Diversionary History

If there’s one thing everybody seems to agree about the history of zero, it’s that it was driven primarily by notational considerations. More specifically, zero was required to enable positional notation. The historical record reinforces this assumption, to such an extent that it becomes apparently obvious, and thus unproblematic.

For instance (grabbing what’s immediately to hand), John D Barrow’s The Book of Nothing organizes its discussion of ‘the Origin of Zero’ by relating how

… the zero sign and a positional significance when reading the value of a symbol, are features that lie at the heart of the development of efficient human counting systems.

Robert Kaplan, when discussing the retardation of Greek arithmetical notation, explains:

… the continuing lack of positional notation meant that [the Greeks] still had no symbol for zero.

As everyone ‘knows’, the Babylonians, and later the Indians, got it right: discovering or inventing a sign for zero to mark the empty place required for unambiguous positional-numerical values. Zero arose, and spread, because it allowed modular number systems to develop. Except that, conceptually, there is no basis to this story at all.

Counting is primarily practical, so that no argument counts for much besides a demonstration. In this case, demonstration is peculiarly simple, especially when it is noted that nobody seems to think it possible.

Modulus-2 is convenient, but there is nothing magical about it in this regard. A decimal demonstration, for instance, would be no more intellectually taxing, although it would be considerably more cumbersome. Any modulus works.

Start with the basics. The positions or places of a modular notational systems represent powers. If we count from zero, the number of each successive place (ascending to the left by our established convention) corresponds to the modular exponent. The zeroth power for a single digit number, the first and then zeroth power for two digits, the second, first and zeroth power for three digits, and so on.

As the accepted story goes, each place must be filled, if only by a marked nothing (zero), if the proper places, and their corresponding (modular exponential) values, are to be read. The places must indeed be filled. There is no need whatsoever for a zero sign to do this.

The demonstration, then. Our non-zero modulus-2 positional system has two signs, 1 and 2, each bearing its familiar values. The places also have their mod-2 values, counting in sixteens, eights, fours, twos, and units as they decline to the right. Here we go, counting from 1 to 31 (watch carefully for the point at which the supposedly indispensable zero sign is needed):

1, 2, 11, 12, 21, 22, 111, 112, 121, 122, 211, 212, 221, 222, 1111, 1112, 1121, 1122, 1211, 1212, 1221, 1222, 2111, 2112, 2121, 2122, 2211, 2212, 2221, 2222, 1111 …

Conclusion: the positional function of zero is wholly superfluous. The Greeks, or anybody else, could have instantiated a simple, fully-functional positional-numerical notation without any need to accommodate themselves to the trauma of zero. In regard to this matter, the history of numeracy is utterly diversionary (not just the historiography, but the substantial history — the facts).

Perhaps this won’t seem puzzling to people, but it puzzles the hell out of me.

ADDED: Mathematical lucidity on the topic from Alan Liddell. Part 2.

May 27, 2013admin 53 Comments »
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53 Responses to this entry

  • craig hickman Says:

    Don’t have an answer for you there, but…

    Adding, subtracting, and multiplying by zero are relatively simple operations. But division by zero has confused even great minds. How many times does zero go into ten? Or, how many non-existent apples go into two apples? The answer is indeterminate, but working with this concept is the key to calculus. For example, when one drives to the store, the speed of the car is never constant – stoplights, traffic jams, and different speed limits all cause the car to speed up or slow down. But how would one find the speed of the car at one particular instant? This is where zero and calculus enter the picture.

    If you wanted to know your speed at a particular instant, you would have to measure the change in speed that occurs over a set period of time. By making that set period smaller and smaller, you could reasonably estimate the speed at that instant. In effect, as you make the change in time approach zero, the ratio of the change in speed to the change in time becomes similar to some number over zero… Newton and Leibniz solved the problem: and, they created Calculus to do it. What’s fascinating is that without calculus most of our modern civilization would not exist: physics, engineering, etc. or at least not in the way it is. Even the supposed idea of progress came out of calculus. Accelerationis as well?

    fascinating …

    [Reply]

    admin Reply:

    That was closer to the topic here. I agree the calculus is the crucial next step (and evidence for the ineluctable importance of division by zero).

    [Reply]

    Posted on May 27th, 2013 at 2:38 pm Reply | Quote
  • raptros_ Says:

    maybe they just figured that it’d be better to have a single symbol for the additive identity.

    [Reply]

    admin Reply:

    What does that have to do with place value?

