8 ms·

Cherokee Numerals

91 points by horseradish 6y ago

exporectomy 6y ago
Sounds like no loss. Invented and used by one person. We surely all invent and use our own personal ad-hoc notations for things when we don't happen to already know a conventional one.
- marksbrown 6y ago
  Radix Positional notation is just one way to represent numbers. A way that is better for arithmetic than arbitrary numerals certainly. However it's important to remember that alternatives give us a comparison. A way of seeing implicit detail. Who knows, maybe someone will invent a better notation one day, and those future humans will look back and wonder why we didn't see the obvious solution!
  - kragen 6y ago
    If you really want a way that is better for everyday arithmetic, you're looking for residue number systems, like Maya tzolk'in–haab dates (if you disregard the wayeb', anyway) or Fliess's cocaine-fueled invention of biorhythms. For example, a five-digit RNS consisting of remainders from division by 5, 7, 8, 9, and 11 can uniquely represent the natural numbers up to what we call 27720. You count as follows: 00000, 11111, 22222, 33333, 44444, 55550, 66661, 77702, 88013, 90124, X1230, 02341, 13452, and so on. Each digit proceeds to its successor independent of all the other digits, but wraps around at a different number. Similarly, when adding, subtracting, or multiplying, each digit is added, subtracted, or multiplied independently, modulo the base for its place; so 74702 + 17301 = 82203, because 7 + 1 = 8, 4 + 7 = 2 (mod 9), 7 + 3 = 2 (mod 8), 0 + 0 = 0, and 2 + 1 = 3. There's nothing to carry. (In our conventional notation, we would say 9247 + 4291 = 13538.) The computational advantage is even bigger for multiplication, although you need five different multiplication tables if you're going to memorize them, but for example X7313 × 38030 = 82030, where each digit is just the product of the corresponding digits. It's easy to overlook the enormous performance advantage this implies for larger numbers. Multiplying two ten-digit numbers in place-value notation with the standard algorithm requires computing 100 two-digit products and then adding them up with 99 two-digit additions, a total of some 298 digit operations. Even with Karatsuba multiplication, it's only a little faster, though the advantage grows for larger numbers. But with an RNS, it only requires ten digit-to-digit multiplications, so you can multiply such pairs of numbers 30 times as fast, and moreover the RNS multiplication is embarrassingly parallel. As you can imagine, RNSs get substantial use in extremely recondite DSP applications. Exact division, requiring divisibility, can be done with per-digit multiplicative inverses. For example, suppose you want to divide 46451 (which we would call 3876) by 48313 (which we would call 323). The first digit is of course 1, because 1 × 4 = 4. To compute 6 ÷ 8 in the 9s place, we need to use the multiplicative inverse of 8 in ℤ/9ℤ, which turns out to be 8 itself (8 × 8 % 9 = 1), so we multiply 6 × 8, most easily done with duplation and mediation (without a multiplication table) as 6 + 6 = 3, 3 + 3 = 6, 6 + 6 = 3. So our quotient is 13.... For the third digit, 4 ÷ 3, we again get lucky and the multiplicative inverse of 3 in ℤ/8ℤ is just 3, so it's 4 × 3: 4 + 4 = 0, 0 + 4 = 4. So it's 434... The fourth digit is 5 ÷ 1 = 5, since 1 is always its own multiplicative inverse. And the final digit is 1 ÷ 3; the multiplicative inverse of 3 mod 5 is 2, so that's 1 × 2 = 2. So our division result is 13452, which as you see above we conventionally call 12, which is correct. Digit division by zero is less problematic than it sounds; 0 ÷ 0 = 0, and ∀x≠0: x ÷ 0 = ⊥, that is, fails to exist. That's because if your dividend has a 0 where the divisor doesn't, the dividend is not a multiple of the divisor at all — it's an exact multiple of one of the bases, and the divisor isn't. Exact square roots are also relatively trivial, though they may involve many possibilities: √91141 = 81354 in the usual sense (√361 = 19), but other possibilities include 81351, 81321, 81554, 81154, and so on. The first of these corresponds to the fact that 16'651 (in our usual notation!) squares to 277'255'801, which is 10'002 × 27'720 + 361. In