The Hell You Say

Cats, days, numbers, and all that
As I write this, in January of 2015, it is still easy to remember that last month, on the thirteenth day of the month, a number of people did a number of important things (such as marrying) because the day was auspicious, since it could be referred to as 12/13/14, numbers in sequence.  Since there is no month whose number is within two of 15 nor any larger number, such a sequence will not occur again for a very long time.

Will it ever happen again? Not until January 2, 2103, by which time our suicidal society will most likely have ended our interest in that and everything else.

Let’s see how it happened to happen this time. (We shall ignore “last time.”)

Two features of our customary dating system played special roles in the event. First, of course, is the fact that we tend to leave off the first two digits of the year unless there is great danger of confusion People would certainly have been less impressed if they had thought of the date as 12/13/2014. Second, here in the USA we have the custom of indicating the month before the date, so that December 13 comes out 12/13 and not 13/12.  In many parts of the world, that custom is reversed. In most of Europe (maybe all of it: I haven’t been there to ask) the date is mentioned before the month. For those people, the day was 13/12/14, which may be special in some way I haven’t though of, but is certainly not the same “natural” sequence that led so many couples to the altar on December 13 of 2014.

Actually, the US and European systems don’t exhaust the logical  possibilities, either. In writing numbers, we put tens to left of units, hundreds to left of tens, thousands to left of hundreds, and so on. That is, we write the larger chunks to the left of the smaller ones. Thus 2014 is so written because 4 is, in that usage, a mere unit, 1 counts tens, we have no hundreds, and the 2 counts thousands of years. If we were to use the same philosophy in writing dates, we would (and actually a small number do) write the date as 2014 December 13, or, in the fully abbreviated form, 14/12/13. (Maya civilization did this, but it’s hard to recognize because their normal system was based on 20, not 10, and when referring to their calender system, an 18 was used  where a 20 would have messed things up. Sort of like our 12.  But few of us are Mayans, so let’s just stick with our own for now.)

The point is that it is not the actual day that earns some sort of respect (such as weddings), but our way of representing the day, which is quite another matter.

Confusion between things and representations of things is not uncommon, but is not universal, either. For example, consider the word cat. It is used in order to mention a certain kind of animal, usually referring to a domestic, or house cat, which has four paws, a tail, and whiskers, among other observable characteristics. The word has none of these. No paws, no tail no whiskers, does not jump into anybody’s lap to be petted, or any of that. On the other hand the word has three letters, which no domesticated cat has. Here we all know the difference and we are not confused. We can tell the denoting word from that which it denotes.

With dates—and, in fact, with numbers generally—confusion is much more easily obtained. We’ve seen an example with dates. Let’s move on to numbers generally. For which we shall also jump to palindromes.

You may not have run into palindromes before, so I’ll interrupt to tell you what they are. They are  verbal expressions such that when written with the letters in reverse order, say the same thing as they did in their original order. In doing this, spacing and punctuation may or may not be ignored. A classic example of a moderately long palindrome consists of what Napoleon Bonaparte might have  said after a period of isolation on Elba: “Able was I ere I saw Elba.” Another widely known one is what  might have been (but was not) attributed to a character in the Book of Genesis, introducing himslf to his appointed helpmeet: “Madam, I’m Adam.”  Of course, a palindrome need not be particularly long. “Bob” and “Sis” are both palindromes.

Now, what about numerical palindromes? Here’s one: 4,004. You want shorter? Well, how about 252? I suppose a single digit might also be called a palindrome, for that matter. Then 8 or 5 or 7 would all be very short (rather dull)  palindromes. Question: if we use a different system to refer to numbers, will the same numbers prove to be palindromes?

Well, take a look at 252 in Roman numerals. It comes out CCLII which, no, is not palendromic. Apparently being a palindrome is not a property of a number itself, but of our way of representing the number.

A kind of linguistic confusion occurs here because we tend to use the same word (“number”) for a number and for its standard representation; at least we do when the representation uses the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 (called “Arabic numerals” for good reasons that we don’t need to go into here). I don’t really know whether I need to keep things simple by retaining that confusion or to simplify-by-complicating by distinguishing numbers proper from their representations, which can be called numerals. There certainly are facts about numbers themselves that are not merely attributable to their numeral representation. For example, three times five comes out fifteen whether we think of it verbally, as just illustrated, or verbally in some other language (drei mal fünf, for example) or in a symbolic form (3 times 5, or III times V). Serious mathematics deals with actual numbers, and not, except as a sideline, with their representations, but recreational mathematics tends to deal with represenations. For the most part, questions about palindromic numbers are really questions about palindromic numerals. That doesn’t mean they can’t be fun, however.

Here’s a suggestion I sent my ten-year-old granddaughter. Consider a digital clock that shows hours and minutes (but not seconds) and cycles through twelve hours twice in one day. Certain of the times it indicates are palindromic, e.g. 3:43 or 10:01 or 5:55. Obviously, going from 3:43 to 3:53 takes ten minutes. The next palindromic time (there is no 3:63, of course) is 4:04, and it takes eleven minutes for the clock  to get there. How many different time-gaps of this sort will occur in a complete cycle? (As a hint, it only takes two minutes to get from 9:59 to 10:01.)

In Europe most clocks, including digital ones, show 24 hours rather than 12. Does this make a huge difference in the distribution of palindromic times?

I’m not revealing my granddaughter’s reply at this time, mainly because it’a not back yet, but I suspect she can only handle part of the complex of questions given and questions that tend to pop up when one works with the ideas for a while. I might add that I did not ask her to distinguish between whether times themselves are palindromic or whether it’s all a matter of representation. You can answer that question, though, if you think  about Roman numerals (even though it’s altogether non-standard to refer to  minutes using Roman numerals.)

Does any of this apply to contemporary politics or to mystery writing? I’ll let you decide.

 

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