Kaktovik Inupiaq numerals
As the Inuit languages use a base-20 counting system, Arabic numeral notation (which is best used for a base-10 counting system) is rendered as inefficient. Students from Kaktovik, Alaska invented a new numbering notation in 1994 to rectify this issue, which has gained wide use among Alaskan Iñupiat, and has been considered in other countries where dialects of the Inuit language are spoken.
The image below shows the numerals 1–19 and then 0. Twenty is written with a one and a zero, forty with a two and a zero, and four hundred with a one and two zeros, etc.
Background and problem
Inuit languages—like some other language groups—use a vigesimal (base-20) counting system, in contrast to decimal numeral system's base-10. Inuit counting has sub-bases at 5, 10, and 15. Arabic numerals, consisting of 10 distinct digits (0-9) are not adequate to represent a base-20 system.
Cultural changes required the Inuit to do long division math, which helped lead to the introduction of a written numerical system.
The far north is a very unforgiving environment, so the numbers are significant to ensure that enough food was collected from surviving the winter. Skins of large animals were grouped in bundles of five, and smaller animals were bundled together in groups of 20, which was how the base-20 nature of the system developed.
Inuit counting also has sub-bases at 5, 10, and 15. Also called quinary (base-5 or pental) this is a numeral system with five as the base. The Kaktovik numerals take these sub-bases into account, as the base of the symbols change after 4, 9 and 14.
As five is a prime number, only the reciprocals of the powers of five terminate, although its location between two highly composite numbers (4 and 6) guarantees that many recurring fractions have relatively short periods.
In 1998, during a math enrichment activity at Harold Kaveolook school in Kaktovik, Alaska, some students mentioned that their own dialect was on a base 20 system, so that when they tried to write the numbers in Arabic they didn't have enough symbols to represent the Iñupiaq numbers.
The students first addressed this by creating ten extra symbols, which made it difficult to remember and were so elaborate that it took a long time to write down the numbers. The entire middle school in the small town had nine students, so it was possible to involve them all in the discussion regarding the creation of the new system. Bartley, who had a more extensive background in linguistics than in mathematics, joined in.
After brainstorming, the students came up with several qualities that the system would have to have:
- The symbols should be "easy to remember".
- There should be a "clear relationship between the symbols and their meanings".
- 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."
- They should "look very different from Arabic numerals," so there would not be any confusion between the two systems.
- They should be pleasing to look at.
When developing the notation, complication came with writing the number 20, which in a vigesimal system would normally be written with a symbol for one followed by a symbol for zero. The Iñupiaq language does not have a word for zero, however. The students decided that zero should look like crossed arms, meaning that nothing was being counted.
When the middle school pupils began to teach their new system to younger students in the school, they noticed a tendency for the younger students to squeeze the numbers down to fit inside of the lines.[clarification needed] In response, the middle students developed number frames to formalize the writing of the numerals, setting a fixed maximum size. In this way, they accidentally invented a full positional numeral system with the bases forming in the top part of the frame. This would also help in the visual aspects of doing long division.
The corresponding spoken forms are:
|tallimat||itchaksrat||tallimat malġuk||tallimat piŋasut||quliŋuġutaiḷaq|
|qulit||qulit atausiq||qulit malġuk||qulit piŋasut||akimiaġutaiḷaq|
|akimiaq||akimiaq atausiq||akimiaq malġuk||akimiaq piŋasut||iñuiññaŋŋutaiḷaq|
The sub-base five shows in the grouping with the post-base of "-gutailaq", which anticipates any number and groups the preceding set of five. This post base means that a number is less than the word stamped which is attached. This can be troublesome at first for those unfamiliar with the system and language, but it does help as a linguistic aid to learn sub-base five mathematics (for example, the number 19 is formed by subtraction from "iñuiññaq" 20, just as nine is formed by subtraction from 10.)
- qulit 
- iñuiññaq 
- iñuiññaq qulit  (20+10)
- malġukipiaq  (2*20)
- malġukipiaq qulit  (2*20+10)
- piŋasukipiaq  (3*20)
- piŋasukipiaq qulit  (3*20+10)
- sisamakipiaq  (4*20)
- quliŋuġutaiḷaq .
One hundred is tallimakipiaq (or qavluun) and 1000 is kavluutit.
Iñupiat count on both their hands and feet. This was a way of keeping track of larger numbers and using the whole body instead of just hands as in the decimal system. For example, the word for five ("tallimat") is derived from the word for "arm" and the word for 10 ("qulit") is derived from the word for "top", meaning both sets of fingers on the top part of the body. The word for 11 ("qulit atausiq") in most Inuit dialects means something like "it goes down" as if it signified starting counting down on the toes. As we advance, the word for 15 ("akimiaq") means something like "it goes across". The number for 20 ("iñuiññaq") in most dialects has something to do with the "entire person" or "one" or "complete person" or "all extremities." 
Doing computing with new symbols
The students that invented the numerals also developed an Iñupiaq abacus in their shop. The abacus helped to convert decimal numbers into the new base-20 numerals. The upper section of the Abacus with three beads representing the sub bases also showed the non-standard positional numeral systems in their upper sectors.
An unusual advantage of this new system was that arithmetic was actually easier than with the Arabic numerals. Adding two symbols together would automatically look like their sum. For example,
It got even easier for subtraction. One could look at the symbol and remove the proper number of legs on the symbol to come to the answer.
Another advantage came in doing long division. The visual aspects and its sub-base five made long division with very large dividends almost as easy as short division problems and didn't require multiplying or subtracting. The students could keep track of the strokes on the paper with colored pencils.
In 1996, the Commission on Inuit History Language and Culture adopted the numerals to represent the numbers in the Inuit language.
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.
As a result, 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.
This dual thinking in base-10 and base-20 might be comparable to advantages that bilingual students have in forming two ways of thinking about the world.
In 1998, 20-month calendars were available with the new numbering system.
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.
The invention of the numeral system showed Alaskan-native students that math was embedded in their own culture and not simply imparted by a western culture. 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 which is a part of ethnomathematics.
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