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Kaktovik numerals

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The 20 digits of the Kaktovik system
The 20 digits of the Kaktovik system

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.

The image here shows the Kaktovik digits 0 to 19. Larger numbers are composed of these digits in a positional notation: Twenty is written as a one and a zero (𝋁𝋀), forty as a two and a zero (𝋂𝋀), four hundred as a one and two zeros (𝋁𝋀𝋀), eight hundred as a two and two zeros (𝋂𝋀𝋀), and so on.

Discover more about Kaktovik numerals related topics

Numerical digit

Numerical digit

A numerical digit is a single symbol used alone or in combinations, to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits of the hands correspond to the ten symbols of the common base 10 numeral system, i.e. the decimal digits.

Iñupiat

Iñupiat

The Iñupiat are a group of indigenous Alaskans whose traditional territory roughly spans northeast from Norton Sound on the Bering Sea to the northernmost part of the Canada–United States border. Their current communities include 34 villages across Iñupiat Nunaat, including seven Alaskan villages in the North Slope Borough, affiliated with the Arctic Slope Regional Corporation; eleven villages in Northwest Arctic Borough; and sixteen villages affiliated with the Bering Straits Regional Corporation. They often claim to be the first people of the Kauwerak.

Iconicity

Iconicity

In functional-cognitive linguistics, as well as in semiotics, iconicity is the conceived similarity or analogy between the form of a sign and its meaning, as opposed to arbitrariness. The principle of iconicity is also shared by the approach of linguistic typology.

Iñupiaq numerals

Iñupiaq numerals

The following are the vigesimal numerals of Iñupiaq language, excepting higher derivatives in -utaiḷaq, which subtracts one from the value. They are transcribed in vigesimal Kaktovik digits and decimal Hindu-Arabic digits. Apart from the subtractive suffix -utaiḷaq, which has no counterpart in Kaktovik notation, and the idiosyncratic case of itchaksrat 'six', Iñupiaq numerals are closely represented by Kaktovik digits.

Arabic numerals

Arabic numerals

Arabic numerals are the ten symbols most commonly used to write decimal numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as computer symbols, trademarks, or license plates. The term often implies a decimal number, in particular when contrasted with Roman numerals.

Kaktovik, Alaska

Kaktovik, Alaska

Kaktovik is a city in North Slope Borough, Alaska, United States. The population was 283 at the 2020 census.

Positional notation

Positional notation

Positional notation usually denotes the extension to any base of the Hindu–Arabic numeral system. More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early numeral systems, such as Roman numerals, a digit has only one value: I means one, X means ten and C a hundred. In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string.

System

Iñupiaq, like other Inuit languages, has a base-20 counting system with a sub-base of 5. That is, quantities are counted in scores (as in Danish, Welsh and in some French numbers 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. For example, the number seven is called tallimat malġuk in Iñupiaq ('five-two'), and the Kaktovik digit for seven is a top stroke (five) connected to two bottom strokes (two): 𝋇. Similarly, twelve and seventeen are called qulit malġuk ('ten-two') and akimiaq malġuk ('fifteen-two'), and the Kaktovik digits are respectively two and three top strokes (ten and fifteen) with two bottom strokes: 𝋌, 𝋑.[3]

