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Re: Do people ever make variant numerical systems for non-primitive cultures?

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  • R A Brown
    On 05/03/2013 01:44, Matthew George wrote: [snip] ... Roman numerals are certainly clunky and old fashioned, but I don t understand why you call them
    Message 1 of 18 , Mar 5, 2013
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      On 05/03/2013 01:44, Matthew George wrote:
      [snip]
      >
      > I haven't looked at all that many systems. Do people
      > ever make clunky, irrational, and old-fashioned systems
      > like the Roman numerals?

      Roman numerals are certainly clunky and old fashioned, but I
      don't understand why you call them irrational. Essentially
      they use a bi-quinary system.

      [snip]

      > I'd love to see counter-examples to the modern
      > place-value system. Can you recommend any?

      Leibniz had a bizarre system which did not use place-value.
      The consonants _b c d f g h l m n_ = 1..9 respectively. To
      these you add a vowel as a multiplier, thus:
      a = x 1
      e = x 10
      i = x 100
      o = x 1000
      u = x 10 000

      Thus 81 374 = mubodilefa

      But the syllables may be written in any order; thus 81 374
      may be written _bodifalemu_, _lemudibofa_ etc. etc. :)

      --
      Ray
      ==================================
      http://www.carolandray.plus.com
      ==================================
      "language … began with half-musical unanalysed expressions
      for individual beings and events."
      [Otto Jespersen, Progress in Language, 1895]
    • Mechthild Czapp
      Rejistanian does a mild example of that: Rejistani numbers have a kind of place value system without the zero: 1, 2, 3, 4, 5, 6, 7, 8, 9, [1](ke), [1](ke)1,
      Message 2 of 18 , Mar 5, 2013
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        Rejistanian does a mild example of that: Rejistani numbers have a kind of place value system without the zero:

        1, 2, 3, 4, 5, 6, 7, 8, 9, [1](ke), [1](ke)1, [1](ke)2, ... [1](ke)9, 2[ke], ... 9(ke)9, [1](ry), [1](ry)1, [1](ry)2, ... [1](ry)[1]ke, [1](ry)[1]ke1 etc.

        angular brackets show an optional component, round brackets indicate a sign in rejistanian, which is not in Unicode.

        On 05.03.2013, at 01:44, Matthew George wrote:

        > I was thinking about Roman numerals, and how terrible for performing
        > mathematics that whole system is. Then it occurred to me that I've never
        > encountered a conlang with a system anything like it. Toki Pona's is
        > somewhat similar, but much simpler, and is obviously related to that lang's
        > design intent. Pretty much all of the other variation I've heard about
        > involves numerical bases. But the place-value system, complete with zero,
        > is always what people seem to choose.
        >
        > I haven't looked at all that many systems. Do people ever make clunky,
        > irrational, and old-fashioned systems like the Roman numerals? Or are the
        > purposes of number systems so practical that most conlangers have no
        > interest in making such a complex (no, baroque) method for doing math?
        > Most other aspects of conlangs seem to be deliberately elaborated and
        > intricate, reflecting how much weirdness is out there and how the
        > complexity of a language isn't related to how materially-advanced its
        > society is. But historically, most peoples had very basic math skills.
        >
        > I'd love to see counter-examples to the modern place-value system. Can you
        > recommend any?
        >
        > Matt G.
      • Leonardo Castro
        ... I once affirmed that to a friend and he told me Let s check! and started trying to make basic operations with Roman numerals. To my surprise, he
        Message 3 of 18 , Mar 5, 2013
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          2013/3/4 Matthew George <matt.msg@...>:
          > I was thinking about Roman numerals, and how terrible for performing
          > mathematics that whole system is.

          I once affirmed that to a friend and he told me "Let's check!" and
          started trying to make basic operations with Roman numerals. To my
          surprise, he succeeded! He succeeded and said: "It's not that
          difficult...". I watched he performing the operations while analyzing
          developing the methods himself and it really didn't looked very
          difficult.

          I don't have time to repeat what he did now, but I challenge you do so.
        • Jörg Rhiemeier
          Hallo conlangers! Old Albic uses base-12 numerals, and the Elves developed positional arithmetics (also base-12, of course). It was developed from counting
          Message 4 of 18 , Mar 5, 2013
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            Hallo conlangers!

