I'm fascinated by the Inuit-Yupik-Unangan languages in Alaska, Canada and Greenland. There are many of these languages: they ring much of the Arctic Ocean. I just learned that they use a base 20 system for numbers, with a 'sub-base' of 5. That is, quantities are counted in scores (twenties) with intermediate numerals for 5, 10, and 15. This makes a lot of sense if you look at your fingers and toes.

But the Inuit didn't have a written form of their number system - until the early 1990s, when high school students in the town of Kaktovik, Alaska invented one! There were just 9 students at this small school, and they all joined in.

They used 5 principles:

• Visual simplicity: The symbols should be easy to remember.

• Iconicity: There should be a clear relationship between the symbols and their meanings.

• Efficiency: It should be easy to write the symbols without lifting the pencil from the paper.

• Distinctiveness: There should be no confusion between this system and Arabic numerals.

• Aesthetics: They should be pleasing to look at.

They decided that the symbol for zero should look like crossed arms, meaning that nothing was being counted.

This was the start of quite a tale!

(1/n)

The students built base-20 abacuses. These were initially intended to help convert decimal numbers to base 20 and vice versa, but soon the students started using them to do arithmetic in base 20.

The upper section of their abacus has 3 beads in each column for the values of the sub-base of 5, while the lower section has 4 beads in each column for the remaining units.

(2/n)

The students discovered their new system made arithmetic easier than it was with Arabic numerals. Adding two digits together often gives a result that *looks* like the combination of the two digits!

(3/n)

The students also found that long division was more fun with Kaktovik numerals! They noticed visually interesting patterns. They discovered that they could keep track of intermediate steps with colored pencils.

(4/n)

After the students of Kaktovik invented their new numerals, their scores on standardized math tests improved dramatically! Before, their average score was down in the 20th percentile. Afterwards, their scores shot above the national average.

Some argue that being able to work in both base 10 and base 20 was helpful - much like being bilingual.

I suspect that even more important was the sheer process of developing their own system of numerals! Getting engaged in mathematics is so much better than learning it passively.

But this was just the start of the story. The new numerals began to catch on... but then:

"Under the federal No Child Left Behind Act, from 2002 to 2015, schools faced severe sanctions—or even closure—for not meeting state standards, provoking a “scare” that some local educators say squeezed the Kaktovik numerals into a marginal role despite the system’s demonstrated educational impact."

For more, read this:

https://www.scientificamerican.com/article/a-number-system-invented-by-inuit-schoolchildren-will-make-its-silicon-valley-debut/

(5/n)

A Number System Invented by Inuit Schoolchildren Will Make Its Silicon Valley Debut

Math is called the “universal language,” but a unique dialect is being reborn

Scientific American

Why is Scientific American talking about Kaktovik numerals just *now*? It's because some linguists working with the Script Encoding Initiative at U.C. Berkeley recently got them added to Unicode! See here:

https://en.wikipedia.org/wiki/Kaktovik_Numerals_(Unicode_block)

For more on the cool mathematical properties of Kaktovik numerals, try the Wikipedia article on them:

https://en.wikipedia.org/wiki/Kaktovik_numerals

I suspect there's more to be said. Maybe someone has worked out all the details somewhere?

Below is another example of long division.

(5/n)

Kaktovik Numerals (Unicode block) - Wikipedia

Finally, here's a map of Inuiit languages! The Kaktovik numerals were invented by high school students in northern Alaska who spoke Iñupiaq, one of these languages.

For more on Iñupiaq:

https://en.wikipedia.org/wiki/I%C3%B1upiaq_language

For more on Inuit languages, try this:

https://en.wikipedia.org/wiki/Inuit_languages

(6/n, n = 6)

Iñupiaq language - Wikipedia

@johncarlosbaez I've been obsessing over this since I read about it earlier this week! I would love to boost this but could you please edit to remove the word Es**** which is recognized now as a slur? I checked Wikipedia and there are other terms for the language and language groups mentioned.

@irisvirus - I'll do it. I know that word is considered bad by some, but there are still Wikipedia pages that talk about the "Eskimo-Aleut" language group, like this one:

https://en.wikipedia.org/wiki/Inuit

That's why I used the term.

The link to "Eskimo-Aleut" on that page now links to a page on "Eskaleut languages":

https://en.wikipedia.org/wiki/Eskaleut_languages

But if "Eskimo-Aleut" is bad, I don't think "Eskaleut" is much better. As you'll see, this language group is also called the "Inuit–Yupik–Unangan" group. That could be better - I'm hoping these are names of languages that are actually words *in those languages*.

In short, I'll fix my article but I hope you fix up some Wikipedia articles!

Inuit - Wikipedia

@johncarlosbaez Thanks, and yeah, I saw that too. I'm not in a position to make a positive statement that Inuit-Yupik-Unangan is OK but I would agree Es****ut is most probably not OK. Inuit is at least a term people use for themselves as far as I've observed.

