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”.
Continue reading “Something Cute Without 9’s and a 6”
I do sometimes read online forums of educators, particularly math educators, since it’s fun to have somewhere to talk shop, and the topics of conversation are constant enough you don’t have to spend much time getting the flavor of a particular group before participating. If you suppose the students are lazy, the administrators meddling, the community unsupportive, and the public irrationally terrified of mathematics you’ve covered most forum threads. I had no luck holding forth my view on one particular topic, though, so I’ll try fighting again here where I can easily squelch the opposition.
The argument, a subset of students-are-lazy (as they don’t wish to understand mathematics), was about a mnemonic technique called FOIL. It’s a tool to help people multiply binomials. Binomials are the sum (or difference) of two quantities, for example, (a + 2) or (b + 5). Here a and b are numbers whose value I don’t care about; I don’t care about the 2 or 5 either, but by picking specific values I avoid having too much abstraction in my paragraph. The product of (a + 2) with (b + 5) is the sum of all the pairs made by multiplying one term in the first binomial by one term in the second. There are four such pairs: a times b, and a times 5, and 2 times b, and 2 times 5. And therefore the product (a + 2) * (b + 5) will be a*b + a*5 + 2*b + 2*5. That would usually be cleaned up by writing 5*a instead of a*5, and by writing 10 instead of 2*5, so the sum would become a*b + 5*a + 2*b + 10.
FOIL is a way of making sure one has covered all the pairs. The letters stand for First, Outer, Inner, Last, and they mean: take the product of the First terms in each binomial, a and b; and those of the Outer terms, a and 5; and those of the Inner terms, 2 and b; and those of the Last terms, 2 and 5.
Here is my distinguished colleague’s objection to FOIL: Nobody needs it. This is true.
Continue reading “In Defense Of FOIL”
I’ve outlined now some of the numbers which grew important enough to earn their own names. Most of them are counting numbers; the stragglers are a handful of irrational numbers which proved themselves useful, such as π (pi), or attractive, such as φ (phi), or physically important, such as the fine structure constant. Unnamed except in the list of categories is the number whose explanation I hope to be the first movement of this blog: e.
It’s an important number physically, and a convenient and practical number mathematically. For all that, it defies a simple explanation like π enjoys. The simplest description of which I’m aware is that it is the base of the natural logarithm, which perfectly clarifies things to people who know what logarithms are, know which one is the natural logarithm, and know what the significance of the base is. This I will explain, but not today. For now it’s enough to think of the base as a size of the measurement tool, and to know that switching between one base and another is akin to switching between measuring in centimeters and measuring in inches. What the logarithm is will also wait for explanation; for now, let me hold off on that by saying it’s, in a way, a measure of how many digits it takes to write down a number, so that “81” has a logarithm twice that of “9”, and “49” twice that of “7”, and please don’t take this description so literally as to think the logarithm of “81” is equal to that of “49”.
I agree it’s not clear why we should be interested in the natural logarithm when there are an infinity of possible logarithms, and we can convert a logarithm base e into a logarithm base 10 just by multiplying by the correct number. That, too, will come.
Another common explanation is to say that e describes how fast savings will grow under the influence of compound interest. A dollar invested at one-percent interest, compounded daily, for a year, will grow to just about e dollars. Compounded hourly it grows even closer; compounded by the second it grows closer still; compounded annually, it stays pretty far away. The comparison is probably perfectly clear to those who can invest in anything with interest compounded daily. For my part I note when I finally opened an individual retirement account I put a thousand dollars into an almost thoughtfully selected mutual fund, and within mere weeks had lost $15. That about finishes off compound interest to me.
Continue reading “Some Names Which e Doesn’t Have”
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.”
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.
Continue reading “How Many Numbers Have We Named?”
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?
Continue reading “What’s Remarkable About Naming Sixty?”
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.
Continue reading “Did King George III pay too little for astronomers or too much for tea?”