So we’ve introduced, with maybe more words than strictly necessary, the idea that we can set up a match between the numbers from 0 to 496 and particular locations on the New York Thruway. There are a number of practical quibbles that can be brought against this scheme. For example: could we say for certain that the “outer” edge of this road, which has roughly the shape of an upside-down u, isn’t loger than the “inner” edge? We may need more numbers for the one side than the other. And the mile markers, which seemed like an acceptable scheme for noting where one was, are almost certainly only approximately located.
But these aren’t very important. We can imagine the existence of the “ideal” Thruway, some line which runs along the median of the whole extent of the highway, so there’s no difference in length running either direction, and we can imagine measuring it to arbitrarily great precision. The actual road approximates that idealized road. And this gives what I had really wanted, a kind of number line. All the numbers from zero to 496 (or so) match a point on this ideal Thruway line, and all the points on this Thruway match some number between zero and 496. That the line wriggles all over the place and changes direction over and over, well, do we really insist that a line has to be straight?
Well, we can at least imagine taking this “ideal” Thruway, lifting it off the globe and straightening it out, if we really want to. Here we invoke a host of assumptions even past the idea that we can move this curvy idealized road around. We assume that we can straighten it out without changing its length, for example. This isn’t too unreasonable if we imagine this curve as being something like a tangled bit of string and that we straighten it out without putting any particular tension on it; but if we imagined the idealized road as being a rubber band, held taut at the New York City and Ripley, New York, ends and pinned in place at the major turns we notice that isn’t actually guaranteed. Let’s assume we can do this straightening-out without distorting the lengths, though.
Straight lines, and number lines that are straight, we’re very familiar with. We see them in any ruler, and have a suggestion of them in any sheet of lined paper. I haven’t been able to establish just how old the idea is; it probably goes back to whoever first noticed a distance could be marked off in the number of paces, or the number of foot lengths, or the number of arm’s spans between two things. At least, we know that as a way to count off a whole number of paces, and take whatever awkward bit left over that might not fit into a whole number of paces as some fraction of a pace.
As we set this “ideal” Thruway up, we have a line with points numbered from 0 through 496, and all the numbers in-between. Is there any reason we have to use that range, though? If we want to use mileage markers, yes, as we don’t like putting negative numbers on highway signs. But if we don’t need to consider that, we could imagine instead of setting the zero point at the New York City/Yonkers border instead putting it, say, at what is now the mile 248 marker, somewhere near Rome. The half of the Thruway from Yonkers to Rome can be matched to the negative numbers, -248 up through zero, and the other half ascending again from zero up through 248.
This may seem like a small achievement, but it identifies one of the lovely things about number lines: they exist for our convenience and we can set them to whatever range is useful. If we wanted to we might set the zero point at mile marker 100, near Saugerties and Woodstock, and let the numbers from from -100 up through positive 396. We could set the zero point near Batavia, and let the numbers run from -390 up through 106, though I agree that range doesn’t see particularly convenient. The 0 to 496 range has obvious value, and -248 to positive 248 has its pleasant symmetry. Probably there are a few others with some convenience.
Since we like measuring distances of about this size in miles, we’re drawn naturally to a 496-digit range. But we aren’t bound to the mile measure by anything besides our bias for working with numbers that are not too big or small. If we were interested in measuring in feet, we could relabel this line with numbers running from 0 up to 2,618,880, or any other interval of just a bit over 2.6 million units in span. If we wanted to measure in millimeters instead, we could have a span of nearly eight hundred million.
We needn’t keep making these numbers bigger, either. We could identify each point with a number between 0 and 100, representing what percentage of the length of this ideal Thruway is between a given point and the starting point. Or we can make it smaller, and go between 0 and 1, the number representing the fraction of the whole length between the given point and the starting point.
This is all fine for relabelling positions along the Thruway. The next thing I’m interested in is trying some ambitious number ranges.