More broadly, the fact that Wikipedia articles use the terms is no defense -- "when you know better, do better" and all that. I'll think about editing the Wikipedia pages. Will need to get better informed first.

@irisvirus thanks for pointing this out; no pressure of course, but it sounds like any change is an improvement over what's currently there!
@johncarlosbaez I boosted your post before realising your following post contained a racial slur. I cannot figure out how to unboost so I can only ask you to please edit the E word out of your post. Thank you. (Inuk Mom)
@campbellb @johncarlosbaez if you Click on the boost Button again it should unboost.

@campbellb - I got rid of that slur. I thought I'd gotten rid of it everywhere about a week ago, but just now I noticed one place where I hadn't. It's gone now.

But if you want, you can unboost my post just by clicking boost again.

@johncarlosbaez It's interesting that they use the ñ sound and symbol
@johncarlosbaez
I think the Mayans also used the base 20 system.
@JFernandoPascualS - yes! And like the Inuit, they also use a sub-base of 5:

@johncarlosbaez

Your findings are really very interesting and I think I might be able to share some facts about the cognitive reasons why base 5 is so effective.

However, I do not want to promote a link that contains a word that is considered offensive by the ethnicity involved.
May I second that you change the E word to #Inuit?
Thanks a lot for your consideration.

@HistoPol - Fixed! I thought I'd gotten rid of that word everywhere a week ago, but one instance survived.

I hope folks change Wikipedia, which is where I got that term.

@johncarlosbaez Here a map from the other side of the world, from Australia, showing the lands of the Aboriginal people here.
Map of Indigenous Australia

The AIATSIS map serves as a visual reminder of the richness and diversity of Aboriginal and Torres Strait Islander Australia.

AIATSIS corporate website
@johncarlosbaez What about a \(\LaTeX\) package?

@rjf_berger - I someone makes one! The new Kaktovic numeral unicode block doesn't yet work on my laptop, so this page is mostly blanks for me:

https://en.wikipedia.org/wiki/Kaktovik_Numerals_(Unicode_block)

Kaktovik Numerals (Unicode block) - Wikipedia

@johncarlosbaez I was also fascinated by this article. If you are a visual learner, it seems like a much easier way to learn math.
@johncarlosbaez
Reminds me of the arguments in favor of Moore method.
@johncarlosbaez my sister, who has been a teacher for decades, likes to call it “no child gets ahead.”

@johncarlosbaez

This is very nice.

But what I like about non-standard pictures like the above is that if you haven't trained yourself to understand them, they look very mysterious. Unfortunately, most people see mathematics (and many other things, such as computing, physics, chemistry, ...) like this - lots of meaningless symbols.

Of course, if you study the link that John Baez is discussing, there is a lot of meaning in the symbols of this particular picture.

@MartinEscardo @johncarlosbaez I think exactly those are not a good example of what you describe. Those numerals are making sense by just looking at them unkike our conventional math symbols.

I always have imagined ideal math symbols for all fields of maths (alg., top., cat. theo.,..) that have their own intrinsic symbolic "calculus" making all theorems trivial dessous d enfants.

@johncarlosbaez. French people will love it
@Catweazle @johncarlosbaez Bretons especially: 40, 60, 80 and sometimes 100, 120 and 140 are counted as multiples of 20.
@graham_knapp @Catweazle - do Bretons use base 20 more than French? In French 80 is "quatre-vingts".
@johncarlosbaez @Catweazle yes in French it is just 80 but in Breton it is
40: daou-ugent (2-20)
60: tri-ugent (3-20)
80: pevar-ugent (4-20)
then 90: dek ha pevar-ugent(10 & 4-20)
50 is hanter kant, a half hundred !
@johncarlosbaez @Catweazle I once made the mistake of deciding to code all these #Breton counting rules in Python https://github.com/dancergraham/num2words #bzh
GitHub - dancergraham/num2words: Modules to convert numbers to words. 42 --> forty-two

Modules to convert numbers to words. 42 --> forty-two - dancergraham/num2words

GitHub
@johncarlosbaez The Kartovik numbers are definitely reminiscent of the abacus system. FWIW, I sometimes drive people crazy by counting to 10 (ok, 9) on one hand: each finger is one, thumb is 5. Two hands gets me to 99. I'm always surprised when somebody doesn't understand what I'm doing.
@johncarlosbaez Interestingly, this is nearly identical to a Chinese abacus, except that allows 0+5 as a synonym for 1*5+0 and uses the columns differently

@johncarlosbaez, it remeber me to this one

https://www.zmescience.com/science/cirstercian-numbers-90432432/

Here also a page to convert the numbers to the cistercian system.
https://www.dcode.fr/cistercian-numbers

What are Cistercian numbers — the forgotten ciphers of Medieval monks

A system of numerals that is written down like tic-tac-toe.

ZME Science
@Catweazle - cool, thanks! I'm starting to really like these lesser-known systems.

