I dislike these pseudo-scientific claims about alternative number systems and methods of paper and pencil arithmetic:
> Because of the tally-inspired design, arithmetic using the Kaktovik numerals is strikingly visual. Addition, subtraction and even long division become almost geometric. The Hindu-Arabic digits are an awkward system, Bartley says, but “the students found, with their numerals, they could solve problems a better way, a faster way.”
I think the students can be praised for having come up with simple to understand and write number system that corresponds to the conventions for counting in Alaskan Inuit language, and it seems appropriate to capture these notations in upcoming Unicode standards.
However, spending time learning base 20 arithmetic has obvious disadvantages that the article ignores. The times tables, memorized in grade school and fundamental to paper and pencil calculations, are now four times larger. Base 20 is not a popular notation for numbers. One important advantage of the number system (Hindu-Arabic) that most of the world uses is that most of the world uses it. I grew up with inches and degrees Fahrenheit and had to learn the metric system to pursue my science education. I'm glad I didn't have to learn how to count as well. We shouldn't make it harder for these kids to enjoy the rest of the world's books, journals, and internet resources about math and science.
Last year someone on HN linked to a sample set of problems from the International Linguistics Olympiad. I really enjoyed working through a few of them and found it an approachable challenge despite having no training in linguistics. About 2/3 of the way down there's a problem on Inuktitut Numbers, which I recommend attempting before reading the attached article if possible: https://ioling.org/booklets/samples.en.pdf
> Math is called the “universal language,” but a unique dialect is being reborn
It is not a unique dialect. It is just a yet another numeral system.
> Because of the tally-inspired design, arithmetic using the Kaktovik numerals is strikingly visual.
Ok, so is there any reason to think that it is better than other similar systems like the Mayan system? I am not even convinced that “strikingly visual” system is any better than our modern way to represent numbers in bases above ten using letters (…, 8, 9, A, B, …). If numbers look similar, you are more likely to mix them up.
I think the benefit, is its confusing to convert between systems. The point is to match the base ti the one the language/culture generally uses. English uses base-10, this particular language/cultural group did not, so constantly converting back and forth to base 10 was confusing.
i agree that it's disappointing that the sciam author, amory tillinghast-raby, knew so little about math that they didn't understand that what's supposed to be universal about math isn't the system of numerals; such ignorance or malicious disregard for truth is astounding in this context
as for why it's better, if we count 0 as 3 strokes (backslash, left, slash) and a base-20 digit as 4.32 bits, the kaktovik digits average 1.18 bits per stroke, versus what I calculate as 1.11 bits per stroke for our western arabic digits (using the stroke counts [3, 1, 3, 4, 3, 4, 3, 2, 4, 3])
averaging the number of strokes required per number up to 268 (a randomly selected number) we get 6.30 strokes per number with the kaktovik numerals or 6.79 strokes per number for western arabic numerals, an 8% advantage for the kaktovik numerals
the mayan base-20 numerals are more immediately comprehensible than the kaktovik numerals but i think they are harder to write and more error-prone to read
a way that base 20 is worse is that the multiplication table is substantially more unwieldy to memorize; however, if you can overcome that, both multiplication and division become more practical. for example, numbers between 1000 and 8000 have four base-10 digits but only three base-20 digits, so multiplying two of them in the usual way in base 10 will require 16 multiplication-table lookups and summing four partial products of usually 5 digits, while doing it in base 20 requires 9 lookups and summing three partial products of usually 4 digits, about 40% less work (aside from the number of strokes required)
in the limit, representing a large number in base 10 requires about 30.1% more digits than base 20, and so about 69% more work in the standard multiplication algorithm, but beyond about 5 digits you should be using karatsuba multiplication anyway
a way in which the kaktovik numerals are worse than western arabic numerals is that you definitely wouldn't want to use them to write a check; all numbers except for 20ⁿ-1 (0, 19, 399, 7999, etc.) can be increased by adding a single extra stroke to an existing digit
Define “better”. It is visual, so at least basic addition and subtraction don’t have to be done as calculation, they are purely visual. This could be a benefit for everyday tasks. Of course the representation nor the base actually matters mathematically, so in that regard it’s useless
> Ok, so is there any reason to think that it is better than other similar systems like the Mayan system?
