Kaktovik Numerals(en.wikipedia.org)
en.wikipedia.org
Kaktovik Numerals
https://en.wikipedia.org/wiki/Kaktovik_numerals
76 comments
Or perhaps engaging with this topic so practically - understanding their native language's counting system and then devising a notation - sparked an interest in what is often perceived as a useless and boring topic.
I had a few "ooh, this is not only useful, but intellectually fun" experiences with math in school. Positional number systems was one of them. Distinct aha moment. So much of what had not made sense before suddenly did, including why we had just spent weeks on logarithms. Without at least a few experiences like that, I am sure I would have been even more tuned out in class, destined to hate math for life. I found the way it was taught most of the time to be a form of low-grade torture.
I had a few "ooh, this is not only useful, but intellectually fun" experiences with math in school. Positional number systems was one of them. Distinct aha moment. So much of what had not made sense before suddenly did, including why we had just spent weeks on logarithms. Without at least a few experiences like that, I am sure I would have been even more tuned out in class, destined to hate math for life. I found the way it was taught most of the time to be a form of low-grade torture.
It's seems much more likely that the new system was easier to use because it's an iconic tally system (vertical stroke = 1, horizontal = 5) that doesn't require leaerning 9 digit symbols or place value.
Yeah, I think the wiki statement might be poorly worded, but the way I interpret it is something like:
- they learned base 20 counting verbally at home, possibly simple additions, multiplications, and such too
- they learned base 10 english counting as well, and eventually written arithmetic with Arabic numerals
- but there was a missing link in having to translate bases that made it harder for them to understand the underlying patterns of written arithmetic or even algebra
- having the new written numerals provided that missing bridge, allowing them to round out their understanding
To be clear that's just armchair speculation on my part, but that's how I interpreted it. They went from having a challenge of a missing mental connection, to having the advantage of two perspectives.
But certainly there's confounders, like perhaps the new numeral system motivated more interest and family support in math.
- they learned base 20 counting verbally at home, possibly simple additions, multiplications, and such too
- they learned base 10 english counting as well, and eventually written arithmetic with Arabic numerals
- but there was a missing link in having to translate bases that made it harder for them to understand the underlying patterns of written arithmetic or even algebra
- having the new written numerals provided that missing bridge, allowing them to round out their understanding
To be clear that's just armchair speculation on my part, but that's how I interpreted it. They went from having a challenge of a missing mental connection, to having the advantage of two perspectives.
But certainly there's confounders, like perhaps the new numeral system motivated more interest and family support in math.
What is a positional number system and how do logarithms help in learning/using it?
This reminds me of the Birkana artificial hexadecimal numerals: https://yawar.blogspot.com/2016/10/the-birkana-hexadecimal-n...
https://en.wikipedia.org/wiki/Maya_numerals
The Mayan numerals are also base 20 but like 2000 years older.
The Mayan numerals are also base 20 but like 2000 years older.
TIL Maya used a mixed radix system for calendars, using base 18 for most significant digits and base 20 for least significant digits.
The real world is messy
The real world is messy
We also use a mixed radix system for calendars, which is why we separate the digits of a date with non-numeric separators.
> The Iñupiaq language has a base-20 numeral system, as do the other Eskimo–Aleut languages of Alaska and Canada (and formerly Greenland). Arabic numerals, which were designed for a base-10 system, are inadequate for Iñupiaq and other Inuit languages. To remedy this problem, students in Kaktovik, Alaska, invented a base-20 numeral notation
Nobody suggest this to the Danes, please.
Nobody suggest this to the Danes, please.
"However, Danish numerals are not vigesimal since it is only the names of some of the tens that are etymologically formed in a vigesimal way. In contrast with e.g. French quatre-vingt-seize, the units only go from zero to nine between each ten which is a defining trait of a decimal system."
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I see similarities with Chinese https://en.wikipedia.org/wiki/Counting_rods
This character set was only added to Unicode last September, as mentioned on the page. I first learned about them via this video https://www.youtube.com/watch?v=EyS6FfczH0Q
That’s how I learned about them—I’m always curious about what’s new in each release of Unicode and how it impacts my Rust Unicode library (finl_unicode).