base 10, it's easy to see if a number is divisible by 2 or 5, and slightly harder to tell if it's divisible by 9 or 11, or any of these multiplied by some power of 10, such as 1100 or 50; but beyond that you end up doing a fair bit of work. In an RNS, it's easy to see if a number is divisible by any factor of any of the bases, in this case 2, 3, 4, 5, 7, 8, 9, or 11, or any product of factors of two or more bases, which in this case is additionally 6, 10, 11, 12, 14, 15, 18, 20, 21, 22, 24, 28, 30, 33, 35, 36, 40, 42, 44, 45, 55, 56, 60, 63, 66, 70, 72, 77, 84, 88, 90, 99, 105, 110, 120, 126, 132, 140, 154, 165, 168, 180, 198, 210, 220, 231, 252, 264, 280, 308, 315, 330, 360, 385, 396, 420, 440, 462, 495, 504, 616, 630, 660, 693, 770, 792, 840, 924, 990, 1155, 1260, 1320, 1386, 1540, 1848, 1980, 2310, 2520, 2772, 3080, 3465, 3960, 4620, 5544, 6930, 9240, 13860 (in our usual notation!), and of course 27720. There are some serious problems. Without some kind of huffman coding, it's a little less efficient than a place-value system because of the need for varying numbers of symbols for the different places, although this problem becomes less serious with larger bases like 255 and 256. At some point—sooner if you're using large bases—you start needing a lot of distinct digits, though the Maya handled the analogous problem by writing each base-20 digit of the Long Count in base 5, and the Babylonians handled it by writing their base-60 digits in base 10. And needing separate multiplication tables and reciprocal tables for each digit is confusing. Worse, though, it's hard to tell which of two numbers is larger than the other. The easiest way is to use the Chinese remainder theorem to convert them to a more conventional notation, which you can then compare lexicographically. In 01999, Brönnimann, Emiris, Pan, and Pion published an open-access article on "Sign determination in residue number systems" https://www.sciencedirect.com/science/article/pii/S0304397598001017/pdf?md5=2c2d432b5a2d6cce46b3c51d12e5cb3f&pid=1-s2.0-S0304397598001017-main.pdf https://www.sciencedirect.com/science/article/pii/S030439759... which claims to have a better algorithm. Unfortunately I do not understand their paper. And that makes it hard to compute everyday things like ⌊x÷3⌋. If you were such a nutcase that you actually wanted to use an RNS for human computation, you'd probably be better off with a very small number of bases, like two (as in Fliess's original delusions about biorhythms and in tzolk'in dates) or three (as in some modern biorhythm scams).
JdeBP 6y ago
It's also interesting how history has taken us to the point that an anthropologist names a numeral system invented in India and brought to Europe via Africa "Western", because readers will understand that name, and uses that name in contrast to a system that was invented in the actual West. Also, not enough is made of the fact that the numeral system was based upon United States decimal coinage. Viewed in terms of 1 cent, 5 cent, and 10 cent coins, with the added notion of a barred gate for 1 to 5 (as explained), the numerals actually make a lot of sense; especially if one considers them as erosions into arcs and squiggles caused by rapidly drawing the original full circles and bars. * 1 to 5 are the five-barred gate with the downward bars mostly elided or reduced to squiggles for speed and the pen not removed from the paper in between bars. * 6 to 10 are a 5 cent piece reduced to an arc plus one to five 1 cent pieces as bars, again mostly elided. * 11 to 15 are a 10 cent piece still mostly a circle plus one to five 1 cent pieces. * The 30 to 90 symbols are the superscript notation described in Lowery's biography, but prefixed instead of suffixed. 50, for example, is a prefixed 5, as reduced to a squiggle, before a 10 cent piece. The text states that numerals above 20 had the numerals from 1 to 9 appended. By implication, therefore: 16 to 25 are a 10 cent piece, plus a 5 cent piece, plus one or two barred gates of 1 cent pieces; and 26 to 29 are a 10 cent piece, plus two 5 cent pieces, plus some 1 cent pieces. It would appear that Sequoyah didn't encounter 25 cent pieces often enough to warrant retaining them in the system as it evolved.
- deleted 6y ago
