The Book of Numbers - by Joel David Hamkins

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The Book of Infinity<br>The Book of Numbers<br>All the numbers, placed in a curious order

Joel David Hamkins<br>Jan 02, 2023

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A curious order on the natural numbers

Consider the natural numbers up to ten, but in the following curious order:<br>8 5 4 9 1 7 6 10 3 2 0<br>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?<br>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:<br>eight, five, four, nine, one, seven, six, ten, three, two, zero<br>OK, well, …how could that possibly help? Am I trolling? Perhaps not every new perspective is helpful.<br>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:<br>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<br>What is this crazy order? Can you observe carefully and discover it? Think about it before reading on…<br>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?<br>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 .<br>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.<br>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.<br>The Book of Numbers<br>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.<br>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?<br>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?<br>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...