TIL a new way to write sheep counts (spoken system: https://en.wikipedia.org/wiki/Yan_tan_tethera )
As employed so hauntingly by Jake Thackray in "Molly Metcalfe": https://youtu.be/TiXINuf5nbI
Lovely story and song! Thanks for sharing!
Seems like it could be generalized to something more like a base 5 where the positions are rotated.
This sort of rotation-and-superposition number system also appears (in base 5 too!) in the game Riven: https://dni.fandom.com/wiki/D%27ni_Numerals
It's sort of funny how both Kaktovik and the D'ni numerals were invented around the same time (1994 for Kaktovik and 1997 for when Riven was released).
It's sort of funny how both Kaktovik and the D'ni numerals were invented around the same time (1994 for Kaktovik and 1997 for when Riven was released).
> Arabic numerals, which were designed for a base-10 system, are inadequate for Iñupiaq and other Inuit languages.
I am curious as to why Arabic numerals (or any number system, for that matter) are inadequate for a language.
I am curious as to why Arabic numerals (or any number system, for that matter) are inadequate for a language.
Because numbers written in the given number system don’t necessarily match up with the words in the given language. Take the number 475, for example. In English that reads as ‘four hundred seventy five’. When you look at the number, the symbols correspond to the words. You have 4 (four, in position of hundreds), 7 (seven, in position of tens), and 5 (five).
Now consider a language which uses a different base, base 12 for example. The same number might be read as ‘three gross three dozen seven’. But when you look at the number, there is no symbol representing three, and no symbol representing seven in the position of units. The number is hard to parse and read out, and it is also hard to write when going by the words.
Now consider a language which uses a different base, base 12 for example. The same number might be read as ‘three gross three dozen seven’. But when you look at the number, there is no symbol representing three, and no symbol representing seven in the position of units. The number is hard to parse and read out, and it is also hard to write when going by the words.
English eleven to nineteen don't have a match in base-10 numerals either.
Although thirteen to nineteen show a clear connection to the older style of saying the ones place before the tens place (“four and twenty blackbirds baked into a pie…”). Spanish has special words for eleven (once—pronounced ohn-say) to fifteen (quince) before moving to a standard form for sixteen (dieciséis—ten and six) through nineteen diecinueve—ten and nine).
The French manage to convert between base 20 language and base 10 numerals.
Not really. French is effectively base-10, but with some vestigial names. English is basically the same -- we say eleven instead of ten-one. The only real difference is that we say eighty instead of quatre-vingt. But neither system maintains any semblance of base-20 beyond that.
A more familiar sense of disorientation of language mismatch would be when metric users encounter imperial units and their arbitrary bases.
A more familiar sense of disorientation of language mismatch would be when metric users encounter imperial units and their arbitrary bases.
You can get a very close analog by comparing English numbers (base 1,000) to Chinese numbers (base 10,000). The terminology coincides (with a couple asterisks) for numbers below 10,000, but above 10,000 it's essentially impossible to do mental conversion of numbers even though all the digits are the same.
Trying to operate simultaneously in a base 20 system and a base 10 system would be much worse than that, since all the digits would be different between representations.
Trying to operate simultaneously in a base 20 system and a base 10 system would be much worse than that, since all the digits would be different between representations.
Some dialects of French have a word for 70, 80 and 90!
I thought nonante (and septante, octante) were only used in Zwitserland and Belgium?
(If true, it would be a bit like saying in some dialects colour is written as color, while the latter is only(?) used in the States?)
Both of those examples sound correct to me, yes. Dialects are usually used in particular geographical areas, there is no contradiction there.
So do the Danish, however the Danish language only counts the 10s in base-20. The 100s, the 1000s, the decimals and so on are still in base-10. So what ends up happening is that the numbers 40-90 are simply numbers with weird names and a perfect mapping to base 10. I don’t know how French count, but perhaps it is similar. This does not appear to be the case with Inuit counting.
You could certainly write numbers above 9 as their base-10 representation. For example, 12. That number can be represented in both Kaktovik numerals and arabic. But you end up with an extra digit in arabic numerals, because you're supposed to carry the 1 once you reach 10.
Instead, we represent hexadecimal (base-16) in programming as 0-9 AND a-f, for example, A is 10, B is 11, C is the same as saying 12.