  [deleted]
- bradrn 6y ago
  > It's also interesting how history has taken us to the point that an anthropologist names a numeral system invented in India and brought to Europe via Africa "Western", because readers will understand that name, and uses that name in contrast to a system that was invented in the actual West. It is true that place value systems originated in India, but I would personally say that calling our numeral system ‘Indian’ would be no less incorrect than calling it ‘Western’. The history is more complicated (and more interesting!) than such names would imply: • The shapes of our modern numerals originated in the Brāhmi script of ancient India, around 300 BCE (𑁦𑁧𑁨𑁩𑁪𑁫𑁬𑁭𑁮𑁯). • Place-value systems are much younger; the first date from around 500 CE. By this time the original shapes of the numerals had already been heavily altered, becoming more similar to the modern Hindu numerals (०१२३४५६७८९). • These systems were then transmitted to Persia, where they evolved into the numeral system used in Persian (۰۱۲۳۴۵۶۷۸۹) and thence Arabic (٠١٢٣٤٥٦٧٨٩). • It was only at this point that the Arabic numerals were transmitted to Europe primarily via Fibonacci; over a few centuries they evolved into their modern forms (0123456789). Wikipedia calls this numeral set ‘Western Arabic’, and I do think this is a more correct name than either ‘Western’ or ‘Indian’.
mod 6y ago
Wow, this is dense. > From the Egyptian hieratic numerals used in almost all the quotidian tasks of the Egyptian state, to the traditional Sinhalese numerals of south India and Sri Lanka, or the Siniform numerals developed for the Jurchin script in 12th-century China, ciphered-additive numeration is cross-culturally recurrent. In that sentence, the article hasn't defined hieratic or quotidian, and the whole sentence is terrifying. I'm fine with looking up definitions, but as a native speaker who's pretty well-read, I find this text really hard to read without a dictionary. There were many words undefined (even by context) in the text that I really don't think even most above-average readers know: syllabary, biscriptal, intelligentsia, interlocutor, grapheme, elided. Made me think of a recent PG essay: http://paulgraham.com/simply.html http://paulgraham.com/simply.html
- jomar 6y ago
  There's a very simple trick: for an overview reading, it doesn't matter what those words mean. Just skip over them: From the Egyptian some-class-of numerals used in almost all the some-kind-of tasks of the Egyptian state, etc. (Also, welcome to reading a text for which you are perhaps not the target audience. Now think about all the computing jargon that we tend to just take for granted in our own writing...)
  - todd8 6y ago
    I had a roommate in college that always looked up every word he didn't know in the dictionary. I adopted this method of improving my vocabulary and recommend it to people that wish to have a broader vocabulary. It is especially easy to use the MacOS dictionary feature by simply right clicking on any word to bring up a context menu to lookup the word.
- statstutor 6y ago
  > In that sentence, the article hasn't defined hieratic or quotidian I suspect the author is a native French speaker? Where "quotidien" is an every-day word.
  - JumpCrisscross 6y ago
pndy 6y ago
Cherokee syllabary [1] is interesting itself - it incorporates variety of characters that look like Latin and other Old World scripts letters but obviously have a different meaning. [1] - https://en.wikipedia.org/wiki/Cherokee_syllabary https://en.wikipedia.org/wiki/Cherokee_syllabary
- protomyth 6y ago
  The other native languages got varying treatments. Dakota basically got the English alphabet with some accented characters added. It mattered quite a lot if a Native was doing the work or some missionaries. Which sometimes might be indicated if things are "Spirit" or "Devil" translated. I do wish we had decided the Yuki got it right and we would all be using octal instead of decimal.
quercusa 6y ago
It's always amazed me that a working writing system was developed by someone who could not himself read!
- Octoth0rpe 6y ago
  If you think about it, every original, isolated writing system (which may only be a handful) was invented by someone who couldn't read :P

