A curious order on the natural numbers
Consider the natural numbers up to ten, but in the following curious order:
8 5 4 9 1 7 6 10 3 2 0
What order is this? Is it random? No, I have placed these numbers in a very definite and logical order, using an underlying order idea with which I am sure that you, gentle reader, are familiar. Can you discover the underlying rule?
Surely one often gets insight to a problem by thinking about it from different perspectives, and so perhaps it may help for me to write out the numbers in a different manner, like this:
eight, five, four, nine, one, seven, six, ten, three, two, zero
OK, well, …how could that possibly help? Am I trolling? Perhaps not every new perspective is helpful.
Let me mention that the underlying order idea here applies just as much with larger numbers, and we can extend the list to the numbers up to one hundred, or one thousand, or one million. The new larger numbers will generally appear between and amongst the original numbers, however, rather than after them. Here are the numbers up to one hundred, for example, ordered according to the same underlying rule, with the original numbers up to ten appearing in the same order as they did before, highlighted in bold:
8 18 80 88 85 84 89 81 87 86 83 82 11 15 50 58 55 54 59 51 57 56 53 52 5 40 48 45 44 49 41 47 46 43 42 4 14 9 19 90 98 95 94 99 91 97 96 93 92 1 100 7 17 70 78 75 74 79 71 77 76 73 72 6 16 60 68 65 64 69 61 67 66 63 62 10 13 30 38 35 34 39 31 37 36 33 32 3 12 20 28 25 24 29 21 27 26 23 22 2 0
What is this crazy order? Can you observe carefully and discover it? Think about it before reading on…
Did you find the underlying rule? Let me provide a further hint by making once again the same trolling suggestion I had made before, namely, writing the numbers out in words. Does it help?
eight, eighteen, eighty, eighty-eight, eighty-five, eighty-four, eighty-nine, eighty-one, eighty-seven, eighty-six, eighty-three, eighty-two, eleven, fifteen, fifty, fifty-eight, fifty-five, fifty-four, fifty-nine, fifty-one, fifty-seven, fifty-six, fifty-three, fifty-two, five, forty, forty-eight, forty-five, forty-four, forty-nine, forty-one, forty-seven, forty-six, forty-three, forty-two, four, fourteen, nine, nineteen, ninety, ninety-eight, ninety-five, ninety-four, ninety-nine, ninety-one, ninety-seven, ninety-six, ninety-three, ninety-two, one, one hundred, seven, seventeen, seventy, seventy-eight, seventy-five, seventy-four, seventy-nine, seventy-one, seventy-seven, seventy-six, seventy-three, seventy-two, six, sixteen, sixty, sixty-eight, sixty-five, sixty-four, sixty-nine, sixty-one, sixty-seven, sixty-six, sixty-three, sixty-two, ten, thirteen, thirty, thirty-eight, thirty-five, thirty-four, thirty-nine, thirty-one, thirty-seven, thirty-six, thirty-three, thirty-two, three, twelve, twenty, twenty-eight, twenty-five, twenty-four, twenty-nine, twenty-one, twenty-seven, twenty-six, twenty-three, twenty-two, two, zero.
What is this order?
Interlude
So I wasn't trolling with the hints at all, for the list is simply in alphabetical order. Thus, eight is first, and the other various numbers whose names start with the letter e, followed by the numbers having names starting with the letter f and so on finally up to zero, which is the very last number as it is the only one starting with z.
We could just as easily alphabetize the numbers up to one thousand, or one million, one billion, and so on, and again the new larger numbers would appear interspersed between and amongst the numbers we already have. For example, the number seven hundred two would appear between seven and seventeen, using the usual dictionary custom that extensions of a short word appear after it, so that zookeeper appears alphabetically after zoo.
The Book of Numbers
Let us consider in our imagination the Book of Numbers, the number dictionary consisting of ALL the natural numbers, placed into their alphabetical order. The ever larger numbers will appear eternally between and amongst the smaller numbers already considered.