It has 'zero'?
It's almost identical to roman numerals (count the strokes and the special symbol for certain multiples of 5 - V, X, etc) so I expect that it has all the downfalls of roman numerals.
I think that these primitive systems are what you get when you optimise for linear and incremental counting - you're optimising for easy and quick recognition of numbers not for convenient arithmetic.
Base-12 is what you get when you optimise for easy and convenient arithmetic. I have no idea what you will get if you optimise for easy and convenient calculus[1] :-)
[1] There's probably a research paper of Phd thesis in that goal.
> "The Alaskan Inuit language, known as Iñupiaq, uses an oral counting system built around the human body. Quantities are first described in groups of five, 10, and 15 and then in sets of 20."
French counting numbers are also still partly 20-based, as are danish and welsh. It's said it's a remnant of an earlier common numerical system. It's then even possible that human languages came from a common origin with a 20-based system, and some kept it.
> The examples listed feel contrived to get the best case results rather than the worst case.
They're cherry-picked. For addition, it only "makes sense visually" the way the article says it does if the answer lies within the sub-base-5 digit (i.e. the answer is, worst case, less than 5 numbers away).
There's also arbitrary rules in the so-called "easy visual arithmetic" - for some divisions (not all), some strokes have to be rotated. For the long division example, the visual indication of the remainder is reversed - i.e. it's a mirror image of the actual digit.
While I like the idea (the base-20 with sub-bases-5 makes counting easier, and having sub-bases means less memory overhead in memorising all 20 digits), the article itself is spinning wildly to make this seem like "the children came up with it on their own".
The title says "A number system invented by schoolchildren", while the article says that this was the result of a teacher-lead class project which came up with symbols for an existing numbering system.
Aside: With the exception of zero this numbering system is only slightly different from roman numerals - use the number of strokes and the special symbol to determine what number you are at. Counting is easier, and simple addition/subtraction/division is easier with roman numerals as well, but as soon as you need to do common things (approximate VAT for any figure[1]) then base-10 is so much easier.
For really easy arithmetic, using a base-12 counting system is even better (hence, the rise and popularity of imperial measures, which layers a base-12 system on top of base-10).
[1] VAT is 15% where I am, so mentally approximating VAT of $FOO is "10% of $FOO + 1/2 of 10% of $FOO). When it was 14% it was just as easy, do the above and remove 1%.
D'ni numerals were base 25, not 20. Each digit is composed of two base-5 "sub-digits" superimposed at 90° to each other -- the digits are designed such that this is unambiguous.
I find it interesting but unsurprising that learning a second base system makes you better at math, much in the same way as learning a second language as a child has benefits.
Grasping the concept of number regardless of base system is an interesting exercise (linked to modular arithmetic), see Mathologer on times tables in any number base:
Kids invent numbering systems all the time. Usually they see the point of numbers very early on and use their fingers to count. But occasionally they're very creative.
I have 3 kids. I remember an instance where the middle one asked for candy. So I said "how much". And she got her box of marbles and stones she collected (quite a collection), and wanted that many.
Wouldn't extending the system to 25 make more sense? I understand that it started as a representation of their previous system, but it stopping before W on W seems counterintuitive.
I see it would be more consistent in base 25.. if we could make up the missing 5 fingers. For me base 20 is already to difficult as I'm not so good in counting with my toes.
That's cool and all, but the computer aspect of this is it just got encoded into unicode and someone is making a font. Great for users of this system of course but ultimately not exactly the most exciting.
What I found interesting is that you could write these right-to-left and bottom-up, as long as you allow the top part to point in either direction, which makes the whole thing a lot easier. For example, 171:
\ (1)
\/\ \ (60 + 1)
>
\/\ \ (70 + 1, add 10 to the right digit)
/ >
\/\ \ (171, add 100 to the left digit)
Hopefully this makes sense, utf-8 will need to catch-up :)
This is more to that!