91 points by horseradish 6y ago

Sounds like no loss. Invented and used by one person. We surely all invent and use our own personal ad-hoc notations for things when we don't happen to already know a conventional one.

marksbrown 6y ago
Radix Positional notation is just one way to represent numbers. A way that is better for arithmetic than arbitrary numerals certainly. However it's important to remember that alternatives give us a comparison. A way of seeing implicit detail. Who knows, maybe someone will invent a better notation one day, and those future humans will look back and wonder why we didn't see the obvious solution!
- kragen 6y ago
  If you really want a way that is better for everyday arithmetic, you're looking for residue number systems, like Maya tzolk'in–haab dates (if you disregard the wayeb', anyway) or Fliess's cocaine-fueled invention of biorhythms. For example, a five-digit RNS consisting of remainders from division by 5, 7, 8, 9, and 11 can uniquely represent the natural numbers up to what we call 27720. You count as follows: 00000, 11111, 22222, 33333, 44444, 55550, 66661, 77702, 88013, 90124, X1230, 02341, 13452, and so on. Each digit proceeds to its successor independent of all the other digits, but wraps around at a different number. Similarly, when adding, subtracting, or multiplying, each digit is added, subtracted, or multiplied independently, modulo the base for its place; so 74702 + 17301 = 82203, because 7 + 1 = 8, 4 + 7 = 2 (mod 9), 7 + 3 = 2 (mod 8), 0 + 0 = 0, and 2 + 1 = 3. There's nothing to carry. (In our conventional notation, we would say 9247 + 4291 = 13538.) The computational advantage is even bigger for multiplication, although you need five different multiplication tables if you're going to memorize them, but for example X7313 × 38030 = 82030, where each digit is just the product of the corresponding digits. It's easy to overlook the enormous performance advantage this implies for larger numbers. Multiplying two ten-digit numbers in place-value notation with the standard algorithm requires computing 100 two-digit products and then adding them up with 99 two-digit additions, a total of some 298 digit operations. Even with Karatsuba multiplication, it's only a little faster, though the advantage grows for larger numbers. But with an RNS, it only requires ten digit-to-digit multiplications, so you can multiply such pairs of numbers 30 times as fast, and moreover the RNS multiplication is embarrassingly parallel. As you can imagine, RNSs get substantial use in extremely recondite DSP applications. Exact division, requiring divisibility, can be done with per-digit multiplicative inverses. For example, suppose you want to divide 46451 (which we would call 3876) by 48313 (which we would call 323). The first digit is of course 1, because 1 × 4 = 4. To compute 6 ÷ 8 in the 9s place, we need to use the multiplicative inverse of 8 in ℤ/9ℤ, which turns out to be 8 itself (8 × 8 % 9 = 1), so we multiply 6 × 8, most easily done with duplation and mediation (without a multiplication table) as 6 + 6 = 3, 3 + 3 = 6, 6 + 6 = 3. So our quotient is 13.... For the third digit, 4 ÷ 3, we again get lucky and the multiplicative inverse of 3 in ℤ/8ℤ is just 3, so it's 4 × 3: 4 + 4 = 0, 0 + 4 = 4. So it's 434... The fourth digit is 5 ÷ 1 = 5, since 1 is always its own multiplicative inverse. And the final digit is 1 ÷ 3; the multiplicative inverse of 3 mod 5 is 2, so that's 1 × 2 = 2. So our division result is 13452, which as you see above we conventionally call 12, which is correct. Digit division by zero is less problematic than it sounds; 0 ÷ 0 = 0, and ∀x≠0: x ÷ 0 = ⊥, that is, fails to exist. That's because if your dividend has a 0 where the divisor doesn't, the dividend is not a multiple of the divisor at all — it's an exact multiple of one of the bases, and the divisor isn't. Exact square roots are also relatively trivial, though they may involve many possibilities: √91141 = 81354 in the usual sense (√361 = 19), but other possibilities include 81351, 81321, 81554, 81154, and so on. The first of these corresponds to the fact that 16'651 (in our usual notation!) squares to 277'255'801, which is 10'002 × 27'720 + 361. In base 10, it's easy to see if a number is divisible by 2 or 5, and slightly harder to tell if it's divisible by 9 or 11, or any of these multiplied by some power of 10, such as 1100 or 50; but beyond that you end up doing a fair bit of work. In an RNS, it's easy to see if a number is divisible by any factor of any of the bases, in this case 2, 3, 4, 5, 7, 8, 9, or 11, or any product of factors of two or more bases, which in this case is additionally 6, 10, 11, 12, 14, 15, 18, 20, 21, 22, 24, 28, 30, 33, 35, 36, 40, 42, 44, 45, 55, 56, 60, 63, 66, 70, 72, 77, 84, 88, 90, 99, 105, 110, 120, 126, 132, 140, 154, 165, 168, 180, 198, 210, 220, 231, 252, 264, 280, 308, 315, 330, 360, 385, 396, 420, 440, 462, 495, 504, 616, 630, 660, 693, 770, 792, 840, 924, 990, 1155, 1260, 1320, 1386, 1540, 1848, 1980, 2310, 2520, 2772, 3080, 3465, 3960, 4620, 5544, 6930, 9240, 13860 (in our usual notation!), and of course 27720. There are some serious problems. Without some kind of huffman coding, it's a little less efficient than a place-value system because of the need for varying numbers of symbols for the different places, although this problem becomes less serious with larger bases like 255 and 256. At some point—sooner if you're using large bases—you start needing a lot of distinct digits, though the Maya handled the analogous problem by writing each base-20 digit of the Long Count in base 5, and the Babylonians handled it by writing their base-60 digits in base 10. And needing separate multiplication tables and reciprocal tables for each digit is confusing. Worse, though, it's hard to tell which of two numbers is larger than the other. The easiest way is to use the Chinese remainder theorem to convert them to a more conventional notation, which you can then compare lexicographically. In 01999, Brönnimann, Emiris, Pan, and Pion published an open-access article on "Sign determination in residue number systems" https://www.sciencedirect.com/science/article/pii/S0304397598001017/pdf?md5=2c2d432b5a2d6cce46b3c51d12e5cb3f&pid=1-s2.0-S0304397598001017-main.pdf https://www.sciencedirect.com/science/article/pii/S030439759... which claims to have a better algorithm. Unfortunately I do not understand their paper. And that makes it hard to compute everyday things like ⌊x÷3⌋. If you were such a nutcase that you actually wanted to use an RNS for human computation, you'd probably be better off with a very small number of bases, like two (as in Fliess's original delusions about biorhythms and in tzolk'in dates) or three (as in some modern biorhythm scams).

