Kaktovik Numerals


I’m fascinated by the Inuit 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 nine 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. So here’s what they came up with:

The students built base-20 abacuses. These were initially intended to help the conversion from decimal 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.

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!

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.

And then something interesting happened: 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 up to above the national average.

Some argue that being able to work in both base 10 and base 20 was helpful—much like being bilingual. Another explanation is that having a written system of numbers that matched the local language was helpful.

But 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. For more, read this:

• Amory Tillinghast-Raby, A number system invented by Inuit schoolchildren will make its Silicon Valley debut, Scientific American, April 10, 2023.

I’ll quote a bit:

At first students would convert their assigned math problems into Kaktovik numerals to do calculations, but middle school math classes in Kaktovik began teaching the numerals in equal measure with their Hindu-Arabic counterparts in 1997. Bartley reports that after a year of the students working fluently in both systems, scores on standardized math exams jumped from below the 20th percentile to “significantly above” the national average. And in the meantime, the board of education in the North Slope Borough’s district seat, Utqiagvik, passed a resolution that spread the numerals almost 500 miles along the Arctic coast. The system was even endorsed by the Inuit Circumpolar Council, which represents 180,000 Inuit across Alaska, Canada, Greenland and Russia.

But 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. “Today the only place they’re really being used is in the Iñupiaq language classrooms,” says Chrisann Justice, the North Slope Borough’s Iñupiaq education department specialist. “We’re just blowing on the coal.”

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:

• Wikipedia, Kaktovik numerals (Unicode block).

For more on the cool mathematical properties of Kaktovik numerals, try this:

• Wikipedia, Kaktovik numerals.

On Mathstodon David Nash wrote:

I read the Wikipedia article about this numeral system and came across this long division example:

My nearly exact thought process while reading the caption:

Article “The divisor goes into the first two digits of the dividend one time, for a one in the quotient.”

Me “OK, sure, got it.”

Article “It fits into the next two digits (red) once if rotated…”

Me: “What do you mean, “rotated”?”

Me “holy *SHIT*, that’s totally genius”

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

The Kaktovik numerals were invented by high school students who spoke Iñupiaq:

• Wikipedia, Iñupiaq language.

As you can see, this endangered language is spoken in northern Alaska:

This map also shows other Inuit languages:

• Wikipedia, Inuit languages.

These are part of a larger group called Inuit-Yupik-Unangan languages:

• Wikipedia, Inuit-Yupik-Unangan languages.

9 Responses to Kaktovik Numerals

  1. Toby Bartels says:

    Your map doesn't show the entire Inuit-Yupik-Unangan family, but just the Inuit branch. (The Yupik languages are spoken on the west coast of Alaska and in far-eastern Siberia, and the Unangan language is spoken in the Aleutian Islands. For this reason, Unangan is also called ‘Aleut’; meanwhile, the Inuit-Yupik branch is often called ‘Eskimoan’, and the entire family is ‘Eskimo-Aleut’ or ‘Eskaleut’. But these are exonyms and sometimes considered slurs, while ‘Inuit’, ‘Yupik’, and ‘Unangan’ are derived from the speakers’ own words for their languages.)

    • John Baez says:

      Thanks for the correction—I fixed my article! I knew that “Eskimo” is considered a slur. There is still a reference to “Eskimo-Aleut” languages on the Wikipedia page Inuit, which someone may want to fix. And the official title of the Wikipedia page on Inuit-Yupik-Unangan languages is “Eskaleut languages”. If “Eskimo” (or “Aleut”) are slurs, I can’t see that this is any better.

      • Toby Bartels says:

        I don’t think that ‘Eskaleut’ is to avoid saying ‘Eskimo’ (much less ‘Aleut’); linguists just like this sort of thing (these are the same people who gave us ‘Amerindian’ for the indigenous languages to the south). And not everyone agrees that ‘Eskimo’ is a slur or that ‘Inuit’ is an appropriate term (since it's taken from Inuktitut and is different in the other Inuit languages). At any rate, not everyone agreed 20 years ago when I read about this; I don’t know where the discourse stands now.

        • John Baez says:

          I wasn’t claiming that anyone says ‘Eskaleut’ to avoid saying ‘Eskimo’. I was just saying that if Wikipedians decide to avoid saying ‘Eskimo’ (except when necessary, like in the article Eskimo), they should probably also avoid ‘Eskaleut’.

          The Wikipedia article Eskimo says:

          Many Inuit, Yupik, Aleut, and other individuals consider the term Eskimo, which is of a disputed etymology, to be unacceptable and even pejorative. Eskimo continues to be used within a historical, linguistic, archaeological, and cultural context. The governments in Canada and the United States have made moves to cease using the term Eskimo in official documents, but it has not been eliminated, as the word is in some places written into tribal, and therefore national, legal terminology. Canada officially uses the term Inuit to describe the indigenous Canadian people who are living in the country’s northern sectors and are not First Nations or Métis. The United States government legally uses Alaska Native for Native Alaskans including the Yupik, Inuit, and Aleut, but also for non-Eskimo Native Alaskans including the Tlingit, the Haida, the Eyak, and the Tsimshian, in addition to at least nine separate northern Athabaskan/Dene peoples. The designation Alaska Native applies to enrolled tribal members only, in contrast to individual Eskimo/Aleut persons claiming descent from the world’s “most widespread aboriginal group”.

  2. arch1 says:

    Thank you John, this is really remarkable. Confronted with a design problem most would have muddled through by rote (sorry), those high schoolers created something of inspiring (and now lasting) beauty and value, in the process exemplifying much of what we admire most about human learning and intellectual exploration.

  3. Wolfgang says:

    This example fits into my impression that notation can be super important to understand things better. It seems to be a fact to me now, by experience, but I think in general this is underestimated. For instance, in a sum (or product) the index is written below the sigma symbol, but in an integral the dx is written somewhere else. Wouldn’t it be more meaningful, to denote it also below the symbol to indicate what the integral is about to integrate in terms of variables? I know that physicists sometimes prefer to write the dx in front of the function they want to integrate, while others prefer to write it at the back, thinking of it as a kind of end delimiter to the start integral sign (as a kind of bracket the thing that is to be integrated).

    • Toby Bartels says:

      I always write \int_{x=a}^b to match \sum_{i=m}^n. Although I still include the dx, since it has a meaning besides identifying the dummy variable. Particularly when changing variables, it's helpful to have both of these.

  4. joshuarbholden says:

    Aren’t these isomorphic to the Maya Numerals? https://en.wikipedia.org/wiki/Maya_numerals Although I have to admit the rotation thing is very cool.

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