Kaktovik numerals

Iñupiaq, like other Inuit languages, has a base-20 counting system with a sub-base of 5. That is, quantities are counted in scores (as in Danish, Welsh and in some French numbers such as quatre-vingts 'eighty'), with intermediate numerals for 5, 10, and 15. Thus 78 is identified as three score fifteen-three.[2]

The twenty digits are:

𝋀	𝋁	𝋂	𝋃	𝋄	𝋅	𝋆	𝋇	𝋈	𝋉	𝋊	𝋋	𝋌	𝋍	𝋎	𝋏	𝋐	𝋑	𝋒	𝋓
0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19

The Kaktovik digits graphically reflect the lexical structure of the Iñupiaq numbering system. For example, the number seven is called tallimat malġuk in Iñupiaq ('five-two'), and the Kaktovik digit for seven, 𝋇, is a top stroke (𝋅 five) plus two bottom strokes (𝋂 two). Similarly, 𝋌 twelve and 𝋑 seventeen are called qulit malġuk ('ten-two') and akimiaq malġuk ('fifteen-two'), and the Kaktovik digits are respectively two and three top strokes (𝋊 ten and 𝋏 fifteen) plus two bottom strokes 𝋂.[3]

Map of Alaska highlighting North Slope Borough, part of Iñupiat Nunaat

In the early 1990s, during a math enrichment activity at Harold Kaveolook school in Kaktovik, Alaska,[4] students noted that their language used a base 20 system and found that, when they tried to write numbers or do arithmetic with Arabic numerals, they did not have enough symbols to represent the Iñupiaq numbers.[5] The students first addressed this lack by creating ten extra symbols, but found these were difficult to remember. The middle school in the small town had nine students, so it was possible for the entire class to work together to create a base-20 notation. Their teacher, William Bartley, guided them.[5]

After brainstorming, the students came up with several qualities that an ideal system would have:

1. Visual simplicity: The symbols should be "easy to remember"
2. Iconicity: There should be a "clear relationship between the symbols and their meanings"
3. Efficiency: It should be "easy to write" the symbols, and they should be able to be "written quickly" without lifting the pencil from the paper
4. Distinctiveness: They should "look very different from Arabic numerals," so there would not be any confusion between notation in the two systems
5. Aesthetics: They should be pleasing to look at[5]

In base-20 positional notation, the number twenty is written with the digit for 1 followed by the digit for 0. The Iñupiaq language does not have a word for zero, and the students decided that the Kaktovik digit 0 should look like crossed arms, meaning that nothing was being counted.[5]

When the middle-school pupils began to teach their new system to younger students in the school, the younger students tended to squeeze the numbers down to fit inside the same-sized block. In this way, they created an iconic notation with the sub-base of 5 forming the upper part of the digit, and the remainder forming the lower part. This proved visually helpful in doing arithmetic.[5]

The Kaktovik numerals have gained wide use among Alaskan Iñupiat. They have been introduced into language-immersion programs and have helped revive base-20 counting, which had been falling into disuse among the Iñupiat due to the prevalence of the base-10 system in English-medium schools.[4][5]

When the Kaktovik middle school students who invented the system graduated to the high school in Barrow, Alaska (now renamed Utqiaġvik), in 1995, they took their invention with them. They were permitted to teach it to students at the local middle school, and the local community Iḷisaġvik College added an Inuit mathematics course to its catalog.[5]

In 1996, the Commission on Inuit History Language and Culture officially adopted the numerals,[5] and in 1998 the Inuit Circumpolar Council in Canada recommended the development and use of the Kaktovik numerals in that country.[6]

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.[5]

The development of an indigenous numeral system helps to demonstrate to Alaskan-native students that math is embedded in their culture and language rather than being imparted by western culture. This is a shift from a previously commonly held view that mathematics was merely a necessity to get into college/university. Non-native students can see a practical example of a different world view, a part of ethnomathematics.[7]

The Kaktovik numerals were added to the Unicode Standard in September, 2022, with the release of version 15.0. A font is available in the external links.

• Maya numerals, a penta-vigesimal system from another American culture

• free Kaktovik font, based on Bartley (1997)
• Kaktovik–Arabic converter, using either images or Unicode
• Grunewald, Edgar (December 30, 2019). "Why These Are The Best Numbers!". YouTube. Archived from the original on December 20, 2021. Retrieved December 30, 2019. The video demonstrates how long division is easier with visually intuitive digits like the Kaktovik ones; the illustrated problems were chosen to work out easily, as the problems in a child's introduction to arithmetic would be.
• Silva, Eduardo Marín; Miller, Kirk; Strand, Catherine (April 29, 2021). "Unicode request for Kaktovik numerals (L2/21-058R)" (PDF). Unicode Technical Committee Document Registry. Retrieved April 30, 2021.