@johncarlosbaez

Reminds me a little bit of the system in Myst.

https://dni.fandom.com/wiki/D%27ni_Numerals

D'ni Numerals

The D'ni used a unique base-25 counting system, rather than the decimal (or base-10) system used by many human societies. They also used numerals unlike any encountered on the surface. Their system had unique symbols for the numbers zero to four and all other numbers are constructed by various arrangements of these. The decimal system has ten digits (0 to 9), and the place value is always a power of ten. This means that the number 37 means 3 sets of 10 with 7 sets of one ((101*3) + (100*7)). The

D'ni
@reziplikativ - thanks, I'll check it out! I've seen Myst but not gotten into it.
@johncarlosbaez @reziplikativ The studio that made Myst has made more recent games, all of which have “naturalistic” puzzles such as arcane machines to operate, and writing systems to decode. It’s almost a running joke at this point that one set of puzzles will revolve around deciphering a non-decimal numeral system.
@johncarlosbaez You are surely not gonna take your shoes off up there to count your toes. Unless you want to go to base 19, then 18, then 17, very quickly.

@johncarlosbaez As a linguist, I can just say that

"THIS IS BRILLIANT!!!!!"

@johncarlosbaez

Oh, forgot to mention: "Eskimo" is a slur, so try "Inuit" and "Metis".

EDIT: You did say "inuit", but still…

EDIT2: Thank you for changing it.

@johncarlosbaez

And, I thought Welsh was difficult! Well done 👍
You have a great deal more understanding than me 😂

@johncarlosbaez

I never knew this about Kaktovik. It's about the smallest Inuit village I've ever visited. This is my favorite pic from the last time I visited there - a sow overlooking the town, which is made up mostly of shipping containers and old double wides...

https://www.seeingbirds.com/Mammals/Bears/Polar-Bear/i-GZmLMzR/A

Polar Bear - Ashok Khosla

Hi. Welcome. I mostly photograph things that quack, gargle, snort and make all sorts of funny sounds. Sometimes the only sound I hear is mother earth. I'm still learning to listen to her. Nature never ceases to amaze me. I hope it never ceases to amaze you. Peace.

@amkhosla - That's a great shot. And it's impressive that you've actually visited this place! Wikipedia says the population is 283:

https://en.wikipedia.org/wiki/Kaktovik,_Alaska

Kaktovik, Alaska - Wikipedia

@johncarlosbaez What I find interesting about this is that both English ('the years of a man's life are three score and ten') and French ('quatre vingt huit") traditionally used base twenty counting.
@simon_brooke @johncarlosbaez Gàidhlig was commonly using a vigesimal system until late C20th too; it's only since the 1980s that decimal has really been taught as the standard eg ceathrad instead of dà fhichead

@HighlandLawyer @simon_brooke @johncarlosbaez
Yup and to be clear (“Gàidhlig was commonly using a vigesimal system until late C20th too…”): most older speakers who didn’t learn that system are STILL using ‘àireamhan tradaiseanta’ (traditional numbers, ie vigesimal system) rather than ‘àireamhan ùr’ (new numbers)!
(Suspect you know that!)

Literally speaking to a bodach last night and we have to translate: he teaches me old and I teach him new…

@HighlandLawyer @simon_brooke @johncarlosbaez
So Simon eighty in Gàighlig in traditional system is almost exactly the same as French: ‘ceithir fichead’ (lit. four twenty)…

…but ‘ochdad’ in new system (as per some French dialects such as Swiss Romande and Belgian French: ‘octante’!)

I always like 50 = leth-cheud (half-hundred … or half a century!)

@HighlandLawyer @simon_brooke @johncarlosbaez

PS to clear up one common misconception that plagues this: people often think this is hard & ‘silly’ & needs maths…
4 x 20 = 80… ??
No!!!
Native speakers just *think* of ‘ceithir fichead’ - or ‘quatre-vingt-dix’ - as numbers. 80, 90.

Just as no EN speaker thinks of nineteen as 9 + 10 = 19!
Or fifty as 5 x 10 = 50!
Sure the spelling in EN disguises it a bit but it’s also bc one takes it for granted as a number…!

@AeonMach @HighlandLawyer @simon_brooke @johncarlosbaez Overall I think the scores system looks easier for learners (or me as a learner anyway) because there's a strong repetitive pattern and actually fewer new words to learn! I found this wiki page with all the numbers helpfully listed out https://gaelicgrammar.org/~gaelic/mediawiki/index.php/Numerals
Numerals - Scottish Gaelic Grammar Wiki

@stephaniejne Ha I like your glass half full and practical approach of realising there are less words in base 20 👌

PS from what I know the ‘air’ approach (eg. ceithir air fhichead) might be a mistaken corruption of ‘(th)ar’ - since as I note there ‘thar’ (over) is normally contracted to ‘ar’ (ceithir ar fhichead).

@AeonMach I hadn't yet seen the air/thar approach anywhere else but on that page. Is it in common usage?