    [Reply]

    Posted on May 27th, 2013 at 2:43 pm Reply | Quote
  • Nick B. Steves Says:

    Unless I am profoundly missing the point here, your modulus-2 counting system (with 1 and 2) doesn’t have a way to represent NO REMAINDER, which seems a rather principal feature of any counting system. Natural numbers are either factors of others or they are not… without zero it is “impossible” to tell… or at least inconvenient. Any modulus-N counting system will have symbols for 0, …, N-1.

    [Reply]

    Alan Liddell Reply:

    Wouldn’t it be simpler to show a remainder only if it exists? So if I read you right, we would have

    11 / 2 = 1r1 and

    21 / 2 = 1

    Let me keep working on this, I see some interesting notational possibilities.

    [Reply]

    admin Reply:

    Yes please.

    [Reply]

    Alan Liddell Reply:

    One emergent property in this set, what I’m calling N_ell, seems to be that when you multiply a number which has a 1 in every place by 2, you get a 2 in every place.

    Examples:

    1 base 10 => 1: 1 * 2 = 2 => 2 base 10
    3 base 10 => 11: 11 * 2 = 22 => 6 base 10
    7 base 10 => 111: 111 * 2 = 222 => 14 base 10
    15 base 10 => 1111: 1111* 2 = 2222 => 30 base 10

    I’ve also found something in multiples of 2 where each digit is 2:

    2 * 2 = 21
    22 * 2 = 212
    222 * 2 = 2212
    2222 * 2 = 22212

    I haven’t gone further than this, but it looks promising.

    admin Reply:

    … but the only counting system with a sign for ‘no remainder’ is zero-based positional numeracy.

    [Reply]

    Nick B. Steves Reply:

    Right. So doesn’t that make it a good thing?

    Okay, so maybe this is giants standing on my shoulders…

    [Reply]

    admin Reply:

    I’m not denouncing the counting system we have, or even suggesting that a neoreactionary revolution needs to throw it out just to show that we can.

    Posted on May 27th, 2013 at 2:45 pm Reply | Quote
  • Alan Liddell Says:

    @Sorry, 21 / 2 = 2

    [Reply]

    Posted on May 27th, 2013 at 5:51 pm Reply | Quote
  • Vladimir Says:

    One big problem with your system seems to be that it doesn’t enable easy shift-and-add multiplication that requires only addition combined with memorization of small (square of base sized) multiplication tables. (The wonderful property of positional systems with zero is that multiplication with a power of the base reduces to a simple left shift.)

    Moreover, the ancient Greeks did have a kind of positional system without zero (the Ionian system). They used a different set of nine symbols for each power of 10 and didn’t mark unused powers at all. This also led to difficulties with multiplication, but probably smaller ones than your system would suffer from.

    [Reply]

    Posted on May 27th, 2013 at 6:03 pm Reply | Quote
  • admin Says:

    With the greatest respect, I think everybody is missing the point here, or at least complicating unduly (prematurely). The primary problem is not whether this counting system advances upon our inherited, zero-based version. There’s no reason at all to think that it does, nor was that suggested. (Given the structure of the two Greek counting systems, and then the Roman, that actually existed, it is quite clear that none of the functional objections noted in the acute commentary here come close to being lethal in principle,)

    The initial question is solely: Given that an extremely simple positional counting system can be constructed without any recourse to zero, how did the historical facts of the history of zero become so completely compacted with conceptual myth (i.e. the inseparability of zero from place value)? The relation of zero to place value is demonstrably synthetic, and perhaps even accidental. This, in turn, means that we have been systematically distracted, and are looking for the significance of zero in the wrong place.

    [Reply]

    Posted on May 27th, 2013 at 10:11 pm Reply | Quote
  • fotrkd Says:

    I’m a maths simpleton so there’s no danger of over complication, but as a 2 in the right hand column of your system is the equivalent of a 1 in the column immediately to the left, haven’t you simply failed to carry it over? If so, isn’t this two just hiding a zero? Take numbers one to six as you’ve got them:

    1, 2, 11, 12, 21, 22

    How do you add 2 + 11 to get 21? What rule do I follow? I’ve no idea. But if 2 is actually a mask for 10 we can set it out as follows:

    10+
    11
    =
    21

    Similarly, how do you multiply 2 * 11 (2*3) to get 22 (6)? Well:

    10*
    11
    =
    110 (=22)

    Now if after awhile the two in the right hand column is consistently seen as an inconveniently written ’10’ (where the zero is simply the ‘no-thing’ or placeholder which appears after the two is transferred), at some point your system is going to be improved to include a no-thing. Otherwise it’s impossible to do basic arithmetic in it, isn’t it? In other words even if your system had been an intermediate step, it’s still begging out for a zero to make it more useful.