JdeBP 6y ago

It's also interesting how history has taken us to the point that an anthropologist names a numeral system invented in India and brought to Europe via Africa "Western", because readers will understand that name, and uses that name in contrast to a system that was invented in the actual West. Also, not enough is made of the fact that the numeral system was based upon United States decimal coinage. Viewed in terms of 1 cent, 5 cent, and 10 cent coins, with the added notion of a barred gate for 1 to 5 (as explained), the numerals actually make a lot of sense; especially if one considers them as erosions into arcs and squiggles caused by rapidly drawing the original full circles and bars. * 1 to 5 are the five-barred gate with the downward bars mostly elided or reduced to squiggles for speed and the pen not removed from the paper in between bars. * 6 to 10 are a 5 cent piece reduced to an arc plus one to five 1 cent pieces as bars, again mostly elided. * 11 to 15 are a 10 cent piece still mostly a circle plus one to five 1 cent pieces. * The 30 to 90 symbols are the superscript notation described in Lowery's biography, but prefixed instead of suffixed. 50, for example, is a prefixed 5, as reduced to a squiggle, before a 10 cent piece. The text states that numerals above 20 had the numerals from 1 to 9 appended. By implication, therefore: 16 to 25 are a 10 cent piece, plus a 5 cent piece, plus one or two barred gates of 1 cent pieces; and 26 to 29 are a 10 cent piece, plus two 5 cent pieces, plus some 1 cent pieces. It would appear that Sequoyah didn't encounter 25 cent pieces often enough to warrant retaining them in the system as it evolved.