            Old Albic uses base-12 numerals, and the Elves developed
            positional arithmetics (also base-12, of course). It was
            developed from counting boards merchants used with columns
            for 1s, 12s, 144s etc, where one would, whenever 12 tokens
            ran up in one column, remove them and place one token in
            the next column. The shapes of the digits have not been
            determined yet, though.

            --
            ... brought to you by the Weeping Elf
            http://www.joerg-rhiemeier.de/Conlang/index.html
            "Bêsel asa Éam, a Éam atha cvanthal a cvanth atha Éamal." - SiM 1:1
          • Daniel Myers
            ... Agreed. Also, tools were developed to facilitate math with Roman numbers - the medieval counting board in particular was very successful. Having used one
            Message 5 of 18 , Mar 5, 2013
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              > -------- Original Message --------
              > From: Leonardo Castro <leolucas1980@...>
              > Date: Tue, March 05, 2013 6:52 am
              >
              > 2013/3/4 Matthew George <matt.msg@...>:
              > > I was thinking about Roman numerals, and how terrible for performing
              > > mathematics that whole system is.
              >
              > I once affirmed that to a friend and he told me "Let's check!" and
              > started trying to make basic operations with Roman numerals. To my
              > surprise, he succeeded! He succeeded and said: "It's not that
              > difficult...". I watched he performing the operations while analyzing
              > developing the methods himself and it really didn't looked very
              > difficult.
              >
              > I don't have time to repeat what he did now, but I challenge you do so.

              Agreed. Also, tools were developed to facilitate math with Roman numbers
              - the medieval counting board in particular was very successful. Having
              used one I can tell you that (with a bit of practice) arithmetic
              computation becomes almost trivial.

              There's a good paper describing them and their use here:
              http://www.amatyc.org/publications/Electronic-proceedings/2005SanDiego/Bell.pdf

              As long as a society doesn't have a need for calculus, they could get
              along just fine without a decimal number sysem.

              - Doc
            • R A Brown
              ... [snip] ... I tried this very many years ago. Certainly addition and subtraction are not exactly difficult. Multiplication was also not difficult, just
              Message 6 of 18 , Mar 5, 2013
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                On 05/03/2013 14:43, Daniel Myers wrote:
                >> -------- Original Message -------- From: Leonardo
                >> Castro Date: Tue, March 05, 2013 6:52 am
                [snip]
                >>
                >> I once affirmed that to a friend and he told me "Let's
                >> check!" and started trying to make basic operations
                >> with Roman numerals. To my surprise, he succeeded! He
                >> succeeded and said: "It's not that difficult...". I
                >> watched he performing the operations while analyzing
                >> developing the methods himself and it really didn't
                >> looked very difficult.

                I tried this very many years ago. Certainly addition and
                subtraction are not exactly difficult. Multiplication was
                also not difficult, just tedious. I could find no
                convenient way, however, of doing division, except, of
                course, by repeated subtraction. It sounds as tho your
                friend probably made a better job of division that I did ;)

                >> I don't have time to repeat what he did now, but I
                >> challenge you do so.
                >
                > Agreed. Also, tools were developed to facilitate math
                > with Roman numbers - the medieval counting board in
                > particular was very successful.

                As, indeed, was the old Roman calculus.

                > Having used one I can tell you that (with a bit of
                > practice) arithmetic computation becomes almost trivial.

                Yep.

                > There's a good paper describing them and their use here:
                > http://www.amatyc.org/publications/Electronic-proceedings/2005SanDiego/Bell.pdf
                >
                > As long as a society doesn't have a need for calculus,
                > they could get along just fine without a decimal number
                > system.

                The ancient Greeks managed famously just using the letters
                of their alphabet, including a couple of obsolete ones.

                --
                Ray
                ==================================
                http://www.carolandray.plus.com
                ==================================
                "language … began with half-musical unanalysed expressions
                for individual beings and events."
                [Otto Jespersen, Progress in Language, 1895]
              • MorphemeAddict
                ... Repeated division by 2, then adding the answers that were even, should work. (I think.) Perhaps less tedious than simple repeated subtraction. Is there any
                Message 7 of 18 , Mar 5, 2013
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                  On Tue, Mar 5, 2013 at 2:55 PM, R A Brown <ray@...> wrote:

                  > On 05/03/2013 14:43, Daniel Myers wrote:
                  >
                  >> -------- Original Message -------- From: Leonardo
                  >>> Castro Date: Tue, March 05, 2013 6:52 am
                  >>>
                  >> [snip]
                  >
                  >
                  >>> I once affirmed that to a friend and he told me "Let's
                  >>> check!" and started trying to make basic operations
                  >>> with Roman numerals. To my surprise, he succeeded! He
                  >>> succeeded and said: "It's not that difficult...". I
                  >>> watched he performing the operations while analyzing
                  >>> developing the methods himself and it really didn't
                  >>> looked very difficult.
                  >>>
                  >>
                  > I tried this very many years ago. Certainly addition and
                  > subtraction are not exactly difficult. Multiplication was
                  > also not difficult, just tedious. I could find no
                  > convenient way, however, of doing division, except, of
                  > course, by repeated subtraction.