    [Reply]

    admin Reply:

    “… haven’t you simply failed to carry it over? If so, isn’t this two just hiding a zero?” — that’s entirely based on a retrospective view, from what we know, surely?
    Yes, it seems strange to us that the same value is recorded in two different ways, but an abacus does the same, and in the early history of numerical symbolism that would be close to the standard of intuition.

    “In other words even if your system had been an intermediate step, it’s still begging out for a zero to make it more useful.” — and then we’d understand far more clearly what notational zero is really for.

    [Reply]

    fotrkd Reply:

    that’s entirely based on a retrospective view, from what we know, surely?

    Well now I’m picturing my sums on an abacus it is, yes. Lucky I ask the dumb questions…

    [Reply]

    admin Reply:

    Not remotely dumb, of course, but actually seriously clarifying.

    Posted on May 27th, 2013 at 11:01 pm Reply | Quote
  • Artxell Knaphni Says:

    You said: “Kant is probably the most elaborate fit of panic in the history of the world.”

    (A Canadian philosopher of mind – analytical tradition – agrees with this. She told me.)

    It could be said that the entire history of Occidental Thought, the metaphysics of substantia and presence, etc., is a similar hysteria. I think it has been remarked that the difference of the Indian and Greek traditions consisted in differing evaluations of the Infinite. The Greeks liked ‘odd’ numbers, because they represented the ‘finite’, limits, and the determinations of ‘form’. They disliked ‘even’ numbers, because they led to ‘infinity’, ‘formlessness’ and ‘chaos’. For the Greeks, the Infinite was bad.

    For the Hindu-Buddhist-Jain traditions, the Infinite was good.

    Does that bother you? Are your inflationary reminders of ‘immanence’ and its ‘horrors’ a covert ‘grounding’ strategy, not to avoid ‘idealism’, but to hide from the various vertigos of infinity? Or is it a Deleuzian/Lyotardian positivity without negation ethos? You want your objects to be substantial, can accept them as economic effects, the ‘play of forces’, but it all has to be nice and tidy, not traversed to its very essence by nothingness? A ‘nothingness’ that renders all ‘immanences’ insecure, opening up transcendence in the every citadel of finitude.

    “And for those in whom the Will has denied itself, this very real world of ours, with all its suns and Milky Ways, is — nothing.” Arthur Schopenhauer

    Notice: not a single numeral.

    [Reply]

    fotrkd Reply:

    Artxell, your posts always confuse me. Why do you think admin might be resistant to the Indian tradition? Inside the front cover of Thirst for Annihilation (which I’ve not read, nor own) is the statement “Zero is immense.” He’s exploring it continuously – and not in order to make it neat and tidy.

    I really think if you tried approaching Outside in without so much inbuilt opposition you might find you’re not as far apart as you think (on a whole range of subjects). That’s intended as encouragement not criticism btw.

    [Reply]

    Artxell Knaphni Reply:

    fotrkd, I have admin’s book, and remember that line, though I haven’t read the whole book.
    I might as well say this here. One of the things I want to do, on my blog, is to work through the colonial thing. Not just because I’m of Indian origin. It’s not that. Rather, it’s because I wish to retain perspectives that I feel have been lost, or haven’t developed. Perspectives which are important and would solve many ‘problems’. Ultimately, these problems could be said to be ‘philosophical’, as it were, but they’re not just that.
    I don’t really see ‘Modernity’ as a radically new emergence, I think admin does. I think it’s more akin to that Chinese emperor who built the Great Wall and burnt all books referring to his predecessors. I’m not saying there haven’t been good things, but there has always been a destructive shadow to it. And the reason for this is that people tend to act without truly thinking. There is a desperation to modernity that afflicts all its productions. And it need not be that way.

    The “opposition” you speak of isn’t really “inbuilt”: it’s methodological. I know the anti- Eurocentric thing has been done for decades now. But I’ve never done it, and I feel that it can be done differently, in a way that needs to be done. Because, essentially, the whole globe is in the grip of that thought, it’s no longer a European problem. One could say that Heiidegger has pointed this out to some degree, but he wasn’t able to come up with anything effective.