Question. What is the nature and order-type of the set of natural numbers when placed into alphabetical order? What is the nature of the Book of Numbers?
For example, will this order be discrete, with each number having a next number and a previous number on the list? Or will it be densely ordered like the rational numbers, with any two numbers having another between them? Will the book partition naturally into chapters?
The order type will not be the same as the usual order on the set of natural numbers ℕ, of course, since there will be infinitely many numbers starting with the letter f, as in forty, four hundred, four million, four thousand, and so on, and these will all be alphabetically preceding two hundred, two million, two thousand and so on. In short, some pages in the Book of Numbers will have infinitely many preceding pages.
Clearly the nature of the order will depend on syntactic features of the naming convention. But for that matter, what actually is the naming convention for extremely large numbers?
Number nomenclature
In order to put all the numbers in alphabetical order, we shall need to know their names. But does every natural number have a definite name? What are the names of the extremely large numbers?
Of course we are all familiar with the standard nomenclature for the smallish natural numbers, such as
one, two, three, …, one hundred, one hundred one, …, fifteen thousand two hundred forty-nine, …
And we all know in principle that the number names continue much higher, using such names as:
one thousand, one million, one billion, one trillion.
But what happens after that? Perhaps some of you are familiar with the number googol and perhaps googol plex, which we shall discuss in a later chapter. But this would be skipping a lot of numbers between one trillion and googol. What are their names?
Question. Does every natural number have a determinate name?
The standard American naming system, the so-called short scale, has terms for naming various milestone powers of ten, as shown in the table.
And so on—I have skipped many entries. The short scale naming system is based on every third power of ten 103n+3, in each case taking the Latin roots for units, tens, and hundreds place in n and appending the suffix —illion.
The alternative long-scale system, in contrast, is based on every sixth power 106n, but confusingly uses the very same names for these different values! The long scale was traditionally used in Britain and British-affiliated territories, but much less so in recent decades, as the short scale is becoming dominant in most English-speaking countries. Much of continental Europe and South America, meanwhile, uses a form of the long scale. So there is a clash in terminology.
Same number different name, same name different number
How absurd and confusing that large numbers have had fundamentally different names in different countries, even in different English speaking countries! It is frankly an intolerable mess. The systems agree on the meaning of one million, which is 106, but they disagree on one billion, which on the short scale is a thousand millions (109), but on the long scale it is a million millions (1012), a thousand times larger. The divergence grows with larger numbers---a trillion on the short scale is a thousand billions (1012), but on the long scale it is a million (long scale) billions (1018), a million times larger. For names above one million, the same number name generally refers on the long scale to a far larger number than on the short scale. The term milliard refers to one billion on the short scale, but is now rarely used, except that UK financiers do use it, sometimes shortened to a 'yard, to refer to what on the short scale would be a billion pounds. A tidy sum.
Sometimes this difference in number names has caused confusion in trans-Atlantic communication, such as with international economic matters or astronomical and other scientific data. On the short scale, for example, you have about 86 billion neurons in your brain, whereas this same number would correctly be described on the long scale as a small fraction of a billion. How embarrassing it would be to be misunderstood about this, say, if you should happen to be discussing the matter on a first date.
Regrettably, since they both consider values of n only up to one thousand, both the short scale and long scale eventually run out of names. In this sense, neither nomenclature is fit for our purpose, since they don't provide names for extremely large numbers.
Meanwhile, one can of course extend the number naming systems in various sensible manners. One such extension was proposed by Conway and Guy (2012), continuing the Latinized root and —illion suffixes, and this system in principle provides a determinate name for every natural number, thereby providing an answer to the question above.