                  Repeated division by 2, then adding the answers that were even, should
                  work. (I think.)
                  Perhaps less tedious than simple repeated subtraction.

                  Is there any equivalent to the 'law of nines' with Roman numerals?

                  stevo


                  > It sounds as tho your
                  > friend probably made a better job of division that I did ;)
                  >
                  >
                  > I don't have time to repeat what he did now, but I
                  >>> challenge you do so.
                  >>>
                  >>
                  >> Agreed. Also, tools were developed to facilitate math
                  >> with Roman numbers - the medieval counting board in
                  >> particular was very successful.
                  >>
                  >
                  > As, indeed, was the old Roman calculus.
                  >
                  >
                  > Having used one I can tell you that (with a bit of
                  >> practice) arithmetic computation becomes almost trivial.
                  >>
                  >
                  > Yep.
                  >
                  > There's a good paper describing them and their use here:
                  >> http://www.amatyc.org/**publications/Electronic-**
                  >> proceedings/2005SanDiego/Bell.**pdf<http://www.amatyc.org/publications/Electronic-proceedings/2005SanDiego/Bell.pdf>
                  >>
                  >> As long as a society doesn't have a need for calculus,
                  >> they could get along just fine without a decimal number
                  >> system.
                  >>
                  >
                  > The ancient Greeks managed famously just using the letters
                  > of their alphabet, including a couple of obsolete ones.
                  >
                  >
                  > --
                  > Ray
                  > ==============================**====
                  > http://www.carolandray.plus.**com <http://www.carolandray.plus.com>
                  > ==============================**====
                  > "language … began with half-musical unanalysed expressions
                  > for individual beings and events."
                  > [Otto Jespersen, Progress in Language, 1895]
                  >
                • Петр Кларк
                  ... The Enamyn numbering system is based on Greek numerals (which were actually Greek letters -- see more at http://en.wikipedia.org/wiki/Greek_numerals), but
                  Message 8 of 18 , Mar 5, 2013
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                    On Monday, 04 March 2013 20:44:20 you wrote:
                    > I haven't looked at all that many systems. Do people ever make clunky,
                    > irrational, and old-fashioned systems like the Roman numerals? Or are the
                    > purposes of number systems so practical that most conlangers have no
                    > interest in making such a complex (no, baroque) method for doing math?
                    > Most other aspects of conlangs seem to be deliberately elaborated and
                    > intricate, reflecting how much weirdness is out there and how the
                    > complexity of a language isn't related to how materially-advanced its
                    > society is. But historically, most peoples had very basic math skills.

                    The Enamyn numbering system is based on Greek numerals
                    (which were actually Greek letters -- see more at
                    http://en.wikipedia.org/wiki/Greek_numerals), but since the Enamyn numbering
                    system is octal (1-8, no zero), some Greek numerals were not used:

                    α = 1 ι = o 11 (d 9) ρ = o 111 (d 73)
                    β = 2 κ = o 21 (d 17) σ = o 211 (d 137)
                    γ = 3 λ = o 31 (d 25) τ = o 311 (d 201)
                    δ = 4 μ = o 41 (d 33) υ = o 411 (d 265)
                    ε = 5 ν = o 51 (d 41) φ = o 511 (d 329)
                    ϛ = 6 ξ = o 61 (d 49) χ = o 611 (d 393)
                    ζ = 7 ο = o 71 (d 57) ψ = o 711 (d 457)
                    η = 8 π = o 81 (d 65) ω = o 811 (d 521)

