Kaktovik numerals


The Kaktovik numerals or Kaktovik Iñupiaq numerals 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. 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

, like other Inuit languages, has a base-20 counting system with a sub-base of 5. That is, quantities are counted in scores, with intermediate numerals for 5, 10, and 15. Thus 78 is identified as three score fifteen-three.
The Kaktovik digits graphically reflect the lexical structure of the Iñupiaq numbering system.
051015
161116
271217
381318
491419

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

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.
nn203n202n201n200n20−1n20−2n20−3
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

The numerals began as an enrichment activity in 1994, when, during a math class exploring binary numbers at Harold Kaveolook middle school on Barter Island Kaktovik, Alaska, students noted that their language used a base-20 system.
They 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.
They first addressed this lack by creating ten extra symbols, but found these were difficult to remember. The small middle school had only nine students so they were all able to work together to create a base-20 notation. Their teacher, William Bartley, guided them.
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.
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.
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.

Computation

Abacus

The students built base-20 abacuses in the school workshop. 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.

Arithmetic

An advantage the students discovered of their new system was that arithmetic was easier than with the Arabic numerals. Adding two digits together would look like their sum. For example,
It was even easier for subtraction: one could simply look at the number and remove the appropriate number of strokes to get the answer. For example,
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 sub-tables for multiplying and subtracting the intermediate steps. The students could keep track of the strokes of the intermediate steps with colored pencils in an elaborated system of chunking.
A simplified multiplication table can be made by first finding the products of each base digit, then the products of the bases and the sub-bases, and finally the product of each sub-base:

1
2
3
4
1
2
3
4


1
2
3
4
5
10
15


5
10
15
5
10
15

These tables are functionally complete for multiplication operations using Kaktovik numerals, but for factors with both bases and sub-bases it is necessary to first disassociate them:
In the above example the factor is not found in the table, but its components, and , are.

Legacy

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.
When the Kaktovik middle school students who invented the system were graduated to the high school in Barrow, Alaska, 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.
In 1996, the Commission on Inuit History Language and Culture officially adopted the numerals, and in 1998 the Inuit Circumpolar Council in Canada recommended the development and use of the Kaktovik numerals in that country.