    If the answers were merely implementational they would have been done by now. The problems are not at the level of implementation. They are foundational. At the level of the roots of Occidental thought and practice. And this is what needs to be addressed. That’s one strand I am hoping to develop, see if it lleads to something else. It does look as if there have always been seeds of this in the Occidental tradition, they can serve as reference points, perhaps,

    Regarding admin and his work, I hope it’s obvious that both Noir Real and I have the greatest respect for both. Noir Real’s admiration is shown every time he posts a “Nick Land, Quote of the Day”. Mine goes back to his appearance in Angelaki, a Deleuze issue, “Deleuze and the Transcendental” or something, which I still haven’t read. Admin and I are of the same generation, he’s a year older, And back then, he was the only one, certainly in England, and within academia, that was actually thinking creatively. His interests were different to mine, but there was noone else really who
    had the courage and integrity to be themselves and see where that would lead, as far as thinking afresh. And, in that, he was a beacon that opened things up considerably. But the main thing is/are the flashes of brilliance he comes up with. So, his work is always going to be of interest, whatever his political itinerary. And, as I’ve said before, politics per se, isn’t really my thing. But I can use it to open up the areas I am interested in, give those areas a political facet, as it were. It’s really those areas that are important to me.

    “Why do you think admin might be resistant to the Indian tradition?”

    I don’t, really. But I’m keeping that trope in play. It seemed like the right thing to say.

    “That’s intended as encouragement not criticism btw.”

    I appreciate that, fotrkd.

    [Reply]

    admin Reply:

    I’m not try to expel zero — quite the ‘opposite’. This is a ground-clearing exercise, designed to demythologize the conceptual basis of its insertion into modern numeracy, in order to approach it accurately. The story of zero as the scullery maid of place value notation isn’t a solid foundation to build upon.

    [Reply]

    Nick B. Steves Reply:

    Well doesn’t it happen all the time that something gets invented for one thing and then people realilze it could be used for a whole lot more? AFAIK, the story of i = -1 is similar…

    Here’s to hoping <SQRT> works… given my track record here, I have my doubts…

    [Reply]

    Nick B. Steves Reply:

    Nope that should’ve read i = sqrt(-1).

    Artxell Knaphni Reply:

    I see that now.
    Just a thought, did Godel have anything to say about it?

    [Reply]

    admin Reply:

    Goedel clicks in more naturally a little further down the rabbit hole …

    Posted on May 28th, 2013 at 12:04 am Reply | Quote
  • admin Says:

    @ Alan Liddell
    From your preliminary results, the first block capture a central feature (roughly equivalent to the ‘strange’ properties of n-1 in Mod-n zero-based systems (e.g. 9 in decimal)). This notation absorbs values up to the modulus (rather than Mod-n -1), so multiplication by the modulus is preserved in numerals, rather than rolled over. (The pattern is consistent for any modulus you choose, as long as you restrict the initial digits to 1s. For e.g., non-zero mod-10 (numeral set 123456789X), multiply any number expressed solely by 1s by 10, and each 1 is replaced by an X.) It might be worth noting that these 1111… type numbers [i.e. Mersenne Numbers] are consistent between zero-based and non-zero based notations, whatever the modulus.

    What the second block indicates is rather less clear to me …

    [Reply]

    Alan Liddell Reply:

    Another property of these numbers is their clustering in pairs if you play fast and loose with the bases, which should be obvious given that it’s base 2:

    1 (+ 0) = 1
    2 (+ 0) = 2
    3 (+ 8) = 11
    4 (+ 8) = 12
    5 (+ 16) = 21
    6 (+ 16) = 22
    7 (+ 104) = 111
    8 (+ 104) = 112
    9 (+ 112) = 121
    10 (+ 112) = 122
    11 (+ 200) = 211
    12 (+ 200) = 212
    13 (+ 208) = 221
    14 (+ 208) = 222

    and so on. If there were an obvious pattern to the additive terms (and I’ve submitted this to get as many eyeballs on it as possible), we would be able to construct an explicit bijection (or at least a forward function if it’s too messy) and thence be able to derive all kinds of good properties.

    *addendum* I just noticed that the difference between 104 and 16 is the same as the difference between 200 and 112. So when the additive terms aren’t going up by 8 (i.e., from 8 to16), they’re going up by 88 (from 16 to 104). I’ll keep investigating.