                    Obviously, the values after η differ from their Greek counterparts, which is
                    likely why the numbering system switched to Enamyn letters in the late fifth
                    century AD, when the written language switched from Greek letters to a newly
                    formulated native script which was a significant improvement over the Greek
                    alphabet.
                    Side note: I'm aware that this is a really *odd* way of counting, and I'm
                    wondering if it's just a little too weird. I know that there are some cultures
                    that count with base-8, but this way of counting seems pretty unique.
                    Comments?
                    Oh, and if you're interested, here's the Python code that will convert
                    from decimal to octal, Enamyn-style:

                    def ash(n):
                    n = int(n) - 1
                    if n >= 0:
                    return str(ash(n / 8)) + str((n % 8) + 1)
                    else: return ""

                    In case you're wondering, "ash" means "eight".
                    :Peter
                  • MorphemeAddict
                    What is the largest number you can write in this system, and how do you write numerals greater than that? stevo
                    Message 9 of 18 , Mar 5, 2013
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                      What is the largest number you can write in this system, and how do you
                      write numerals greater than that?

                      stevo

                      On Tue, Mar 5, 2013 at 10:50 PM, Петр Кларк <pyotr.klark@...> wrote:

                      > On Monday, 04 March 2013 20:44:20 you wrote:
                      > > I haven't looked at all that many systems. Do people ever make clunky,
                      > > irrational, and old-fashioned systems like the Roman numerals? Or are
                      > the
                      > > purposes of number systems so practical that most conlangers have no
                      > > interest in making such a complex (no, baroque) method for doing math?
                      > > Most other aspects of conlangs seem to be deliberately elaborated and
                      > > intricate, reflecting how much weirdness is out there and how the
                      > > complexity of a language isn't related to how materially-advanced its
                      > > society is. But historically, most peoples had very basic math skills.
                      >
                      > The Enamyn numbering system is based on Greek numerals
                      > (which were actually Greek letters -- see more at
                      > http://en.wikipedia.org/wiki/Greek_numerals), but since the Enamyn
                      > numbering
                      > system is octal (1-8, no zero), some Greek numerals were not used:
                      >
                      > α = 1 ι = o 11 (d 9) ρ = o 111 (d 73)
                      > β = 2 κ = o 21 (d 17) σ = o 211 (d 137)
                      > γ = 3 λ = o 31 (d 25) τ = o 311 (d 201)
                      > δ = 4 μ = o 41 (d 33) υ = o 411 (d 265)
                      > ε = 5 ν = o 51 (d 41) φ = o 511 (d 329)
                      > ϛ = 6 ξ = o 61 (d 49) χ = o 611 (d 393)
                      > ζ = 7 ο = o 71 (d 57) ψ = o 711 (d 457)
                      > η = 8 π = o 81 (d 65) ω = o 811 (d 521)
                      >
                      > Obviously, the values after η differ from their Greek counterparts, which
                      > is
                      > likely why the numbering system switched to Enamyn letters in the late
                      > fifth
                      > century AD, when the written language switched from Greek letters to a
                      > newly
                      > formulated native script which was a significant improvement over the Greek
                      > alphabet.
                      > Side note: I'm aware that this is a really *odd* way of counting,
                      > and I'm
                      > wondering if it's just a little too weird. I know that there are some
                      > cultures
                      > that count with base-8, but this way of counting seems pretty unique.
                      > Comments?
                      > Oh, and if you're interested, here's the Python code that will
                      > convert
                      > from decimal to octal, Enamyn-style:
                      >
                      > def ash(n):
                      > n = int(n) - 1
                      > if n >= 0:
                      > return str(ash(n / 8)) + str((n % 8) + 1)
                      > else: return ""
                      >
                      > In case you're wondering, "ash" means "eight".
                      > :Peter
                      >
                    • Alex Fink
                      ... I think a more germane question is this: how do you write numbers using this system if they don t end in a bunch of ones? The actual Greek system gets to
                      Message 10 of 18 , Mar 5, 2013
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                        On Tue, 5 Mar 2013 22:50:22 -0500, Петр Кларк <pyotr.klark@...> wrote:

