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.