60 is https://en.wikipedia.org/wiki/Superior_highly_composite_numb... which means that lot of fractional numbers can be represnted in nice way.
And because one year is around 360 days and 12 (another superior highly composite number) is roughly correspond to number of lunar months in the year, people used those to based their calendar and angular measurments around that.
And 60 is very close to 64, so it could be nicely represented in binary... (10 is kinda akward, too big for 8 and too small for 16)
The visual logic sure seems nice, and the test results seem to speak for themselves (would be interesting to see studies on its impact). But I can't help but feel that if we are going to introduce a new number system with a different base at all, then that base should be 12. It's a highly composite number, it's divisible by 2, 3, 4 and 6, meaning that 1/2, 1/3, 1/4 and 1/6 (and their multiples) all have finite-digit representations, which would simplify a lot of everyday calculations. Several pop science articles have been written about the case for base 12:
If we're happy to use 11 new symbols instead of just 2, we could even keep the ideas from this system of using ticks and sub-bases to make computations more 'visual'.
Sorry, but I don't understand the basis for the opinion presented by a couple of people that the Kaktovik numeral system is similar to the Roman one. Is that because both of them use short lines as the sole element to build digits/numbers?!
But Kaktovik is a positional system while Roman is not!
Could our computers handle it? Makes me remember the count by 5 system. In this case, the 4 sticks are connected into a W. And the tick across the 4 sticks becomes a bar on top. The bar on top can count from 1-4. And then it goes to the next digit. So instead of base 16, we have a base 20.
Much like mayan, a base 20 system, very nice for addition subtraction, need to be a bit cleverer for multiplication. I made a little javascript module to convert arab to mayan numerals and render an SVG to go with them. Here's a fun demo
It's been a very long time since computers were anything other than binary, and unless quantum computing takes a, haha, quantum leap forward it will be a very long time yet until they're doing anything else.
But they can handle converting to and from this numeral system for the few humans who find it more natural to work with just fine.
Yes, computers can handle base 20. The IBM 1401 business computer (1959) could optionally support pounds/shillings/pence in hardware, where there were 20 shillings in a pound and 12 pence in a shilling. (Britain switched to decimal currency in 1971 for reasons that should be clear.) It could add, subtract, multiply, divide, and format pounds/shillings/pence as well as its normal base-10 arithmetic. So it was doing base-12 and base-20 arithmetic.
I should point out that this was implemented in hardware with transistors (lots of germanium transistors), not microcode or software. In other words, the three fundamental hardware datatypes of the IBM 1401 were arbitrary-length decimal numbers, arbitrary-length strings, and pounds/shillings/pence. Of course there were two conflicting standards on how to represent pounds/shillings/pence, so there was a knob on the computer's front panel to select the standard.
(This isn't directly related to the Inuit base-20, but I'm sure IBM would have supported Inuit base-20 if customers would pay for it.)
The decimal system is already based on the human body, most people are just not great at bending their toes one by one and find it convenient to have them covered by footwear. If you want a special symbol for one full hand, just use Roman numerals.
This reminds me of retro computing. Amiga or BeOS had some amazing concepts for the time, and quite possibly Wintel dominance was achieved by predatory tactics. It can be interesting to study old platforms and some enjoy creating new software up to this day. But if you limit yourself to these, don't expect modern living. At best you can hook up an old computer to a modern one as a thin client and fool yourself into thinking that harpooning a whale from a motor boat is traditional living. For whatever reason the world have move on and it's not possible for could have been possibilities to ever catch up with limited number of participants, since the rest of the world will also not stand still.
As a retro computing aficionado, this is where I disagree. Learning about old computers has always taught me something useful about modern systems. Going back to the first IBMs and Apple architectures has a lot of merit. It's like reading a book from the beginning, instead of jumping in the middle and trying to make sense of everything. You can probably continue reading, but you won't know why some things are the way they are.
Obviously, developing software for obsolete systems is not a great income stream, but as an exercise, this too has merit. You learn a lot about constraints and limitations, which today are rarely considered, but could still teach you how to optimize code. You learn how to produce software which can run surprisingly fast on machines from 30 or 40 years ago, and that is transferrable to modern coding.
