I want to do some more tricky examples of using this ε idea, where I show two numbers have to be the same because the difference between them is smaller than every positive number. Before I do, I want to put out a problem where we can show two numbers are not the same, since I think that makes it easier to see why the proof works where it does. It’s easy to get hypnotized by the form of an argument, and to not notice that the result doesn’t actually hold, particularly if all you see are repetitions of proofs where things work out and don’t see cases of the proof being invalid.
Now here’s another great tool Chiaroscuro did, in figuring out what number raised to the fifth power would be 1/6000. Besides trying out a variety of numbers which were judged to be a little bit low or a little bit high, he eventually stopped.
Wisely, too. The number he really wanted was the fifth root of 1/6000, and while there is one, it’s not a rational number. It goes on forever without repeating and without falling into any obvious patterns. But neither he nor anyone else is really interested in any but the first couple of these digits. We’d wanted to know whether this number was close to 0.25, and it’s closer to 0.17 instead. What the tenth digit past the decimal was we don’t really care about. It’s fine to be close enough to the right answer.
This runs a little against the stereotype of the mathematician. To the extent that popular culture notices mathematicians at all, it’s as people who have a lot of digits past a decimal point. But a mathematician is, in practice, much more likely to be interested in saying something that’s true, even if it isn’t so very precise, and to say that the fifth root of 1/6000 is somewhere near 0.17, or better, is between 0.17 and 0.18, is certainly true. Probably — and I’m attempting here to read Chiaroscuro’s mind, as the only guidance I’ve gotten from him is the occasional confirmation about what my guesses to his calculation were — he found that 0.17 was a little low, and 0.18 was a little high, and the actual value had to be somewhere between the two. The Intermediate Value Theorem, discussed in the previous non-Gemini-Chronology entry, guarantees that between those two is an exactly correct answer. (It’s conceivable that there would be more than one, in fact, although for this problem there’s not.)
Chiaroscuro specifically judged the fifth root of 1/6000 to be 0.176, or 17.6%, and I doubt anyone would seriously argue with that claim. This is even though the actual number is a little bit less than that: it’s nearer 0.175537, but even that is only an approximation. We are putting one of those big ideas into play, subtly, when we accept saying one number is equal to another in this way.
Back to the theme of divisibility of numbers. Since we have the idea of writing numbers with a small set of digits, and with the place of those digits carrying information about how big the number is, we can think about what’s implied by that information.
In the number 222, the first two is matched to blocks (hundreds) that are ten times as large as those for the second two (tens), and the second two is matched to units (tens) which are ten times as large as those for the third two (units). It is now extremely rare to have the size of those blocks differ from one place to the next; that is, a number before the initial two here we take without needing it made explicit to represent ten times that hundreds unit, and a number after the final two (and therefore after the decimal point) would represent units which are one-tenth that of the final two’s size.
It has also become extremely rare for the relationship between blocks to be anything but a factor of ten, with two exceptions which I’ll mention next paragraph. The only block other than those with common use which comes to my mind is the sixty-to-one division of hours or degrees into minutes, and then of minutes into seconds. Even there the division of degrees of arc into minutes and seconds might be obsolete, as it’s so much easier on the computer to enter a latitude and longitude with decimals instead. So blocks of ten, decimals, it is, or in the way actual people speak of such things, a number written in base ten.
The cute little thing about a string of 9’s followed by a 6 being a number divisible by 6 inspired my Dearly Beloved, who spent some time looking for other patterns in this kind of number. I’m glad for that; this sort of pattern, while it may not be terribly important, is often fun to play with. And interesting things can be found in play.
I don’t know a good name for this kind of number, and admit it feels awkward to say just “this kind of number”. If I have to talk about them much longer some group name is probably worth devising. Unfortunately the only names which come to my mind come there through organic chemistry, where it’s reasonably common to have an arbitrarily long chain of carbon atoms terminated with some distinctly different group. For example, an alcohol is a string of carbons ending with an oxygen and hydrogen molecule. But an “alcoholic number”, while an imagination-capturing name, doesn’t quite fit. I suppose aldehydes, which end on a double-bond to an oxygen atom, preserves the metaphor, but no one knows the adjective form of aldehyde.
My Dearly Beloved’s experiments found no other numbers for which a repeated string, terminated by a 6, would produce a number divisible by 6. This overlooked the obvious case, though: a string of 6’s, followed by another 6, is itself divisible by 6. Obvious cases are like that, and many people would think of a uniform string of 6’s not part of the pattern “an arbitrary number of one digit, followed by a 6”.