# 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.

We pick up another name, by my standards and by my dictionary’s, at a thousand, and a million, a billion, a trillion. I want here to mention an old-fashioned usage I find charming, since I have some lovely examples in the Robert Benchley essay “If These Old Walls Could Talk!”, about the many imaginatively boring after-dinner speeches he endure as a reporter in the old Waldorf-Astoria ballroom, with sample excerpts: “annual imports for the year 1915 running into tens of millions of dollars”; “leaving six billions of dollars which this country”; “making a total of three hundreds of millions of dollars”. I can find no graceful way to use these examples, and should not have brought them up at all, except that the last case seems to particularly prove whatever my point might be.

In principle we have a system that can go out arbitrarily far; in practice, I can’t remember anyone speaking of an “illion” number past “an octillion”. That we would have to go into the really obscure prefixes to get above that says to me there isn’t any name anyone would recognize for 1,000,000,000,000,000,000,000,000,000,000. But we have few enough things to count in the range of the nonillions that we aren’t hurt by naming problems.

There are a few other big numbers for which it’s convenient to have names: Avogadro’s number, telling how many atoms or molecules it takes to have a useful tabletop amount of something; the googol and googolplex, which serve as nice outposts for number-bigness; Skewes’s Number, at one time the largest number used in a mathematical proof (about how common prime numbers are); Graham’s Number, at another time the largest number used in a mathematical proof (about graph theory, that is, ways to connect points together); and some other numbers vying for the title of largest number used in a mathematical proof.

Then there are some fractions with names: a half, a third, a quarter, and then we swiftly move into just taking the ordinal version of the reciprocal: a 68th. We multiply those by a counting number if we need two thirds, or twenty-five 68ths, but I can’t remember any distinct name for any of these fractions except the base. H L Resnikoff and R O Wells’s Mathematics In Civilization notes in reviewing the ancient Egyptian system of arithmetic they had symbols for a half, a third, a quarter, and so on, but not for any higher fractions except for two-thirds; I find it interesting that in language we still have this limit. But then we are comfortable saying “three quarters”; the ancient Egyptian system would put it “a half and a quarter”, and would be more convoluted for something like 25/68.

All these examples have been rational numbers. There are some named numbers which are not: π is unquestionably the most famous, an irrational number which humanity first found interesting because it told us how far it is around a circle if we know how far across it is. It keeps turning up, in ever-more surprising places. If you were to drop toothpicks on the square tiles of a kitchen floor, you could find π in how many toothpicks cross the lines of grout between tiles. It’s challenging to find things which hide no π within.

There is the fine structure constant, likely an irrational number, which describes (among many interpretations) how strongly photons and electrons will interact. There is the Boltzmann constant, connecting how much energy a gas has to what its temperature is. These and several more named numbers describe obviously interesting physical things.

There are a few numbers which get names because they have some attractive properties. The golden ratio, φ, a little bit more than 1.618, gets its name because one divided by it is φ minus one, a little bit more than 0.618, and so is strangely hypnotic to play with on a pocket calculator that has a 1/x key. I think that number overrated, but it is certainly named.

There is one novelty, i, dubbed the “imaginary number” because if one multiplies i by itself one gets minus one as the product. This idea is newer even than negative numbers, and the name shows the suspicions it arouses. Numbers made from adding one of our familiar real numbers to an imaginary number are even called “complex”, making them sound frightening, although work with them turns out to be simple.

If that is not staggering enough there are things called “quaternions”, built with not just i but also the numbers j and k, which multiplied by themselves are minus one again, but which are not equal to each other or to i, and where one of them times a second will be not minus one but rather plus or minus the third. Quaternions have many strange features besides existing; perhaps the most exciting is these exotic instruments are extremely convenient ways to describe rotations, and work their way into video games or, in dignified form, computer simulations, because of that.

So there is my argument: there are a handful of numbers which get names. Most of those names are convenient blocks to use for counting, and a few that have physical significance or mathematical beauty make the cut. There is one which I have not yet named, but which is in the set of two or three most important non-rational numbers. It’s less important than π; whether it is more or less important than i may be a matter of taste. But I want to explain it at decent length.

## Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

## 29 thoughts on “How Many Numbers Have We Named?”

1. In my youth in Yorkshire in the 1970s, it was still common to hear older folk saying “five and twenty to six” for 5:35, for example. I don’t think it was particularly a London construction.

Thanks for an interesting article.

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1. Huh. Well, I’ll defer with ill-grace and sullen resentment toward actual experience compared to what I remember reading in some book about the English language somewhere. I’m not sure which it was (McCrum-MacNeil-Cran, possibly?) but I do recall the “one-and-twenty” versus “twenty-one” difference being mentioned as a case where the dialect of Crown and Court lost to that of the outskirts, and it’s easily possible I got mixed up what the old ranges were.