                        >On Monday, 04 March 2013 20:44:20 you wrote:
                        >> I haven't looked at all that many systems. Do people ever make clunky,
                        >> irrational, and old-fashioned systems like the Roman numerals? Or are the
                        >> purposes of number systems so practical that most conlangers have no
                        >> interest in making such a complex (no, baroque) method for doing math?
                        >> Most other aspects of conlangs seem to be deliberately elaborated and
                        >> intricate, reflecting how much weirdness is out there and how the
                        >> complexity of a language isn't related to how materially-advanced its
                        >> society is. But historically, most peoples had very basic math skills.
                        >
                        > The Enamyn numbering system is based on Greek numerals
                        >(which were actually Greek letters -- see more at
                        >http://en.wikipedia.org/wiki/Greek_numerals), but since the Enamyn numbering
                        >system is octal (1-8, no zero), some Greek numerals were not used:
                        >
                        >α = 1 ι = o 11 (d 9) ρ = o 111 (d 73)
                        >β = 2 κ = o 21 (d 17) σ = o 211 (d 137)
                        >γ = 3 λ = o 31 (d 25) τ = o 311 (d 201)
                        >δ = 4 μ = o 41 (d 33) υ = o 411 (d 265)
                        >ε = 5 ν = o 51 (d 41) φ = o 511 (d 329)
                        >ϛ = 6 ξ = o 61 (d 49) χ = o 611 (d 393)
                        >ζ = 7 ο = o 71 (d 57) ψ = o 711 (d 457)
                        >η = 8 π = o 81 (d 65) ω = o 811 (d 521)

                        I think a more germane question is this: how do you write numbers using this system if they don't end in a bunch of ones?

                        The actual Greek system gets to make straightforward use of addition: if you wish to write 438, well, υ 400 and λ 30 and η 8 appear in the table, and you catenate them to add and get υλη 438.

                        But in your system, say I want to write the octal number 438 (= decimal 288). Do I still write υλη, even though there's no arthmetic sense in which υ means 411 and λ means 31 here? Do I decompose it as 411 + 21 + 6 and write υκϛ, even though that's not parallel to your expression 438 for it in Arabic digits?

                        Alex
                      • Logan Kearsley
                        ... Sadly, I cannot provide examples of conlangs with entirely non-place-value systems. However, Klingon uses a place-value system that s rather different from
                        Message 11 of 18 , Mar 7, 2013
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                          On 4 March 2013 18:44, Matthew George <matt.msg@...> wrote:
                          > I was thinking about Roman numerals, and how terrible for performing
                          > mathematics that whole system is. Then it occurred to me that I've never
                          > encountered a conlang with a system anything like it. Toki Pona's is
                          > somewhat similar, but much simpler, and is obviously related to that lang's
                          > design intent. Pretty much all of the other variation I've heard about
                          > involves numerical bases. But the place-value system, complete with zero,
                          > is always what people seem to choose.
                          >
                          > I haven't looked at all that many systems. Do people ever make clunky,
                          > irrational, and old-fashioned systems like the Roman numerals? Or are the
                          > purposes of number systems so practical that most conlangers have no
                          > interest in making such a complex (no, baroque) method for doing math?
                          > Most other aspects of conlangs seem to be deliberately elaborated and
                          > intricate, reflecting how much weirdness is out there and how the
                          > complexity of a language isn't related to how materially-advanced its
                          > society is. But historically, most peoples had very basic math skills.
                          >
                          > I'd love to see counter-examples to the modern place-value system. Can you
                          > recommend any?

                          Sadly, I cannot provide examples of conlangs with entirely
                          non-place-value systems. However, Klingon uses a place-value system
                          that's rather different from what most people are familiar with; it's
                          got three numerals, but the digits run 1 to 3, not 0 to 2.
                          Counting goes as follows:
                          1 - 1
                          2 - 2
                          3 - 3 (three ones)
                          11 - 4 (one three and one ones)
                          12 - 5 (one three and two ones)
                          13 - 6 (one three and 3 ones; this is the kind of weird part)
                          21 - 7 (two threes and one one)
                          etc.

                          Most representations are the same as in a regular place value system
                          with zero, except for powers of the radix, which have one less digit,
                          eliminating the zero. A mechanical transformation can be done on any
                          regular place-value system to turn it into a zero-less system like
                          this.

                          I came up with a mixed system (intending to use it for Mev Pailom, but
                          that hasn't worked out very well as of yet) with basic numerals for 1,
                          2, and 5. Counting goes

                          1
                          2
                          1 and 2
                          2 2s
                          5
                          5 and 1
                          5 and 2
                          5 with 2 and 1
                          5 with 2 2s
                          2 5s
                          ...
                          2 and 1 of 5 = 15
                          ...
                          2 2's of 5 = 20

                          I don't have any sort of extra-linguistic numeral-only notation for it
                          yet. And it gets very clumsy when trying to describe numbers much
                          larger than 80 (2 2s of 2 2s of 5). So it needs a bit work still.

                          -l.
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