Values

In the 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 𝋁,𝋀𝋀𝋀
8,000
𝋁𝋀𝋀
400
𝋁𝋀
20
𝋁
1
𝋀.𝋁
0.05
𝋀.𝋀𝋁
0.0025
𝋀.𝋀𝋀𝋁
0.000125
2 𝋂,𝋀𝋀𝋀
16,000
𝋂𝋀𝋀
800
𝋂𝋀
40
𝋂
2
𝋀.𝋂
0.1
𝋀.𝋀𝋂
0.005
𝋀.𝋀𝋀𝋂
0.00025
3 𝋃,𝋀𝋀𝋀
24,000
𝋃𝋀𝋀
1,200
𝋃𝋀
60
𝋃
3
𝋀.𝋃
0.15
𝋀.𝋀𝋃
0.0075
𝋀.𝋀𝋀𝋃
0.000375
4 𝋄,𝋀𝋀𝋀
32,000
𝋄𝋀𝋀
1,600
𝋄𝋀
80
𝋄
4
𝋀.𝋄
0.2
𝋀.𝋀𝋄
0.01
𝋀.𝋀𝋀𝋄
0.0005
5 𝋅,𝋀𝋀𝋀
40,000
𝋅𝋀𝋀
2,000
𝋅𝋀
100
𝋅
5
𝋀.𝋅
0.25
𝋀.𝋀𝋅
0.0125
𝋀.𝋀𝋀𝋅
0.000625
6 𝋆,𝋀𝋀𝋀
48,000
𝋆𝋀𝋀
2,400
𝋆𝋀
120
𝋆
6
𝋀.𝋆
0.3
𝋀.𝋀𝋆
0.015
𝋀.𝋀𝋀𝋆
0.00075
7 𝋇,𝋀𝋀𝋀
56,000
𝋇𝋀𝋀
2,800
𝋇𝋀
140
𝋇
7
𝋀.𝋇
0.35
𝋀.𝋀𝋇
0.0175
𝋀.𝋀𝋀𝋇
0.000875
8 𝋈,𝋀𝋀𝋀
64,000
𝋈𝋀𝋀
3,200
𝋈𝋀
160
𝋈
8
𝋀.𝋈
0.4
𝋀.𝋀𝋈
0.02
𝋀.𝋀𝋀𝋈
0.001
9 𝋉,𝋀𝋀𝋀
72,000
𝋉𝋀𝋀
3,600
𝋉𝋀
180
𝋉
9
𝋀.𝋉
0.45
𝋀.𝋀𝋉
0.0225
𝋀.𝋀𝋀𝋉
0.001125
10 𝋊,𝋀𝋀𝋀
80,000
𝋊𝋀𝋀
4,000
𝋊𝋀
200
𝋊
10
𝋀.𝋊
0.5
𝋀.𝋀𝋊
0.025
𝋀.𝋀𝋀𝋊
0.00125
11 𝋋,𝋀𝋀𝋀
88,000
𝋋𝋀𝋀
4,400
𝋋𝋀
220
𝋋
11
𝋀.𝋋
0.55
𝋀.𝋀𝋋
0.0275
𝋀.𝋀𝋀𝋋
0.001375
12 𝋌,𝋀𝋀𝋀
96,000
𝋌𝋀𝋀
4,800
𝋌𝋀
240
𝋌
12
𝋀.𝋌
0.6
𝋀.𝋀𝋌
0.03
𝋀.𝋀𝋀𝋌
0.0015
13 𝋍,𝋀𝋀𝋀
104,000
𝋍𝋀𝋀
5,200
𝋍𝋀
260
𝋍
13
𝋀.𝋍
0.65
𝋀.𝋀𝋍
0.0325
𝋀.𝋀𝋀𝋍
0.001625
14 𝋎,𝋀𝋀𝋀
112,000
𝋎𝋀𝋀
5,600
𝋎𝋀
280
𝋎
14
𝋀.𝋎
0.7
𝋀.𝋀𝋎
0.035
𝋀.𝋀𝋀𝋎
0.00175
15 𝋏,𝋀𝋀𝋀
120,000
𝋏𝋀𝋀
6,000
𝋏𝋀
300
𝋏
15
𝋀.𝋏
0.75
𝋀.𝋀𝋏
0.0375
𝋀.𝋀𝋀𝋏
0.001875
16 𝋐,𝋀𝋀𝋀
128,000
𝋐𝋀𝋀
6,400
𝋐𝋀
320
𝋐
16
𝋀.𝋐
0.8
𝋀.𝋀𝋐
0.04
𝋀.𝋀𝋀𝋐
0.002
17 𝋑,𝋀𝋀𝋀
136,000
𝋑𝋀𝋀
6,800
𝋑𝋀
340
𝋑
17
𝋀.𝋑
0.85
𝋀.𝋀𝋑
0.0425
𝋀.𝋀𝋀𝋑
0.002125
18 𝋒,𝋀𝋀𝋀
144,000
𝋒𝋀𝋀
7,200
𝋒𝋀
360
𝋒
18
𝋀.𝋒
0.9
𝋀.𝋀𝋒
0.045
𝋀.𝋀𝋀𝋒
0.00225
19 𝋓,𝋀𝋀𝋀
152,000
𝋓𝋀𝋀
7,600
𝋓𝋀
380
𝋓
19
𝋀.𝋓
0.95
𝋀.𝋀𝋓
0.0475
𝋀.𝋀𝋀𝋓
0.002375

