Kaktovik numerals

Kaktovik numerals are a featural positional numeral system created by Alaskan Iñupiat.

The 20 digits of the Kaktovik system

Arabic numeral notation, which was designed for a base-10 numeral system, is inadequate for the Inuit languages, which use a base-20 numeral system. Students in Kaktovik, Alaska, invented a base-20 numeral notation in 1994 to rectify this issue,[1] and this system spread among the Alaskan Iñupiat and has been considered in other countries where Inuit languages are spoken.[2]

The image at right shows the digits 0 to 19. Twenty is written as a one and a zero (\ɤ), forty as a two and a zero (Vɤ), four hundred as a one and two zeros (\ɤɤ), eight hundred as a two and two zeros (Vɤɤ), etc.

System
See also: Inupiaq language § Numerals

Iñupiaq, like other Inuit languages, has a base-20 counting system with a sub-base of 5. That is, scores are indicated by multiplication (as in French or Danish) with additional numerals for 5, 10, and 15. Arabic numerals, consisting of 10 distinct digits (0-9), are not appropriate for a base-20 system.

Development

In the early 1990s, during a math enrichment activity at Harold Kaveolook school in Kaktovik, Alaska,[1] students said that their language used a base 20 system and that when they tried to write numbers with Arabic numerals, they didn't have enough symbols to represent the Iñupiaq numbers.[3]

Map of Alaska highlighting North Slope Borough, part of Iñupiaq Nunauruat

The students first addressed this by creating ten extra symbols, which made it difficult to remember, and elaborated that it took a long time to write down the numbers. The middle school in the small town had nine students, so it was possible to involve them in the discussion regarding creating the new system. The teacher, William Bartley, helped.[3]

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

  1. The symbols should be "easy to remember."
  2. There should be a "clear relationship between the symbols and their meanings."
  3. It should be "easy to write" the symbols. For example, being able to be written without lifting the pencil and should be able to be "written quickly."
  4. They should "look very different from Arabic numerals," so there would not be any confusion between the two systems.
  5. They should be pleasing to look at.[3]

In base-20 positional notation, the number for 20 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 digit 0 should look like crossed arms, meaning that nothing was being counted.[3]

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 a notation with the sub-base forming the top part of the digit. This proved visually helpful in doing arithmetic.[3]

Doing computing with new symbols

Abacus
Traditional abacus showing 52 in decimal
Inupiaq abacus to use with the Kaktovik numerals

The students also developed an Iñupiaq abacus in their shop.[1][4] The abacus helped to convert decimal numbers into the new base-20 numerals. The upper section of the Abacus with three beads represents the sub bases also shows the non-standard positional numeral systems in their upper sectors.[3]

Arithmetic

An unusual advantage of this new system was that arithmetic was actually easier than with the Arabic numerals.[3] Adding two symbols together would automatically look like their sum. For example,

is



It was even easier for subtraction. One could look at the symbol and remove the proper number of legs on the symbol to answer.[3]

Another advantage came in doing long division. The visual aspects and its sub-base five made long division with huge dividends almost as easy as short division problems and didn't require multiplying or subtracting.[5] The students could keep track of the strokes on the paper with colored pencils.[3]

Cuisenaire rods such as those used in the Montessori method were developed to help and teach the system to the younger students. Popsicle sticks and rubber bands represented the sub bases.[3]

The students continued to make discoveries. For example, one discovered complements of sets by seeing what was missing visually in the image of the numbers.[3]

One student discovered set theory on his own

Legacy

The numeral system has helped revive counting in Inuit languages, which had been falling into disuse among Inuit speakers due to the prevalence of the base-10 system in schools.[1][4]

In 1996, the Commission on Inuit History Language and Culture adopted the numerals to represent the Inuit language numbers.[3]

In 1995, the middle school students moved over to the high school in Barrow (now renamed Utqiagvik), Alaska, and took their invention with them. The high school students were permitted to teach the middle school students this system, the local community Iḷisaġvik College added an Inuit mathematics course to its catalog.[3]

In 1997, the student scores in the middle school on the California Achievement Test in mathematics, which was used to measure student success, increased dramatically. Previously, the average score was in the 20th percentile, and after the introduction of the new numerals, the scores rose to be above the national average.[3]

This dual thinking in base-10 and base-20 might be comparable to the advantages that bilingual students have in forming two ways of thinking about the world.[3]

In 1998, 20-month calendars were available with the new numbering system.[6]

The system has since gained wide use among Alaskan Iñupiat and has been considered in other countries where dialects of the Inuit language are spoken.[2]

Significance

This numeral system's development showed Alaskan-native students that math was embedded in their own culture and not simply imparted by western culture.[7] Those students going on to college saw studying mathematics as a necessity to get into college. Also, non-native students can see a practical example of a different world view, a part of ethnomathematics.[7]

Encoding

The Kaktovik numerals have been provisionally assigned a block in the Unicode Supplementary Multilingual Plane (U+1D2C0-1D2DF).

References
  1. Bartley, Wm. Clark (January–February 1997). "Making the Old Way Count" (PDF). Sharing Our Pathways. 2 (1): 12–13. Archived (PDF) from the original on June 25, 2013. Retrieved February 27, 2017.
  2. Regarding Kaktovik Numerals. Resolution 89-09. Inuit Circumpolar Council. 1998. http://www.inuitcircumpolar.com/resolutions7.html Archived February 2, 2017, at the Wayback Machine
  3. Hankes, Judith Elaine; Fast, Gerald R. (2002). Changing the Faces of Mathematics. pp. 225–235. ISBN 978-0873535069.
  4. Hankes, Judith Elaine; Fast, Gerald R. (2002). Perspectives on Indigenous People of North America. p. 255. ISBN 978-0873535069.
  5. Grunewald, Edgar (December 30, 2019). "Why These Are The Best Numbers!". YouTube. Retrieved December 30, 2019.
  6. Noble, Abbey (February 28, 1998). "Native Numbers". New Moon. p. 36.
  7. Engblom-Bradley, Claudette (January 1, 2009). The Alaska Native Reader. p. 244. ISBN 9780822390831.
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