Kaktovik Inupiaq numerals
The Inuit languages use a base-20 counting system. Arabic numeral notation is best used for a base-10 counting system. Students from Kaktovik, Alaska invented a new numbering notation in 1994 which has gained wide use among Alaskan Iñupiat, and has been considered in other countries where dialects of the Inuit language are spoken.
Background and problem
Inuit languages, like other Eskimo–Aleut languages (and several other languages such as Celtic and Mayan languages), use a vigesimal or base 20 counting system. The decimal numeral system use a base ten. 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.
How far a numbering system goes depends on the need in the culture. Normally it was not needed to count higher than 50 or 100 later when reindeer herding was introduced the herders needed larger numbers. Again later the Inuit needed to do long division math helped lead to the introduction of a written numerical system.
The far north is a very unforgiving environment the numbers are very important to ensure that enough food had been collected to survive the winter. Skins of a large animals were grouped in bundles of five and smaller animals were bundled together in groups of 20. 
Inuit counting 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 numeral system considered this as 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.
William Clark Bartley's education was in Anthropology and a background in Latin, Greek, and Sanskrit. He had a Master's degree in linguistics. With no background in math he supported the development of this new system. While a teacher in September 1994 at Harold Kaveolook school in Kaktovik, Alaska Bartley was teaching the middle school kids math. The students mentioned that their own dialect was on a base 20 and 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 10 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.
After brainstorming the students came up with a number of 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 written without lifting the pencil and easy to write quickly.
- They should look very different from Arabic numerals so there would not be any confusion.
- They should be pleasing to look at.
It is possible that the Iñupiaq have very high visual perceptual skills and this might be the connection why the numeral system is also very symbolic visually.
A complication came with writing the number 20. Since it was a base for the system, 20 should be written with a symbol for one and zero but the Iñupiaq language does not have a word for zero. Again, going back to their language that the counting had been done on fingers or toes  the students decided that zero should look like crossed arms meaning that nothing was being counted.
When the middle school kids went to teach their new system two younger students in the school they noticed a tendency for the younger students to squeeze the numbers down to fit inside of the lines. In response, the middle students went back to create number frames to help formalize the system in how to write the numerals. in this way they accidentally invented a positional numeral system with the bases forming in the top part of the frame. A positional system is a numeral system in which the contribution of a digit to the value of a number is the product of the value of the digit by a factor determined by the position of the digit.
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 subbase five shows in the grouping with this postbase 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 but it does help as a linguistic aid to learn the sub base 5 math
(19 is formed by subtraction from iñuiññaq 20, just as 9 is formed by subtraction from 10. See Inupiat language.)
Subtraction continues along the lines of French or Danish for the tens.
- 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.
In Greenlandic Inuit language:
Iñupiat count on both their hands and feet. This was certainly a way of keeping track 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 it now we start counting down on the toes. Going forward 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 developed a Iñupiaq abacus in their shop. The abacus also 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 subbase 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, this middle school invention moved over to the high school in Barrow (now renamed Utqiagvik), Alaska. The local community Iḷisaġvik College added an Inuit mathematics course to its catalog.
In 1997, the middle school student scores on the California achievement test in mathematics increased dramatically. Previously, the average score was in the 20th percentile and one year later their scores rose to be above the national average.
This dual thinking in base 10 and based 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 culture and not just a product of 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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