Discover more about System related topics

Iñupiaq numerals

Iñupiaq numerals

The following are the vigesimal numerals of Iñupiaq language, excepting higher derivatives in -utaiḷaq, which subtracts one from the value. They are transcribed in vigesimal Kaktovik digits and decimal Hindu-Arabic digits. Apart from the subtractive suffix -utaiḷaq, which has no counterpart in Kaktovik notation, and the idiosyncratic case of itchaksrat 'six', Iñupiaq numerals are closely represented by Kaktovik digits.

Inuit languages

Inuit languages

The Inuit languages are a closely related group of indigenous American languages traditionally spoken across the North American Arctic and adjacent subarctic, reaching farthest south in Labrador. The related Yupik languages are the two main branches of Eskaleut, a primary language family. The Inuit live primarily in three countries: Greenland, Canada, and the United States.

Vigesimal

Vigesimal

A vigesimal or base-20 (base-score) numeral system is based on twenty. Vigesimal is derived from the Latin adjective vicesimus, meaning 'twentieth'.

Quinary

Quinary

Quinary is a numeral system with five as the base. A possible origination of a quinary system is that there are five digits on either hand.

20 (number)

20 (number)

20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score.

Danish language

Danish language

Danish is a North Germanic language spoken by about six million people, principally in and around Denmark. Communities of Danish speakers are also found in Greenland, the Faroe Islands, and the northern German region of Southern Schleswig, where it has minority language status. Minor Danish-speaking communities are also found in Norway, Sweden, the United States, Canada, Brazil, and Argentina.

Welsh language

Welsh language

Welsh is a Celtic language of the Brittonic subgroup that is native to the Welsh people. Welsh is spoken natively in Wales, by some in England, and in Y Wladfa. Historically, it has also been known in English as "British", "Cambrian", "Cambric" and "Cymric".

Origin

Map of Alaska highlighting North Slope Borough, part of Iñupiat Nunaat
Map of Alaska highlighting North Slope Borough, part of Iñupiat Nunaat

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

Iñupiaq abacus designed for use with the Kaktovik numerals
Iñupiaq abacus designed for use with the Kaktovik numerals

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

Simple long division: 30,561 ÷ 61 = 501 (vigesimal 3,G81 ÷ 31 = 151). The divisor  (black) goes into the first two digits of the dividend (purple) one time, for a one in the quotient (purple). It fits into the next two digits (red) once if rotated, so the next digit in the quotient (red) is a one rotated (a five). The last two digits are matched once for a final one in the quotient (blue).
Simple long division: 30,561 ÷ 61 = 501 (vigesimal 3,G81 ÷ 31 = 151). The divisor Simple long division: 30,561 ÷ 61 = 501 (vigesimal 3,G81 ÷ 31 = 151). The divisor  (black) goes into the first two digits of the dividend (purple) one time, for a one in the quotient (purple). It fits into the next two digits (red) once if rotated, so the next digit in the quotient (red) is a one rotated (a five). The last two digits are matched once for a final one in the quotient (blue).Simple long division: 30,561 ÷ 61 = 501 (vigesimal 3,G81 ÷ 31 = 151). The divisor  (black) goes into the first two digits of the dividend (purple) one time, for a one in the quotient (purple). It fits into the next two digits (red) once if rotated, so the next digit in the quotient (red) is a one rotated (a five). The last two digits are matched once for a final one in the quotient (blue). (black) goes into the first two digits of the dividend (purple) one time, for a one in the quotient (purple). It fits into the next two digits (red) once if rotated, so the next digit in the quotient (red) is a one rotated (a five). The last two digits are matched once for a final one in the quotient (blue).
Long division with more chunking: 46,349,226 ÷ 2,826 = 16,401 (vigesimal E9D,D16 ÷ 716 = 2,101). The divisor  goes into the first three digits of the dividend twice (traced in red and blue), for a two in the quotient (red and blue), into the next three once (green), does not fit into the next three digits (thus zero in the quotient), and fits into the remaining pink digits once.
Long division with more chunking: 46,349,226 ÷ 2,826 = 16,401 (vigesimal E9D,D16 ÷ 716 = 2,101). The divisor Long division with more chunking: 46,349,226 ÷ 2,826 = 16,401 (vigesimal E9D,D16 ÷ 716 = 2,101). The divisor  goes into the first three digits of the dividend twice (traced in red and blue), for a two in the quotient (red and blue), into the next three once (green), does not fit into the next three digits (thus zero in the quotient), and fits into the remaining pink digits once.Long division with more chunking: 46,349,226 ÷ 2,826 = 16,401 (vigesimal E9D,D16 ÷ 716 = 2,101). The divisor  goes into the first three digits of the dividend twice (traced in red and blue), for a two in the quotient (red and blue), into the next three once (green), does not fit into the next three digits (thus zero in the quotient), and fits into the remaining pink digits once.Long division with more chunking: 46,349,226 ÷ 2,826 = 16,401 (vigesimal E9D,D16 ÷ 716 = 2,101). The divisor  goes into the first three digits of the dividend twice (traced in red and blue), for a two in the quotient (red and blue), into the next three once (green), does not fit into the next three digits (thus zero in the quotient), and fits into the remaining pink digits once. goes into the first three digits of the dividend twice (traced in red and blue), for a two in the quotient (red and blue), into the next three once (green), does not fit into the next three digits (thus zero in the quotient), and fits into the remaining pink digits once.