deleted 6y ago
[deleted]
bradrn 6y ago
> It's also interesting how history has taken us to the point that an anthropologist names a numeral system invented in India and brought to Europe via Africa "Western", because readers will understand that name, and uses that name in contrast to a system that was invented in the actual West. It is true that place value systems originated in India, but I would personally say that calling our numeral system ‘Indian’ would be no less incorrect than calling it ‘Western’. The history is more complicated (and more interesting!) than such names would imply: • The shapes of our modern numerals originated in the Brāhmi script of ancient India, around 300 BCE (𑁦𑁧𑁨𑁩𑁪𑁫𑁬𑁭𑁮𑁯). • Place-value systems are much younger; the first date from around 500 CE. By this time the original shapes of the numerals had already been heavily altered, becoming more similar to the modern Hindu numerals (०१२३४५६७८९). • These systems were then transmitted to Persia, where they evolved into the numeral system used in Persian (۰۱۲۳۴۵۶۷۸۹) and thence Arabic (٠١٢٣٤٥٦٧٨٩). • It was only at this point that the Arabic numerals were transmitted to Europe primarily via Fibonacci; over a few centuries they evolved into their modern forms (0123456789). Wikipedia calls this numeral set ‘Western Arabic’, and I do think this is a more correct name than either ‘Western’ or ‘Indian’.

mod 6y ago

Wow, this is dense. > From the Egyptian hieratic numerals used in almost all the quotidian tasks of the Egyptian state, to the traditional Sinhalese numerals of south India and Sri Lanka, or the Siniform numerals developed for the Jurchin script in 12th-century China, ciphered-additive numeration is cross-culturally recurrent. In that sentence, the article hasn't defined hieratic or quotidian, and the whole sentence is terrifying. I'm fine with looking up definitions, but as a native speaker who's pretty well-read, I find this text really hard to read without a dictionary. There were many words undefined (even by context) in the text that I really don't think even most above-average readers know: syllabary, biscriptal, intelligentsia, interlocutor, grapheme, elided. Made me think of a recent PG essay: http://paulgraham.com/simply.html http://paulgraham.com/simply.html

jomar 6y ago
There's a very simple trick: for an overview reading, it doesn't matter what those words mean. Just skip over them: From the Egyptian some-class-of numerals used in almost all the some-kind-of tasks of the Egyptian state, etc. (Also, welcome to reading a text for which you are perhaps not the target audience. Now think about all the computing jargon that we tend to just take for granted in our own writing...)
- todd8 6y ago
  I had a roommate in college that always looked up every word he didn't know in the dictionary. I adopted this method of improving my vocabulary and recommend it to people that wish to have a broader vocabulary. It is especially easy to use the MacOS dictionary feature by simply right clicking on any word to bring up a context menu to lookup the word.
statstutor 6y ago
> In that sentence, the article hasn't defined hieratic or quotidian I suspect the author is a native French speaker? Where "quotidien" is an every-day word.
- JumpCrisscross 6y ago

pndy 6y ago

Cherokee syllabary [1] is interesting itself - it incorporates variety of characters that look like Latin and other Old World scripts letters but obviously have a different meaning. [1] - https://en.wikipedia.org/wiki/Cherokee_syllabary https://en.wikipedia.org/wiki/Cherokee_syllabary

protomyth 6y ago
The other native languages got varying treatments. Dakota basically got the English alphabet with some accented characters added. It mattered quite a lot if a Native was doing the work or some missionaries. Which sometimes might be indicated if things are "Spirit" or "Devil" translated. I do wish we had decided the Yuki got it right and we would all be using octal instead of decimal.

quercusa 6y ago

It's always amazed me that a working writing system was developed by someone who could not himself read!

Octoth0rpe 6y ago
If you think about it, every original, isolated writing system (which may only be a handful) was invented by someone who couldn't read :P