot = 1 inch.
So far, so good. Now what if I divide both sides, not by 12, but by 12 inches? (It is awkward to say that when we divide an inch by inches, both the inches and the inch cancel.)
We would write this:
(3) 1 foot / 12 inches = 1
Or we could divide equation (2) by 1 inch, and then also we would have:
(4) 1 foot / 12 / 1 inch = 1.
So now is 12 multiplied by 1 inch equal to 12 inches? What does it mean to multiply a number by an inch? I know what it means to multiply a number by 1, but I am not sure of this. And why when we multiply 12 by 1 inch do we get 12 inches and not 12 inch? (Perhaps that's just a linguistic problem.)
It makes far more sense, as far as I can see, to present the matter this way. We take equation (2) seriously -- that is to say, we hold that 1 foot *is* 12 inches. But then we have to divide both sides of this equation (which is really an identity) by 12 to get that 1/12 of a foot is an inch. Now how can we divide something which does not seem to be a number by 12?
So my confusion comes down to these two points:
(1) What is the difference between 12 and 12 inches?
(2) Why is it that we can divide a measure (12 inches) by a number (12)?
It seems we can go back and forth from numbers to measures and from measures to numbers.