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.