    [Reply]

    Alan Liddell Reply:

    Another obvious property: the number of digits changes at 2^n -1. So at 2^1 – 1 = 1 we start out with 1 digit, at 2^2 – 1 = 3 we now get 2 digits, at 2^3 – 1 = 7 we get 3 digits and so on. This should also prove useful in nailing down this counting system, but I’ll stop polluting your comments and post about it elsewhere.

    [Reply]

    admin Reply:

    In no way is this pollution!

    Alan Liddell Reply:

    Interestingly enough, these digit-changing numbers, each one a Mersenne number, are the same numbers with the strange multiply-by-2 property.

    admin Reply:

    8 = 10 – 2 = 8
    16 = 10 + 10 – 2 – 2 = 8 + 8
    104 = 100 + 10 – 4 – 2 = 96 + 8
    112 = 100 + 10 + 10 – 4 – 2 – 2 = 96 + 8 + 8
    200 = 100 + 100 + 10 – 4 – 4 – 2 = 96 + 96 + 8
    … that’s what’s happening, isn’t it? The numbers in the final phase of the equations are differences, between decimal and binary numbers of equivalent power, which are then accumulated by the relevant digits (in this case, either 1 or 2).
    The series of differences proceeds: 0, 8, 96, 992, 9984, 99968, 999936, 9999872 …
    So we can anticipate that:
    255 (+ 0 + 8 + 96 + 992 + 9984 + 99968 + 999936 + 9999872 = 11110856) = 11111111
    and because
    1 (+ 0) = 1, that
    256 (+ 11110856) = 11111112

    [Reply]

    Alan Liddell Reply:

    Ah, I see what you mean:
    1 – 1 = 10 ^ 0 – 2^0
    10 – 2 = 10^ 1 – 2^1
    100 – 4 = 10^2 – 2^2
    1000 – 8 = 10^3 – 2^3
    1000 – 16 = 10^4 – 2^4
    10000 – 32 = 10^5 – 2^5

    Differences between decimal and binary, yes.

    This is promising.

    So if f is our bijection, then f(2^n – 1) = 2^n – 1 + sum(10^i – 2^i)_{i=1}^{n-1}. I wonder what else we can make explicit like that.

    admin Reply:

    “… these digit-changing numbers, each one a Mersenne number …”
    Yes, we had a moment of synchronous convergence on that observation.

    [Reply]

    Posted on May 28th, 2013 at 12:43 am Reply | Quote
  • craig hickman Says:

    @adminIn other words, it’s about absence and presence: zero and one were derivative of theological structures, economic structures of the Babylonian empire. The empty place was the vacated god-king – or maybe Heraclitus “Absent while present.” It’s not about math, its about our brain… this is your implication – the others seem to be literalists, wired to the surface of numbers rather than thought.

    [Reply]

    admin Reply:

    I’m not sure. My tendency is to see the theology as superficial, compared to the numeracy. The questions come first, though.

    [Reply]

    craig hickman Reply:

    Well, I got to thinking about metrology and error: Mistakes can make measurements and counts incorrect. Even if there are no mistakes, nearly all measurements are still inexact. The term ‘error’ is reserved for that inexactness, also called measurement uncertainty. The ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BC and the Babylonians derived their knowledge of these systems from the Sumerians… just a thought.

    Since most of these early systems were tax counting, gold counting etc. It may be the zero was a place holder for that uncertainty or error in the process of counting large amounts. They needed a marker, but one that was a separator, a divider, rather than a positive number. As time went on zero became that mismeasure of that inexactness in the system of numbers. Since the Babylonians and Indians had a true place-value system, where digits written in the left column represented larger values (much as in our base ten system: 734 = 7×100 + 3×10 + 4×1). The Sumerians and Babylonians were pioneers in this respect as you’ve already stated. Just a crossover in conceptual thought that brought zero into being.

    And, of course, math, number, geometry, etc. were all sacred arts to these people… and, since they used sexagesimal notation, and tables with extensive lists of these reciprocals to for division of large quantities they must have needed an empty place holder to account for certain discrepancies. One also knows they kept accurate and detailed records of the movement of the stars, sun, and moon and that the idea of an empty place holder for let’s say ‘eclipses’ the Unaccounted or Absent One that vacated the sky during these temporal fluctuations… all guesses… but makes you wonder…

    [Reply]

    admin Reply:

    “… a crossover in conceptual thought that brought zero into being.” — I’d rather say, encountered zero from a specific (practical) angle.

