Something Cute With 9’s and a 6.


After the last few essays I’d like to take a moment for a distinct, cute little problem of no practical use but cute.

Write down as many 9’s as you like, and when finished with that place a 6 at the right end. The result is divisible by 6.

That is, whatever number you’ve written, divided by 6, produces a whole number. Divisibility is one of those things which turns up whenever you have a collection of things which can be multiplied, and one thing is divisible by the second if you can find something in your collection so that the second multiplied by your find equals the first. It’s most often used to talk about the integers — the positive counting numbers, their negative counterparts, and zero if we didn’t include that already — and if it isn’t said divisible-with-respect-to-what then integers are what is usually meant. Partly that’s because integers are the first thing where divisibility stands out: if we look at the real numbers, everything is divisible by everything else (as long as that “else” is not zero), and a property that’s (almost) always true is usually too dull to mention. The next topic where divisibility gets mentioned much is usually polynomials, with a few eccentrics holding out for the complex numbers where the real part and the imaginary part are both integers.

There are several ways to prove this string of 9’s followed by 6 is divisible by 6. Here’s a proof which I like.

Continue reading “Something Cute With 9’s and a 6.”

How Many Numbers Have We Named?


I want to talk about some numbers which have names, and to argue that surprisingly few of numbers do. To make that argument it would be useful to say what numbers I think have names, and which ones haven’t; perhaps if I say enough I will find out.

For example, “one” is certainly a name of a number. So are “two” and “three” and so on, and going up to “twenty”, and going down to “zero”. But is “twenty-one” the name of a number, or just a label for the number described by the formula “take the number called twenty and add to it the number called one”?

It feels to me more like a label. I note for support the former London-dialect preference for writing such numbers as one-and-twenty, two-and-twenty, and so on, a construction still remembered in Charles Dickens, in nursery rhymes about blackbirds baked in pies, in poetry about the ways of constructing tribal lays correctly. It tells you how to calculate the number based on a few named numbers and some operations.

None of these are negative numbers. I can’t think of a properly named negative number, just ones we specify by prepending “minus” or “negative” to the label given a positive number. But negative numbers are fairly new things, a concept we have found comfortable for only a few centuries. Perhaps we will find something that simply must be named.

That tips my attitude (for today) about these names, that I admit “thirty” and “forty” and so up to a “hundred” as names. After that we return to what feel like formulas: a hundred and one, a hundred and ten, two hundred and fifty. We name a number, to say how many hundreds there are, and then whatever is left over. In ruling “thirty” in as a name and “three hundred” out I am being inconsistent; fortunately, I am speaking of peculiarities of the English language, so no one will notice. My dictionary notes the “-ty” suffix, going back to old English, means “groups of ten”. This makes “thirty” just “three tens”, stuffed down a little, yet somehow I think of “thirty” as different from “three hundred”, possibly because the latter does not appear in my dictionary. Somehow the impression formed in my mind before I thought to look.
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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?
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Did King George III pay too little for astronomers or too much for tea?


In the opening pages of his 1998 biography George III: A Personal History, Christopher Hibbert tosses a remarkable statement into a footnote just after describing the allowance of Frederick, Prince of Wales, at George III’s birth:

Because of the fluctuating rate of inflation and other reasons it is not really practicable to translate eighteen-century sums into present-day equivalents. Multiplying the figures in this book by about sixty should give a very rough guide for the years before 1793. For the years of war between 1793 and 1815 the reader should multiply by about thirty, and thereafter by about forty.

“Not really practical” is wonderful understatement: it’s barely possible to compare the prices of things today to those of a half-century ago, and the modern economy at least existed in cartoon back then. I could conceivably have been paid for programming computers back then, but it would be harder for me to get into the field. To go back 250 years — before electricity, mass markets, public education, mass production, general incorporation laws, and nearly every form of transportation not muscle or wind-powered — and try to compare prices is nonsense. We may as well ask how many haikus it would take to tell Homer’s Odyssey, or how many limericks Ovid’s Metamorphoses would be.
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