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The Kaktovik numerals or Kaktovik Iñupiaq numerals^[1] are a base-20 system of numerical digits created by Alaskan Iñupiat. They are visually iconic, with shapes that indicate the number being represented.

The Iñupiaq language has a base-20 numeral system, as do the other Eskimo–Aleut languages of Alaska and Canada (and formerly Greenland). Arabic numerals, which were designed for a base-10 system, are inadequate for Iñupiaq and other Inuit languages. To remedy this problem, students in Kaktovik, Alaska, invented a base-20 numeral notation in 1994, which has spread among the Alaskan Iñupiat and has been considered for use in Canada.

System

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

The Kaktovik digits graphically reflect the lexical structure of the Iñupiaq numbering system.^[3]

The twenty digits

0		kisitchisaġvik	5		tallimat	10		qulit	15		akimiaq
1		atausiq	6		itchaksrat	11		qulit atausiq	16		akimiaq atausiq
2		malġuk	7		tallimat malġuk	12		qulit malġuk	17		akimiaq malġuk
3		piŋasut	8		tallimat piŋasut	13		qulit piŋasut	18		akimiaq piŋasut
4		sisamat	9		quliŋŋuġutaiḷaq	14		akimiaġutaiḷaq	19		iñuiññaġutaiḷaq

Larger numbers are composed of these digits in a positional notation:

Decimal	Vigesimal
Arabic	Arabic	Kaktovik
20	1020
40	2020
400	10020
800	20020

Values

In the following table are the decimal values of the Kaktovik digits up to three places to the left and to the right of the units' place.^[3]

Decimal values of Kaktovik numbers

n	n × 203	n × 202	n × 201	n × 200	n × 20−1	n × 20−2	n × 20−3
1	, 8,000	400	20	1	. 0.05	. 0.0025	. 0.000125
2	, 16,000	800	40	2	. 0.1	. 0.005	. 0.00025
3	, 24,000	1,200	60	3	. 0.15	. 0.0075	. 0.000375
4	, 32,000	1,600	80	4	. 0.2	. 0.01	. 0.0005
5	, 40,000	2,000	100	5	. 0.25	. 0.0125	. 0.000625
6	, 48,000	2,400	120	6	. 0.3	. 0.015	. 0.00075
7	, 56,000	2,800	140	7	. 0.35	. 0.0175	. 0.000875
8	, 64,000	3,200	160	8	. 0.4	. 0.02	. 0.001
9	, 72,000	3,600	180	9	. 0.45	. 0.0225	. 0.001125
10	, 80,000	4,000	200	10	. 0.5	. 0.025	. 0.00125
11	, 88,000	4,400	220	11	. 0.55	. 0.0275	. 0.001375
12	, 96,000	4,800	240	12	. 0.6	. 0.03	. 0.0015
13	, 104,000	5,200	260	13	. 0.65	. 0.0325	. 0.001625
14	, 112,000	5,600	280	14	. 0.7	. 0.035	. 0.00175
15	, 120,000	6,000	300	15	. 0.75	. 0.0375	. 0.001875
16	, 128,000	6,400	320	16	. 0.8	. 0.04	. 0.002
17	, 136,000	6,800	340	17	. 0.85	. 0.0425	. 0.002125
18	, 144,000	7,200	360	18	. 0.9	. 0.045	. 0.00225
19	, 152,000	7,600	380	19	. 0.95	. 0.0475	. 0.002375

Origin

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]

Computation

Abacus

The students built base-20 abacuses in the school workshop.^[4][5] These were initially intended to help the conversion from decimal to base-20 and vice versa, but the students found their design lent itself quite naturally to arithmetic in base-20. The upper section of their abacus had three beads in each column for the values of the sub-base of 5, and the lower section had four beads in each column for the remaining units.^[5]

Arithmetic

An advantage the students discovered of their new system was that arithmetic was easier than with the Arabic numerals.^[5] Adding two digits together would look like their sum. For example,

2 + 2 = 4

+ =

It was even easier for subtraction: one could simply look at the number and remove the appropriate number of strokes to get the answer.^[5] For example,

4 − 1 = 3

− =

Another advantage came in doing long division. The visual aspects and the sub-base of five made long division with large dividends almost as easy as short division, as it didn't require writing in subtables for multiplying and subtracting the intermediate steps.^[4] The students could keep track of the strokes of the intermediate steps with colored pencils in an elaborated system of chunking.^[5]