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.
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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.
> 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
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...)
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’.
I suspect the author is a native French speaker? Where "quotidien" is an every-day word.
> We don’t know what impetus exists for developing a ciphered-additive system where the signs for 20 through 90 have their own distinct signs, and where there is a multiplier for 10 in place of a zero
Romantic languages have an obvious difference when reading the words representing them when in comes to numbers under ten, 11-15, and multiples of ten to 90. English’s 11-12 differs from other teens. 20-90 may perhaps be hybrids of words and numerals to eliminate confusion or seem more familiar to speakers.
It’s also obvious why a multiplier for 10 would replace zero, as that’s what it does.
Am I missing something here??
French speakers are likely to use the word fairly commonly; whereas, people with English as their first language are unlikely to know it.
So, via translation, the readability level has changed considerably.
Consider quotidian (meaning daily), this is an old word that has become more popular in past forty years and has now made its way out of academic writing into ordinary English and appeared about 35 times in HN comments in the past year. It is roughly as common in ordinary English as the word cryptography, but "cryptography" appeared about 49 times in HN comments in the past month.
Google ngram viewer gives a nice comparison of word frequencies in Google's corpus of scanned books. See [1] for a frequency comparison of the some of the words mentioned and the CS words cryptography, Turing machine, Unicode. (Sorry about the link size!)
[1] https://books.google.com/ngrams/graph?content=cryptography%2...
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.
I do wish we had decided the Yuki got it right and we would all be using octal instead of decimal.
The context of the use is pretty strong in this case, too, where "Egyptian hieratic numerals" [numerals used in Egyptian hieratic] is grammatically coordinated with "traditional Sinhalese numerals" [numerals related to Sinhala] and "Siniform numerals" [numerals similar to Chinese [numerals]].
> Place-value systems are much younger; the first date from around 500 CE
No, although place-value systems do seem to be younger than the non-place-value system the article calls "ciphered-additive", place-value numerals date from 02000 BCE: https://en.wikipedia.org/wiki/Babylonian_cuneiform_numerals#...
However, there seems to be no line of descent from Babylonian numerals, which used an empty space to represent zero, to the Hindu numerals about 2000 years later. (The oldest occurrence of zero is actually at Gwalior, somewhat later than the Brahmi script.) Similarly, the Maya vigesimal place-value numeral system dates from the first century BCE, and the decimal place-value numeral system of the khipu is normally considered to date from around the same time, although Ruth Shady has reported a tantalizing "proto-khipu" find that may predate even the Babylonian system. However, there is no plausible route of cultural transmission from these systems to Classical India.
Even the Hindu decimal place-value numeral system is likely older than 00500 CE; the Bakhshali manuscript, which uses a dot for zero like modern Eastern Arabic numerals, probably dates from about 00300 CE. https://en.wikipedia.org/wiki/Bakhshali_manuscript#Contents
> 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).
It turns out that Arabic-speakers in Africa were using a divergent set of digits for the Hindu–Arabic place-value system, not descended from ٠١٢٣٤٥٦٧٨٩, and these are entirely readable to modern European-educated eyes, except for the 4, which is rotated 90° and has a tail: https://en.wikipedia.org/wiki/Arabic_numerals#Adoption_in_Eu... https://commons.wikimedia.org/wiki/File:The_Brahmi_numeral_s...
This divergent evolution is why many of "0123456789" bear a closer resemblance to the numerals from Gwalior and in one case the Brahmi script you see in that graphic than to the Eastern Arabic "٠١٢٣٤٥٦٧٨٩". It's not that the Europeans evolved "٣" and "٦" back into the Gwalior "3" and the Brahmi "6" by coincidence; it's that the Africans they copied their numbers from were already using numerals that looked like "3" and "6".
These numerals are called "Western Arabic" because the Western Arabs in the Maghreb used them, as opposed to the Eastern Arabs in Arabia itself. It's not a juxtaposition of the names of two separate polities, "Western" and "Arabic".
I agree with the grandparent that it's terribly amusing that an anthropologist railing against "coopt[ing] Native American accomplishments by claiming them as generically American", "American imperial project[s]", and "the dominant American narrative of the primitivity of indigenous Americans" would so completely erase the African origin of what he repeatedly calls "Western numerals" in a region recently invaded and currently partly occupied by the US Air Force.