For very large numbers, however, the system calls for one to repeat the primitive Latinized roots many times, as in sesexagintasescentilli-sesexagintasescentilli-sesexagintasescentillion. Worse, in my view, is that this absurd iterative naming might be required many times within a single number name. The system is principally designed to provide names for the milestone powers of ten, but when a number has many nonzero digits, the corresponding absurd milestone names are needed for every block of three digits (or six on the long scale) with a nonzero digit, and so even a moderately sized large number, with say a several hundred nonzero digits, could have a name involving hundreds of different absurdly iterated Latinized root names. For this reason, I find the system unwieldy. A simpler number naming system is needed.
Question. Is there a simple, sensible, efficient naming system that provides a canonical name for every natural number?
The digit-pronunciation naming system
Let me propose a simple possible answer, what I call the digit-pronunciation naming system, by which one simply pronounces the decimal digits of a number in order, so that 7216 is pronounced seven-two-one-six and so on for any number. Thus, we obtain a naming system of the numbers, and while it does not extend the standard Latinized nomenclature, nevertheless I find it to be perfectly sensible and efficient, while also providing a definite unique name for every natural number. The digit-pronunciation system is efficient mainly because it entirely avoids the need for the Latinized roots to name the milestone powers of ten, as these powers are determined simply by their position in the name, just as the decimal notation system itself is a place value system. The digit-pronunciation system is also easily modified for any base.
Let me add, furthermore, that the digit-pronunciation naming system is already in common use for very large numbers, such as reciting a phone number or reading off the number on a credit card, and it is also commonly used to help disambiguate small numbers, such as 50 and 15. So I find it to be a reasonable naming system, efficient, already in use, and providing a definite name for any given number.
The Book of Numbers
So let us finally place the natural numbers in alphabetical order. We shall form the Book of Numbers, the number dictionary, listing all the natural numbers in alphabetical order according to the digit-pronunciation naming system. The ten digits themselves appear in alphabetical order like this:
eight, five, four, nine, one, seven, six, three, two, zero
Larger numbers are placed into alphabetical order by comparing the first difference in letters in the iterated digit-pronunciation names, combined with the standard proviso that extensions of a given word appear after it in the dictionary order.
Can you put the numbers 40, 87, 88, 873, 882 in alphabetical order, using the digit-pronunciation naming system?
Interlude
The alphabetical order would be 88, 882, 87, 873, 40, in light of the digit-pronunciation names of these five numbers, which are ordered alphabetically like this:
eight-eight, eight-eight-two, eight-seven, eight-seven-three, four-zero
Notice that the ten fundamental digit names are prefix-free—none of them is an initial segment of another. Thus, when comparing the names of two numbers, we will never be in a situation where part of one digit is combined with part of another in order to make the alphabetical comparison. Rather, the alphabetical order is determined completely by comparing the digits themselves, according to the alphabetical order on the ten fundamental digits.
Let us begin to understand the nature of the alphabetical order on the numbers by making several elementary observations.
The last number
Although there is no final, largest number in the usual order on the natural numbers, nevertheless in the alphabetical order there actually is a last number. Namely, the number zero appears last in the alphabetical order, the final entry on the very last page of the Book of Numbers, because no other number starts with the letter z, and so this number will appear as the very last entry alphabetically.
The first number
Next, observe also that there is a first number alphabetically---the number 8 is first, since it begins with the letter e, and the only other numbers beginning with e also begin with 8, followed possibly by additional digits, and thus will appear after the single-digit 8.
The first chapter
For a similar reason, the next number after 8 alphabetically is 88 and then 888 and 8888 and so on. There is no number alphabetically between any of these numbers, since eight is the very first number that could be added to the end alphabetically. Thus, the beginning of the alphabetical order on the natural numbers, what might be seen as the first chapter of the Book of Numbers, looks like this:
8 88 888 8888 88888 888888 · · ·
Successors
Pressing this idea a bit harder, what I claim is that every number (except 0) has an alphabetical successor, which is obtained simply by adding a digit 8 at the end of the decimal representation of the number.