You learn a lot about memory, how to use it efficiently and what can be achieved with just 640K, which as we all know, "is all the memory anyone should ever need". You learn that by introducing limitations a sort of game happens in which you need to be more creative to implement things which have since become obvious. And this makes you a better problem solver.
There is a lot to learn from old computers, and while some people will always disagree, I think it makes you a better software engineer.
[+] [-] todd8|2 years ago|reply
> Because of the tally-inspired design, arithmetic using the Kaktovik numerals is strikingly visual. Addition, subtraction and even long division become almost geometric. The Hindu-Arabic digits are an awkward system, Bartley says, but “the students found, with their numerals, they could solve problems a better way, a faster way.”
I think the students can be praised for having come up with simple to understand and write number system that corresponds to the conventions for counting in Alaskan Inuit language, and it seems appropriate to capture these notations in upcoming Unicode standards.
However, spending time learning base 20 arithmetic has obvious disadvantages that the article ignores. The times tables, memorized in grade school and fundamental to paper and pencil calculations, are now four times larger. Base 20 is not a popular notation for numbers. One important advantage of the number system (Hindu-Arabic) that most of the world uses is that most of the world uses it. I grew up with inches and degrees Fahrenheit and had to learn the metric system to pursue my science education. I'm glad I didn't have to learn how to count as well. We shouldn't make it harder for these kids to enjoy the rest of the world's books, journals, and internet resources about math and science.
[+] [-] n4r9|2 years ago|reply
[+] [-] majou|2 years ago|reply
[+] [-] LudwigNagasena|2 years ago|reply
It is not a unique dialect. It is just a yet another numeral system.
> Because of the tally-inspired design, arithmetic using the Kaktovik numerals is strikingly visual.
Ok, so is there any reason to think that it is better than other similar systems like the Mayan system? I am not even convinced that “strikingly visual” system is any better than our modern way to represent numbers in bases above ten using letters (…, 8, 9, A, B, …). If numbers look similar, you are more likely to mix them up.
[+] [-] bawolff|2 years ago|reply
I think the benefit, is its confusing to convert between systems. The point is to match the base ti the one the language/culture generally uses. English uses base-10, this particular language/cultural group did not, so constantly converting back and forth to base 10 was confusing.
[+] [-] atlas_hugged|2 years ago|reply
[+] [-] kragen|2 years ago|reply
i agree that it's disappointing that the sciam author, amory tillinghast-raby, knew so little about math that they didn't understand that what's supposed to be universal about math isn't the system of numerals; such ignorance or malicious disregard for truth is astounding in this context
as for why it's better, if we count 0 as 3 strokes (backslash, left, slash) and a base-20 digit as 4.32 bits, the kaktovik digits average 1.18 bits per stroke, versus what I calculate as 1.11 bits per stroke for our western arabic digits (using the stroke counts [3, 1, 3, 4, 3, 4, 3, 2, 4, 3])
averaging the number of strokes required per number up to 268 (a randomly selected number) we get 6.30 strokes per number with the kaktovik numerals or 6.79 strokes per number for western arabic numerals, an 8% advantage for the kaktovik numerals
the mayan base-20 numerals are more immediately comprehensible than the kaktovik numerals but i think they are harder to write and more error-prone to read
a way that base 20 is worse is that the multiplication table is substantially more unwieldy to memorize; however, if you can overcome that, both multiplication and division become more practical. for example, numbers between 1000 and 8000 have four base-10 digits but only three base-20 digits, so multiplying two of them in the usual way in base 10 will require 16 multiplication-table lookups and summing four partial products of usually 5 digits, while doing it in base 20 requires 9 lookups and summing three partial products of usually 4 digits, about 40% less work (aside from the number of strokes required)
in the limit, representing a large number in base 10 requires about 30.1% more digits than base 20, and so about 69% more work in the standard multiplication algorithm, but beyond about 5 digits you should be using karatsuba multiplication anyway
a way in which the kaktovik numerals are worse than western arabic numerals is that you definitely wouldn't want to use them to write a check; all numbers except for 20ⁿ-1 (0, 19, 399, 7999, etc.) can be increased by adding a single extra stroke to an existing digit
the chinese (base 10) system, which has a less extreme version of this problem, has a separate set of high-security "大写" or "financial" numerals for contexts where this matters https://en.wikipedia.org/wiki/Chinese_numerals#Standard_numb...