But maybe after studying different languages I'm less terrified by seeing things I don't understand. I mean, I already know I'm ignorant; why should further evidence of that be terrifying?
Is there a good browser extension for looking things like this up in Wikipedia? I recall when I was in Venezuela sometimes I found cyber-cafés that had a browser extension that would pop up a Spanish translation of any English word you moused over.
https://en.wikipedia.org/wiki/Hieratic https://en.wiktionary.org/wiki/quotidian (though Wikipedia is happy to redirect you here if you try to look up "quotidian" in Wikipedia instead) https://en.wikipedia.org/wiki/Sinhala_script (though this is sort of defined in the article) https://en.wikipedia.org/wiki/Sinhala_script#Numerals https://en.wikipedia.org/wiki/Sinhala_numerals https://en.wikipedia.org/wiki/Sri_Lanka ("For the American alternative rock band, see Sri Lanka (band).") https://en.wikipedia.org/wiki/Grapheme https://en.wikipedia.org/wiki/Elide https://en.wikipedia.org/wiki/Interlocutor https://en.wikipedia.org/wiki/Intelligentsia https://en.wikipedia.org/wiki/Digraphia (a Wikipedia search result for "biscriptal") https://en.wikipedia.org/wiki/Syllabary
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/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).
The old African numeral glyphs resemble today's European styles more closely than Caslon resembles blackletter, more closely than either resembles Palmer cursive, and more closely than any of the three resembles Roman monumental inscriptions (except, of course, for the upper case of Caslon). Yet we consider Caslon, blackletter, Palmer, and the inscriptions all to be the same Latin alphabet.
Suppose you find that an Italian professor of computer science has published a computer science paper in Italian using Knuth's cmr10 font from California. Moreover, he has previously published papers in English, some also using cmr10, in part because he wants people in California to be able to read them, and Italian is mainly treated as a language of speaking and non-serious writing — or at any rate not for computer science.
Would you therefore say, "Italy today doesn't use its own native Latin alphabet that diverged from that used in Greece; it uses the Californian system borrowed wholesale, not just local glyphs that bear a great resemblance to Californian glyphs"? To me this sounds absurd, even though the cmr10 glyphs being used were designed in California.
(Perhaps the answer in this case is too obvious because "cmr10" is an abbreviation for "Computer Modern Roman 10 point", and Rome is in Italy, but on the other hand, we are discussing a system of numerals almost universally known as "Arabic".)
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†At the time that the French adopted it, political boundaries in the Maghreb were of course very different, so at the time Fibonacci might have said his numbers were the numbers used in the Almohad Caliphate; but before the Almohads conquered the Maghreb and al-Andalus, they were the Berber rulers of Tinmel, so I don't think it's a stretch to call them Moroccan. The numerals were already in use throughout the Maghreb before the Almohad conquest, though.
This is of course correct; I should have mentioned it. (Though I was specifically talking about the numeral system 0123456789, which has no direct connection to the Babylonian numerals.)
> Even the Hindu decimal place-value numeral system is likely older than 00500 CE; the Bakhshali manuscript, which uses a dot for zero like modern Eastern Arabic numerals, probably dates from about 00300 CE.
As it happens, an earlier draft of my post did indeed say ‘300–500 CE’, but I cut it out because I figured the extra precision didn’t really add anything to the post. (Also, because no-one would care about exact dates; looks like I was wrong there!)
> It turns out that Arabic-speakers in Africa were using a divergent set of digits for the Hindu–Arabic place-value system, not descended from ٠١٢٣٤٥٦٧٨٩, and these are entirely readable to modern European-educated eyes, except for the 4, which is rotated 90° and has a tail
Interesting, I hadn’t known this — thanks for the correction!
> These numerals are called "Western Arabic" because the Western Arabs in the Maghreb used them, as opposed to the Eastern Arabs in Arabia itself. It's not a juxtaposition of the names of two separate polities, "Western" and "Arabic".
I am aware of this, and I never meant to imply otherwise.
You can always count on me to be a gigantic pedant! That was a really important time in mathematics—that's when we invented the Pulverizer Algorithm, for example.
> I am aware of this, and I never meant to imply otherwise.
I'm sorry I misread your comment!