For example, the next number alphabetically after 532 is 5328, because any other digit sequence alphabetically after the first number must either extend it or deviate from one of those digits. But 5328 will be below any other higher deviation or extension, and so it is a successor. Similarly, 53288 and 532888 are the next few numbers, simply adding more 8s at the end. In this way, locally, there is a chapter beginning with 532 and its iterated-8 successors:
532 5328 53288 532888 5328888 ⋯
Every number except 0 in the alphabetical order has a successor, and a next successor, and a next successor after that, making a little copy of the natural numbers proceeding forward alphabetically just from that number. Locally, one simply tacks on additional 8s to move to later numbers alphabetically.
Predecessors
What comes just before 532? Is there a predecessor number? Well, the digit immediately before 2 is 3 (remember three is alphabetically before two), and so one natural candidate for the predecessor might be 533, which is certainly earlier than 532 alphabetically. But 533 is not the predecessor of 532, since 5338 and 53388 and even 53345634738 are all after 533, but before 532 alphabetically. Indeed, what I claim is that there is no number at all that is an immediate predecessor of 532, since it would have to start with 533⋯, but then tacking on another 8 would make an alphabetical increase, but still below 532. So there is no immediate predecessor.
The same argument works with any number that does not end with digit 8. If a number does not end with 8, then for any number preceding it alphabetically, we can tack an extra 8 on the end, and this would be between them. If in contrast a number does end with 8, then it will have a predecessor simply by removing the 8.
Thus we see that every number (except zero) is sitting in its own little chain of successors, a chapter initiated by a number not ending with digit 8, followed by the iterated-8 successor sequence. For example, the entire chapter starting with 532 appears before the chapter starting with 5325, like this:
Density of the chapters
Let us now become a bit more sophisticated. I should like to ask whether there is anything between these two chapters?
Yes, indeed, there is—there is another totally different and entire iterated-8 successor sequence strictly between these. Simply consider the sequence starting with 53287, for example, which is after 532, but before 5325. And since this number does not end in 8, its entire iterated-8s sequence will also appear in the gap, like this:
And indeed, there are entire further chapters between any two of these, and so on infinitely. For example, the sequence beginning with 53284 is after the 532 chapter and before the 53287 chapter. What I claim is that between any two iterated successor sequence chapter there is another such chapter. Thus, the chapters of the Book of Numbers are themselves densely ordered, which means that strictly between any two of them we can find another.
No last chapter before zero
There is no last chapter before the number zero, because to any number alphabetically before zero we can append a 7, or 5, 4 or what have you, which will be later alphabetically, but since it doesn't end in 8, it starts a new chapter, the beginning of its own iterated-8s successor sequence, which will lie entirely before zero. So there is no last chapter before zero.
The overall order type
The analysis we have given so far is enough to conclude the overall order type of the Book of Numbers. Namely, we have divided the book into the iterated-8 successor sequence chapters, each starting with a number whose decimal representation does not end with 8, except for the very first chapter, which consists of 8 88 888 ⋯ and so on. After that first chapter, the chapters continue in a densely ordered manner, with no final chapter before the single final number zero appears on the last page of the book.
From this information, it follows that the order type of the Book of Numbers is exactly
I recognize that this notation may not be familiar to all my readers, and so let me try to explain a little about it. The symbol ℚ refers to the order of the rational numbers, which is a countable dense linear order, and 1+ℚ is such a dense linear order with an initial point. This is the order type of the chapters, since we have a first chapter, but then after that first chapter they are ordered as a dense linear order, just like the rational numbers. (It is a remarkable theorem of Georg Cantor, which I won't get into here, that all such orders built from the natural numbers are realized as a copy of the rational order like this.) So, we have 1+ℚ many chapters, but each chapter consists of an iterated-8s successor sequence, which is ordered just like the natural numbers. The product ℕ·(1+ℚ) refers to 1+ℚ many copies of ℕ, and this is exactly what the book consists of, plus the number zero by itself at the very end, and this explains the +1 in the order type expression. So the order type of the Book of Numbers is precisely ℕ·(1+ℚ)+1.