[+] [-] moi2388|2 years ago|reply
[+] [-] lelanthran|2 years ago|reply
It has 'zero'?
It's almost identical to roman numerals (count the strokes and the special symbol for certain multiples of 5 - V, X, etc) so I expect that it has all the downfalls of roman numerals.
I think that these primitive systems are what you get when you optimise for linear and incremental counting - you're optimising for easy and quick recognition of numbers not for convenient arithmetic.
Base-12 is what you get when you optimise for easy and convenient arithmetic. I have no idea what you will get if you optimise for easy and convenient calculus[1] :-)
[1] There's probably a research paper of Phd thesis in that goal.
[+] [-] cuteboy19|2 years ago|reply
[+] [-] photochemsyn|2 years ago|reply
So, sounds fundamentally like Mayan numerals?
https://en.wikipedia.org/wiki/Maya_numerals
[+] [-] kzrdude|2 years ago|reply
[+] [-] galleywest200|2 years ago|reply
[+] [-] tragomaskhalos|2 years ago|reply
[+] [-] 8note|2 years ago|reply
I think the most interesting checks is with exponentials though? How does this represent e? Pi? Complex number rotations?
[+] [-] lelanthran|2 years ago|reply
They're cherry-picked. For addition, it only "makes sense visually" the way the article says it does if the answer lies within the sub-base-5 digit (i.e. the answer is, worst case, less than 5 numbers away).
There's also arbitrary rules in the so-called "easy visual arithmetic" - for some divisions (not all), some strokes have to be rotated. For the long division example, the visual indication of the remainder is reversed - i.e. it's a mirror image of the actual digit.
While I like the idea (the base-20 with sub-bases-5 makes counting easier, and having sub-bases means less memory overhead in memorising all 20 digits), the article itself is spinning wildly to make this seem like "the children came up with it on their own".
The title says "A number system invented by schoolchildren", while the article says that this was the result of a teacher-lead class project which came up with symbols for an existing numbering system.
Aside: With the exception of zero this numbering system is only slightly different from roman numerals - use the number of strokes and the special symbol to determine what number you are at. Counting is easier, and simple addition/subtraction/division is easier with roman numerals as well, but as soon as you need to do common things (approximate VAT for any figure[1]) then base-10 is so much easier.
For really easy arithmetic, using a base-12 counting system is even better (hence, the rise and popularity of imperial measures, which layers a base-12 system on top of base-10).
[1] VAT is 15% where I am, so mentally approximating VAT of $FOO is "10% of $FOO + 1/2 of 10% of $FOO). When it was 14% it was just as easy, do the above and remove 1%.
[+] [-] Xorakios|2 years ago|reply
Write pi in ancient Roman numerals, Greek, Japanese, Han Mandarin.
Business and science didn't adopt Indo/Arabic numerals just for fun. They just work.
[+] [-] sp332|2 years ago|reply
(2) . (14) (7) (6) (5) (1) (17) (0) (8) (11) (0) (12) (9) (5)...
(3) . (2) (16) (12) (14) (16) (9) (16) (11) (17) (19) (9) (13) (2)...
[+] [-] nathandaly|2 years ago|reply
If you ever played Riven, I think the Kaktovik numerals inspired the numbering system from the game.
