What’s Remarkable About Naming Sixty?

Here’s the astounding thing Christopher Hibbert did with his estimate of how much prices in 18th century Britain had to be multiplied to get an estimate for their amount in modern times: he named it.

Superficially, I have no place calling this astounding. If Hibbert didn’t have an estimate for how to convert 1782 prices to 1998 ones he would have not mentioned the topic at all. But consider: the best fit for a conversion factor could be from any of, literally, infinitely many imaginable numbers. That it should happen to be a familiar, common number, one so ordinary it even has a name, is the astounding part.

Part of that is a rounding-off, certainly. Perhaps the best possible fit to convert those old prices to the modern was actually a slight bit under 62, or was 57 and three-eighteenths. But nobody knows what £200 times 57 and three-eighteenths would be, as evaluating it would require multiplying by sevens, which no one feels comfortable doing, and dividing by eighteen, which makes multiplying by seven seem comfortable, unless we remember where we left the calculator, and why would we dig out a calculator to read about King George III?

No, better that we suppose the number is near enough sixty and that given the other ways this number can be wrong, the difference between the truly perfect fit and this one is not enough to bother with. This is another important mathematical principle, and another one that must be understood to master calculus. Every real number is near another real number, and often it’s near some number easier to work with. It’s also a good one for understanding a difference between mathematics as non-mathematicians see it and mathematics as mathematicians do it.

Without bothering to actually support my claim with evidence, I claim the defining characteristic of mathematics, as non-mathematicians see it, is that mathematics is something which has a great many digits, ideally running past the decimal place. If someone on a TV show says the trip will take two hours, fourteen minutes, 6.185 seconds, the viewer knows this is supposed to be a person very comfortable doing math, and the viewer is supposed to be impressed. Clearly, hewing all those digits out of raw number-stuff is hard work, and the claimant has thought of something as hard as 6.185 just in case it came up.

And yet that precision is meaningless: surely something will happen during the trip to make its duration vary from the estimate by more than one two-thousandths of a second. Furthermore, this meaningless precision should not even be impressive: it is as easy to get eight digits out of a calculator as it is to get three. Possibly easier. It isn’t much harder to get eighteen digits out of a computer. A mathematician would likely argue that it is better to say something perfectly true, such as that the trip will be about two and a quarter hours, but we should not be surprised if it is five minutes shorter or longer. (Every field suffers its popular-representation stereotypes. Chemists I think have it worst, as they are apparently people who mix together colorful liquids to make them explode. 6.185 is pleasant compared to an explosion.) At least it’s less hard work, unless the mathematician wants to show off hewn digits.

So this is one interesting aspect of actually naming the conversion factor. We accept that it is too much bother for too little gain to pin down the exact number, and will take something close enough and which is easier to work with. But there is power in that. Running through all of analysis, where we prove that arithmetic works the way we thought arithmetic worked, and where we show that calculus works the way it always did for the professor and usually did for the teaching assistant but never quite did for us on the exam, are proofs which run along the lines of “this must equal that plus-or-minus some error margin, and the error margin can be as small as we like, so therefore this must equal that because otherwise we would like an error margin smaller than the difference between them”. But that isn’t what I find remarkable about naming sixty.

Here I need to argue that we have actually few names for numbers. Some readers are probably leaping past me, observing that there are rational numbers which we can describe as some counting number like ‘twelve’ or a fraction like ‘seven-eights’ or a combination, like, ‘twelve and seven-eighths’; and there are irrational numbers which we can describe, such as ‘the square root of twelve and seven-eights’ but which hasn’t got a name quite the way that ‘twelve’ has. And there are enormously more irrational number than there are rationals, so that it is in some sense impossibly unlikely to ever encounter a rational number.

This is true, and not even very difficult to show. A student in the higher grades of elementary school could follow the argument. Possibly it would do some good for mathematics education if students on the brink of the bewilderment of algebra got to shiver at the sublime discovery that there is not just an infinity, but there are uncountably many infinities, all different from the others, and that the best of our collected imaginations can barely start thinking of what they might be. But I mean something simpler.