    Posted on May 28th, 2013 at 12:48 am Reply | Quote
  • spandrell Says:

    We might need a secret notational system to hide our bitcoins from Leviathan.

    [Reply]

    admin Reply:

    Something more like this. (I’m going to hash up a less esoteric mode of presentation.)

    [Reply]

    Posted on May 28th, 2013 at 3:04 am Reply | Quote
  • James A. Donald Says:

    Alternate history: The Babylonians developed zero and positional notation, independently, but used it only for angles between heavenly bodies, that being the only case where people made heavy use of large numbers.

    Everyone, describing the heavens, followed Babylonian convention, more or less, though at some point they drifted from base sixty to the more convenient base ten.

    Zero followed positional notation around, because both Babylonian and both used for astronomy, not because of any logical connection between the two.

    [Reply]

    admin Reply:

    None of that strikes me as controversial. Yet a logical connection has been assumed to exist between the two, to such an extent that the very idea of positional notation is generally taken to imply adoption of zero. Working the problem from the other side, it seems passing strange that classical European antiquity failed so pitifully to innovate a place value notation, given the widespread use of the abacus. The conventional narrative — pinning this failure on Greek allergy to zero and its paradoxes — clearly lacks the force ascribed to it.

    [Reply]

    craig hickman Reply:

    Haha… true and on target!

    [Reply]

    craig hickman Reply:

    Yea, I was just using it as example, not implying a connection between the two, which as you point out has no justification.

    [Reply]

    Posted on May 28th, 2013 at 4:20 am Reply | Quote
  • Chevalier de Johnstone Says:

    Perhaps I am missing something, but we seem to be operating under the assumption here that if, say, the Babylonians had a counting system that needed a placeholder, they could have just scrapped it for one that doesn’t. Isn’t it more likely that the notation system, once invented and adopted, displays first-mover technology advantages and is resistant to replacement by some other technology, even if better? Since in the Babylonian base-60 cuneiform system it is not possible to represent some numbers without a placeholder, they invented a placeholder. Yes, they could have thrown out the existing notation system and invented something new, just as we could all be typing on Dvorak keyboards right now.

    To say that the positional function of zero is historically superfluous because it’s possible to invent a notation system that doesn’t use it seems to me like saying the function of the accent mark in, say, Spanish or French is superfluous because people could just change their pronunciation to avoid needing an accent mark notation. If we adopt this version of an objective analytical approach then a whole lot of cultural relics are superfluous…but so what?

    [Reply]

    admin Reply:

    “… we seem to be operating under the assumption here that if, say, the Babylonians had a counting system that needed a placeholder, they could have just scrapped it for one that doesn’t.” — I don’t think anybody is arguing (or assuming) that.
    Rather:
    (a) Place value is conceptually independent of zero.
    (b) There is no logical reason why a simple positional notation, without zero, could not have arisen.

    [Reply]

    James A. Donald Reply:

    The Babylonians started off with an oral system where they had a word for a hundred, a word for a sixty, a word for a ten, and a word for a one, or at least that is how they did things when they were writing words in text as text, much as we would say “three hundred and seven” in text, and 307 in a numeric context.

    Where we would say one hundred and ninety two, they would say hundred, sixty, ten, ten, ten, one, one.

    Perhaps if they had had words for the numbers one to sixty, or words for the numbers one to ten, they would have wound up doing things your way, and today, we would all be doing things that way.

    So if we, having words all the way up to a score, had invented positional notation, would have wound up with your system, base twenty.

    [Reply]

    admin Reply:

    The Babylonians had inconsistent modules, starting with decimal, and then switching to 60. Without rigidly consistent modules, an efficient positional system is impossible.

    Posted on May 28th, 2013 at 4:30 pm Reply | Quote
  • northanger Says:

    Zero as Messiah.

    [Reply]

    admin Reply:

    ‘Zero Messiah’ makes the point more provocatively.

    [Reply]

    Posted on May 28th, 2013 at 11:14 pm Reply | Quote
  • Alan Liddell Says:

    @admin
    General formula (more or less) found. It’s piecewise, depending on odd or even, and has (several) other quirks, but we can definitely tease base-10 natural numbers into these “superbinaries”.

    [Reply]

    Handle Reply:

    But how do you represent 0?

    [Reply]

    Alan Liddell Reply:

    You don’t. Zero is not a counting number. Some definitions of the set N of natural numbers contain zero, some don’t — the crux is the application.

    [Reply]

    Posted on May 29th, 2013 at 12:52 am Reply | Quote

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