Spoilers: https://lparchive.org/Riven/Update%2015/
[+] [-] duskwuff|2 years ago|reply
[+] [-] waldothedog|2 years ago|reply
[+] [-] readthenotes1|2 years ago|reply
[+] [-] foota|2 years ago|reply
[+] [-] photochemsyn|2 years ago|reply
https://youtu.be/qhbuKbxJsk8
[+] [-] readthenotes1|2 years ago|reply
The new system does seem more visually accessible for arithmetic. I wouldn't be surprised if it's easier to teach children than the Arabic numerals
[+] [-] cat_plus_plus|2 years ago|reply
[+] [-] candiodari|2 years ago|reply
I have 3 kids. I remember an instance where the middle one asked for candy. So I said "how much". And she got her box of marbles and stones she collected (quite a collection), and wanted that many.
[+] [-] boomboomsubban|2 years ago|reply
No clue if that'd ruin the arithmetic benefits.
[+] [-] thro1|2 years ago|reply
[+] [-] bawolff|2 years ago|reply
[+] [-] jonny_eh|2 years ago|reply
What was the last novel number system that impressed you?
[+] [-] ricardobeat|2 years ago|reply
[+] [-] andrewstuart|2 years ago|reply
[+] [-] numbol|2 years ago|reply
[+] [-] kneebonian|2 years ago|reply
3 * 5 * 4 = 60
Actually quite intuitive and impressive.
[+] [-] thro1|2 years ago|reply
[+] [-] c7b|2 years ago|reply
https://www.scienceabc.com/eyeopeners/why-we-should-already-... https://gizmodo.com/why-we-should-switch-to-a-base-12-counti...
If we're happy to use 11 new symbols instead of just 2, we could even keep the ideas from this system of using ticks and sub-bases to make computations more 'visual'.
[+] [-] asjfka|2 years ago|reply
[+] [-] witrak|2 years ago|reply
But Kaktovik is a positional system while Roman is not!
[+] [-] nashashmi|2 years ago|reply
[+] [-] wolfram74|2 years ago|reply
https://wolfram74.github.io/ArabIntToMayaInt/countdown.html
[+] [-] codebje|2 years ago|reply
It's been a very long time since computers were anything other than binary, and unless quantum computing takes a, haha, quantum leap forward it will be a very long time yet until they're doing anything else.
But they can handle converting to and from this numeral system for the few humans who find it more natural to work with just fine.
[+] [-] kens|2 years ago|reply
I should point out that this was implemented in hardware with transistors (lots of germanium transistors), not microcode or software. In other words, the three fundamental hardware datatypes of the IBM 1401 were arbitrary-length decimal numbers, arbitrary-length strings, and pounds/shillings/pence. Of course there were two conflicting standards on how to represent pounds/shillings/pence, so there was a knob on the computer's front panel to select the standard.
(This isn't directly related to the Inuit base-20, but I'm sure IBM would have supported Inuit base-20 if customers would pay for it.)
[+] [-] cat_plus_plus|2 years ago|reply
This reminds me of retro computing. Amiga or BeOS had some amazing concepts for the time, and quite possibly Wintel dominance was achieved by predatory tactics. It can be interesting to study old platforms and some enjoy creating new software up to this day. But if you limit yourself to these, don't expect modern living. At best you can hook up an old computer to a modern one as a thin client and fool yourself into thinking that harpooning a whale from a motor boat is traditional living. For whatever reason the world have move on and it's not possible for could have been possibilities to ever catch up with limited number of participants, since the rest of the world will also not stand still.
[+] [-] magic_hamster|2 years ago|reply
Obviously, developing software for obsolete systems is not a great income stream, but as an exercise, this too has merit. You learn a lot about constraints and limitations, which today are rarely considered, but could still teach you how to optimize code. You learn how to produce software which can run surprisingly fast on machines from 30 or 40 years ago, and that is transferrable to modern coding.
You learn a lot about memory, how to use it efficiently and what can be achieved with just 640K, which as we all know, "is all the memory anyone should ever need". You learn that by introducing limitations a sort of game happens in which you need to be more creative to implement things which have since become obvious. And this makes you a better problem solver.
There is a lot to learn from old computers, and while some people will always disagree, I think it makes you a better software engineer.
[+] [-] anthk|2 years ago|reply
[+] [-] OJFord|2 years ago|reply
[+] [-] c3534l|2 years ago|reply