If you have multiple digits, like A4C in hex, it gets more complicated to figure out what number you're talking about in base-10. A is 10, 4 is 4, C is 12. To convert base-10 number you need to do this equation: (10 * 16 * 16) + (4 * 16) + (12) = 2636
Instead, we represent hexadecimal (base-16) in programming as 0-9 AND a-f, for example, A is 10, B is 11, C is the same as saying 12.
If you have multiple digits, like A4C in hex, it gets more complicated to figure out what number you're talking about in base-10. A is 10, 4 is 4, C is 12. To convert base-10 number you need to do this equation: (10 * 16 * 16) + (4 * 16) + (12) = 2636
It explains the problem: they are not a good way to express a base-20 numbering system used in the language.
(Now, you could easily augment or modify them to do that—and the creators [0] of this system initially tried that but were unsatisfied—the common way of expressing base-16 using arabic numerals plus the first six letters of the alphabet as added numerals is an example, but if you aren't using a language whose existing writing system conventionally users Arabic numerals, why would you?)
[0] middle school students!
(Now, you could easily augment or modify them to do that—and the creators [0] of this system initially tried that but were unsatisfied—the common way of expressing base-16 using arabic numerals plus the first six letters of the alphabet as added numerals is an example, but if you aren't using a language whose existing writing system conventionally users Arabic numerals, why would you?)
[0] middle school students!
I guess I was confused about that. I didn't think that a numbering system was considered part of a language. I always thought that a numbering system was more of an ADAPTER pattern (like Hexadecimal, binary, and octal are all "addons" to the English language).
The way that a language constructs number words can have an implied base; English has an implied based 10 in most of the language (though 0-19 uses a different structure that could imply base 20 if it continued.) Most germanic and romance languages I think do something similar, though, e.g. French (FR-fr, but not some other dialects) breaks back into an implicit base-20 at 60-99.
Having numerals that map well to words reduces friction for practical arithmetic.
Having numerals that map well to words reduces friction for practical arithmetic.
It used to be common to count by scores (twenties) in English, though this has mostly disappeared other than for poetical/literary usage (e.g., "Four score and seven years ago").
Dozens (twelves) were also quite common, and are still in everyday use for certain commodities (eggs, donuts, etc.).
Dozens (twelves) were also quite common, and are still in everyday use for certain commodities (eggs, donuts, etc.).
> It used to be common to count by scores (twenties) in English, though this has mostly disappeared
Counting by scores isn't enough to be base 20. The special quantities designated by the system are still 10, 100, and 1000, an obvious sign that the numbers are conceived of in base 10. If the system pivoted around 20, 400, and 8000 (as the dozen/gross system you mention does), then you could (and should) call it a base 20 system.
Counting by scores isn't enough to be base 20. The special quantities designated by the system are still 10, 100, and 1000, an obvious sign that the numbers are conceived of in base 10. If the system pivoted around 20, 400, and 8000 (as the dozen/gross system you mention does), then you could (and should) call it a base 20 system.
> The special quantities designated by the system are still 10, 100, and 1000, an obvious sign that the numbers are conceived of in base 10.
I actually gave an example of this. "Four score and seven" = 4 twenties, 7 units. That's base 20.
I actually gave an example of this. "Four score and seven" = 4 twenties, 7 units. That's base 20.
No, it isn't. It's base ten.
Dozens even go out to another level of base 12, now that I think about it. A gross (144) is twelve dozen, so you could have something like 6 gross, 5 dozen, and 3.
There is at least one more level, “great gross” is 12³, 1728.
There is also, per Wikipedia, apparently the mixed-base “small gross” / “great hundred” = 120, which I have never encountered in the wild.
There is also, per Wikipedia, apparently the mixed-base “small gross” / “great hundred” = 120, which I have never encountered in the wild.
The long (=great) hundred is 120, while the long thousand is 1200. It seems pretty understanding, so I wonder if anyone ever needed to use a long hundred thousand and worked out what that was supposed to be.
Why stop at base-20 when base-1000 is available? https://en.wikipedia.org/wiki/Cistercian_numerals
How inefficient! It only takes 10 distinct black-and-white pixels to represent 1000 distinct glyphs. Even low-res bitmap fonts are usually at least 6x6 pixels in size, meaning that we can use them to represent numbers in base 2^36. And with modern high-DPI screens we have easily enough room for glyphs that are 30 pixels square, letting us represent numbers in base 2^900. But why limit ourselves to black and white? We have the full 8-bit color space to work with, letting us devise a numeral system that can easily represent (2^24)^900 numbers using the space of a single glyph. :)
Base e would be the most efficient, if one assumes that the cost of representing a number in a particular radix is proportional to the radix. If you restrict the radix to natural numbers (e-state devices being rather hard to construct), base 3 would be more efficient.