Questions for further thought
What comes next: O T T F F S S E ?
Carry out the analysis of the nature of the order type of the natural numbers, but using the digit-pronunciation system in another base, say, base sixteen (octal), or base eight, or base two, known as binary. Does using a different base affect the features of the overall order? Of course, in a different base, the numbers will be in a different order, but how much change occurs?
Explore the nature of the natural numbers in alphabetical order according to the digit pronunciation system, but using a language other than English. Is the order type the same? How much of the analysis given here carries over to your other language? Is there a language where there is no final number?
Consider the natural numbers ordered alphabetically according to their usual
representation written in base ten (or another base), that is, using the numeral
digits as the lexical sorting characters, rather than writing out the names of those
digits in English. So 123 would precede 123254 alphabetically, which would
precede 13, and so on. Does every number have an immediate successor? Is
there a least number? Can you say exactly what is the order type?
Write an essay explaining the number nomenclature system of Conway and Guy and explaining how and why, in particular, it provides a definite number name for every natural number.
Write an essay explaining the number naming system proposed by John Gowers in the further reading section, and compare the advantages over the Conway-Guy system.
What can you say about the alphabetical order of the real numbers under the digit-pronunciation naming system, whereby π would be pronounced three point one four one five nine and so on? Is this a dense order? Is it complete?
Further reading
Conway and Guy, Book of Numbers, 2012, Springer. A romp through fascinating number ideas and issues, led by some leading mathematical minds. (Kindly note that despite the titles being the same, this suggested reference is not the imaginary Book of Numbers that I consider in the chapter.)
John Gowers, How big is a trillion anyway? A proposal for a more efficient naming system than the Conway-Guy system.
Credits
This chapter is based on ideas I had presented on Math.StackExchange at https://math.stackexchange.com/q/3047540/413, which grew out of a conversation I had had over beers with Stefan Hoffelner and Stefan Mesken following a talk I gave at the Logic Oberseminar at the University of Münster in 2019. The list of numbers up to one hundred in alphabetical order is from Hervé Graumann 1988 (http://www.graumann.net/echo/graumann/aaa_pageshtml/dladnE.html), although I have used “one hundred” rather than plain “hundred.” The short-scale names were obtained on Wikipedia. Question 4 was suggested by Nikita Danilov.
A natural generalization of this question (more specifically, your further question 4) is this: let (P, <) be any poset. We can define a new poset Dict(P) whose elements are finite lists of elements of P, with similar dictionary relations (i.e. (x_n) < (y_n) if some initial segment of (x_n) is less than (in particular, comparable to) some initial segment of (y)). (This construction is functorial, too! Do with that what you will.)
Whenever P is a total order, so is Dict(P), and I believe that's an if and only if. If P is a discrete poset (i.e. no elements are comparable), then Dict(P) is an n-ary tree, where n = |P|. I can't seem to figure out how to describe Dict(P) in the next most basic non-linear example: when P = {a, b, c} with a < b and a < c, no other relations. Do you have any sense of what it looks like?
Further questions then are (a) which posets can appear as dictionaries of other posets and (b) do dictionaries of posets actually show up "in nature"?
Sorry, I have to comment before reading further:
"For example, the number seven hundred two would appear between seven and seventeen, using the usual dictionary custom that extensions of a short word appear after it, so that zookeeper appears alphabetically after zoo."
HTML, and various other things, would allow you to stack extensions of a short word with that word, instead of in front of or behind it. You'd end up with a list of bifurcating sets (e.g. {seven,{seventeen, seventeenhundred},{{seventy, {seventyone,seventyonethousand},seventythousand}}). It reminds me of multiple displacement amplification from biology ( https://en.wikipedia.org/wiki/Multiple_displacement_amplification )