I hadn’t thought of this before, but I remember times sometimes being given as “half nine” for 9:30, and wonder if that’s a last echo of the one-and-twenty construction. I like the way it sounds, but I also like having the “o’clock” appended to times that aren’t on the hour, too.

Also, thanks for your kind words.

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1. “Half nine” for 9:30 (other half-pasts) is still alive and well here. I’ll sometimes use it myself in informal chat with other locals.

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2. Geoffrey says:

Lovecraft invokes “vigintillions” at some point, but sensible people have shifted to scientific notation well before then. Note that the English “billion” used to be 1E12, although I think it’s almost extinct these days.

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1. I hadn’t encountered “vigintillions”, although I haven’t read much Lovecraft. (In one of those many odd circumstances I’m better versed in his imitators and parodies than the actual thing.)

The point where people switch to scientific notation would probably be a guide to what range of numbers people actually like working in. I wouldn’t be surprised if people typically don’t care for the range to go past about a thousand or ten thousand.

I do regret the loss of the former British billion, considering the logic it does make (and the corresponding loss of milliard, although billiard would be confusing and I don’t think even French has found something to use trilliard for) in counting groups of six places. I suppose it’s too late to recover without a determined effort at forcing a change in the language, and it seems like all the people with the energy and determination to do that have decided to instead be angry about the use of “decimate” for “destroy a huge part of” rather than “destroy one-tenth of”, which has to be the English language’s slightest change of meaning ever.

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3. Chiaroscuro says:

Hmm. I wonder if ‘A fifth’ is still clearly enough different from ‘a five-th’ to count. It’s a tight case. ‘Eighth’ and ‘ninth’, not so much. Also a ‘cent’ being a bit archaic, but a good term for a hundredth of something.

In Spanish, the numbers go up to fifteen, then ‘ten and six’, ‘ten and seven’, etc.Even in English , here’s clear echo of ‘Fourteeh’ coming from ‘four ten’ the way ‘Forty’ would.

Let’s also not forget a myriad. A gross. A dozen. A score. But a dozen or score are subtly not number names any more, in the ways that a ‘pair’, ‘duo’, or ‘triad’ aren’t.

–Chi

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1. I think that a fifth doesn’t stand up as a separate word; it’s the ordinal name again, the fifth in a series or the fifth part of a whole. At least, a half and a quarter hide their connection to two and four. But there’s a lot of arbitrariness in where the line gets drawn.

I haven’t got enough familiarity with Spanish to draw comparisons, but I did learn enough French to think about the French-of-France versions of 80 and 90, quatre-vingt and quatre-vingt et dix. Swiss French adds huitante for 80, and Swiss and Belgian French put in nonante for 90, and the differences run up the number system.

Myriad, gross, and score I had forgotten. Score survives in a famous quote, at least, but that does represent a fossil version of its use as a number name.

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4. If I weren’t already thoroughly familiar with numeric notation, this article would’ve frustrated me. The title and first few paragraphs led me to expect a full enumeration of professional English number-words (one, two, thousand, quadrillion), but it rapidly diverged and wandered through personal taste and local usages. It’s an awkward beast, neither fully humorous nor entirely studious.

Where you use Greek letters, you may want to include the name in parentheses: Π (pi), to make it clear it’s not just a square “n” (as it appears in this typeface). I have just had cause to learn that the HTML character entity “phi” renders as a loopy shape I thought was “koppa,” but isn’t. Koppa, according to Wikipedia, is variously depicted like “G” or as a fat lollipop. (Does this blog dis/allow HTML markup in comments?)

Modern Japanese has, for a historical reasons, a couple of different counting systems and notations, but mostly it constructs in straightforward base ten: 1, 2, … 9, 10, 10+1, 10+2, … 10+9, 2*10, 2*10+1, … 9*10+9, 100, etc. The words for 10^2/10^3/10^4 are hyaku, sen and man; larger magnitudes are based around 10^4, instead of (as in English) 10^3. This also seems to be the case with (Sanskrit?) as I discovered when I had cause to interpret some Bangladeshi economic statistics reported in “crore taka;” “taka” is the currency, and it turns out “crore” is a 10^4 multiplier.

A recent issue of Scientific American has a feature article on the evolution from complex numbers to quaternions (in the 1840s) and octonions, and how the latter seem to be related to string theory and M-theory. It mentions how the inventor’s friend was unsettled by the idea of j and k: roots of -1 that somehow weren’t the same as i.

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1. My original notion for this essay was actually to try coming up with a count for how many named numbers there are. But I got stuck on the problem of how I could say that “twenty” was a name while “twenty-one” was not, and then whether “nineteen” was, and supposed my best alternative was to list the things that felt like named numbers to me, and things which didn’t, and see if there was a plausible dividing line. (Comparisons across languages I flirted with a little bit, but realized my experience was awfully limited.) But this is a new kind of writing for me; finding the right level of familiarity as well as finding the right approach to mathematics are going to take some time.