As it happens, we've learned to make two-state devices way more cheaply than three-state devices, so binary wins in the real world, but if we figured out how to make three-state devices for at most 1.5 times the cost of two-state devices, base three would win.
It's fun to prove that base e and base 3 are theoretically more efficient than base 2 (and not all that difficult...only basic calculus is necessary).
As it happens, we've learned to make two-state devices way more cheaply than three-state devices, so binary wins in the real world, but if we figured out how to make three-state devices for at most 1.5 times the cost of two-state devices, base three would win.
It's fun to prove that base e and base 3 are theoretically more efficient than base 2 (and not all that difficult...only basic calculus is necessary).
> Base e would be the most efficient
Only in one dimension.
Only in one dimension.
Font that supports Kaktovik numerals:
https://languagetools-153419.appspot.com/ik/downloads/
https://languagetools-153419.appspot.com/ik/downloads/
so these numerals notation were invented in 1994. How does the Iñupiaq people do numeric calculation in the past without numerals notation?
Calculation was invented well before writing.
Probably like a lot of other hunter gatherer groups, worked with 10 numbers at most- what they could count on their fingers.
I don't think they'd necessarily have a base-20 system built into the language if they didn't work with numbers larger than 10.
There's another set of countables on feet, for a total of 20.
Base-20 works well on one hand.
Using your thumb, start counting on the first bone in each finger. That gets you to four. Continue with the first joint (8), then the second bone (12), then the second joint (16) and finally the tips (20).
Using your thumb, start counting on the first bone in each finger. That gets you to four. Continue with the first joint (8), then the second bone (12), then the second joint (16) and finally the tips (20).
makes me think of how to count with the base 60 babylonian number system.
You'd use the thumb of one hand to count to 12 on the phalanges of the other 4 fingers, and keep track of every 5 passes like we would do with the second hand
You'd use the thumb of one hand to count to 12 on the phalanges of the other 4 fingers, and keep track of every 5 passes like we would do with the second hand
5 fingers have 14 segments. Using segments on both sides of both hands = 2 sides of 2 hands = 56 finger segment slots. Counting the palms and backs of both hands = 4 hand sides = 60 segment slots and hand sides. Along with the 14 joints on 4 sides = 56 joint nodes = 116 joint nodes, segment slots and hand sides. Including the 10 finger (tips) gets a total of 126 and two arms is 128. Palms, backs of hands, arms, etc. could be further subdivided. Exercise: determine count of combinations.
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And 32 with your fingers only, if you use each finger as a bit :)
31 really, because a fist is zero. Four also presents a minor difficulty.
I don't think Alaskans spend a lot of time looking at their toes. It's cold there.
The long division example in there is a massive WTF, getting towards the level of the inverse square root float/int thing.
Sub-base 5 makes the multiplication table actually smaller than decimal. Nice.
I remember this being in a NACLO problem last year.
I'd be in support of stopping at 15 and moving to a base 16 metric. Hexadecimal numbers being discrete from alpha glyphs would be a bonus.
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Stopping at 10 is nice too and also an option.
16 is not a colossally abundant number
Nice for steganography applications.
Your comment made me really curious, could you please elaborate how this number system can be used in Steganography? I have a basic idea of steganography, but I want to know more
These glyphs are not immediately recognizable as numbers by most people. This can assist the purpose of hiding information visually.
It can also be used to avoid computer vision solutions from detecting numbers in an image, avoiding OCR, etc.
It can also be used to avoid computer vision solutions from detecting numbers in an image, avoiding OCR, etc.
I think we can assume that parent made a typo, or suffered autocorrect , and most likely meant stenography
> Scores on the California Achievement Test in mathematics for the Kaktovik middle school improved dramatically in 1997 compared to previous years. Before the introduction of the new numerals, the average score had been in the 20th percentile; after their introduction, scores rose to above the national average. It is theorized that being able to work in both base-10 and base-20 might have comparable advantages to those bilingual students have from engaging in two ways of thinking about the world.