You’ve got a good point about the use of Greek letters, and I’ll see about making them clearer in future essays. Koppa, as I remember it, was one of those letters that they kept trying to drop from the alphabet, so it went through a lot of permutations.

I think the basic HTML is allowed in comments. At least the π substitution came through, and shortly, I should learn if the em tag is allowed.

Quaternions are a little bit bizarre to start with. Octonions go far stranger, and I must admit the only use I’ve seen for them is that they’re a example of a particular kind of structure in group theory that otherwise hasn’t got anything obvious. But I hadn’t seen the Scientific American article — I’m pretty magazine-oblivious — and appreciate the headsup.

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1. I understand that the plot didn’t work this way, but I’m entertained by the notion of Ponyland’s butterflies being out to psychologically torment mammals.

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5. “neither fully humorous nor entirely studious”

That’s just what I’m reading for.

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1. I am still experimenting to find my voice for this. It’s rapidly turning less formal than lectures, which is probably fitting, but a little more formal than hanging around the department lounge complaining about the undergraduates would be.

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1. I’d read it if it had less of your entertaining voice in it, but I wouldn’t read it as eagerly. You can’t please everyone; don’t even try, or you’ll probably please no one.

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1. I don’t expect to please everyone or make a serious effort at trying. But I am experimenting and if something isn’t working for some readers, I’ll at least listen to what they feel doesn’t work. I may not change what I’m doing based on that, but listening can only hurt my ego.

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6. Geoffrey says:

On a tangent, one thing I find interesting is the shift from “5:8” to “5:08” for times shortly after the hour. It feels a little like the shift from pounds-shillings-and-pence to decimal currency, where we go from presenting a value as a combination of different units to expressing it as fractions of a single unit.

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1. I don’t think I’ve seen 5:8 as a representation for times just past the hour, but I bet now that I’m primed for it I’ll see that all over the place. You may be on to something about it representing a change from conceiving of a thing as combinations of units to a unit and subunit. We need to get an under-occupied etymologist in.

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7. The naming of numbers in Danish has an unusual characteristic in that it employs the irreducible fraction ½ (“halv”) as a means of naming odd multiples of ten from 50 to 90. For example, the modern name for 50, “halvtreds”, is a contraction of the original name “halvtredsindtyve” which translates as “half way to three (meaning.2½) times 20”. Seventy and ninety are similarly named as 3½ x 20 and 4½ x 20.
Danish also uses ½ in clock time expressions. But whereas “half nine” is taken to mean 09:30 in English, the linguistically equivalent “halv ni” in Danish means “half to nine” or 08:30. It causes lots of misunderstanding regarding meeting times!
PS Fascinating site, your educated English is a joy to read, and thanks for liking my post on Carnot’s Dilemma.

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1. I had no idea Danish used halves in that way. It’s an interesting method. To me, without actually knowing, it looks like a variation on counting by groups of twenty, the way “score” is occasionally used in English, or the way French supports “quatre-vingt” for “eighty”.

The difference in “half nine” meaning half past or half before nine also interests me. I wonder if there’s a particular reason for it, or if it’s just the luck of the draw.

And thanks kindly for your praise. I’m delighted to find someone trying to make thermodynamics more accessible. It’s a field at least as fascinating as quantum mechanics, but far more obscure apart from the popularity of “entropy” as a concept.

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1. Quite true, although if someone asked you to name a number and you offered “a nonillion” you’d get resistance the way you don’t for “nine”. (I remember in an elementary school word-and-spelling puzzle getting ruled out for offering “Yttrium” as an English word, although since I put it forth because I wanted to show off I suppose I was asking for it.)

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8. I wonder if you would have been ruled out for offering “Tungsten” as an English word? I suspect not, although it is even more Swedish than the semi-Latinised Yttrium (“tung” and “sten” being plain Swedish for “heavy” and “stone”, a reference to the density of the mineral in which it was found)

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1. In this particular instance I would have, since the rules of the game demanded a word beginning with ‘y’ then. (It was something like, each person identified and spelled a word, and the next person in turn had to provide a word that started with whatever ended the last one.) The teacher wouldn’t accept a word starting with “ytt” as possible English and wouldn’t look it up, which is, obviously, an injustice I still feel.

Still, there’s a fair question to come up about how widely used something has to be to count as “an English word”. I’d be sympathetic, at least, to the claim that some terms are just jargon, useful in a narrow context or field of study but not escaped into the general language. Particle physics, for example, offers the hypothetical “gluino”, which my spell checker rejects, but which I can find on Wikipedia, and which is probably on those Science Channel shows about pop speculative science narrated by Morgan Freeman. Maybe it’s a word; maybe not. But surely there was a point when so few people used it, or understood what another could mean by it, that it fell below the threshold of word-ness.

I would think the names of elements, at least the naturally occurring elements, should be above that threshold, but the teacher wasn’t hearing any appeals at the time.

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