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

is

𝋂 + 𝋂 = 𝋄

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

is

𝋄𝋁 = 𝋃


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]

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:

× 𝋁
1
𝋂
2
𝋃
3
𝋄
4
× 𝋁
1
𝋂
2
𝋃
3
𝋄
4
× 𝋅
5
𝋊
10
𝋏
15
1 𝋁 𝋁 𝋂 𝋃 𝋄 5 𝋅 𝋅 𝋊 𝋏 𝋁𝋀 5 𝋅 𝋁𝋅 𝋂𝋊 𝋃𝋏
2 𝋂 𝋂 𝋄 𝋆 𝋈 10 𝋊 𝋊 𝋁𝋀 𝋁𝋊 𝋂𝋀 10 𝋊 𝋂𝋊 𝋅𝋀 𝋇𝋊
3 𝋃 𝋃 𝋆 𝋉 𝋌 15 𝋏 𝋏 𝋁𝋊 𝋂𝋅 𝋃𝋀 15 𝋏 𝋃𝋏 𝋇𝋊 𝋋𝋅
4 𝋄 𝋄 𝋈 𝋌 𝋐

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:

6 * 3 = 18

is

𝋆 * 𝋃 = (𝋁 * 𝋃) + (𝋅 * 𝋃) = 𝋒

In the above example the factor 𝋆 (6) is not found in the table, but its components, 𝋁 (1) and 𝋅 (5), are.

Discover more about Computation related topics

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.[4][5]

When the Kaktovik middle school students who invented the system graduated to the high school in Barrow, Alaska (now renamed Utqiaġvik), 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.[5]

In 1996, the Commission on Inuit History Language and Culture officially adopted the numerals,[5] and in 1998 the Inuit Circumpolar Council in Canada recommended the development and use of the Kaktovik numerals in that country.[6]

Significance

Scores on the California Achievement Test in mathematics for the Kaktovik middle school improved dramatically in 1997 compared to previous years. Before the introduction of the new numerals, the average score had been in the 20th percentile; after their introduction, scores rose to above the national average. It is theorized that being able to work in both base-10 and base-20 might have comparable advantages to those bilingual students have from engaging in two ways of thinking about the world.[5]

The development of an indigenous numeral system helps to demonstrate to Alaskan-native students that math is embedded in their culture and language rather than being imparted by western culture. This is a shift from a previously commonly held view that mathematics was merely a necessity to get into college/university. Non-native students can see a practical example of a different world view, a part of ethnomathematics.[7]

In Unicode

The Kaktovik numerals were added to the Unicode Standard in September, 2022, with the release of version 15.0. A font is available in the external links.

Kaktovik Numerals[1][2]
Official Unicode Consortium code chart (PDF)
  0 1 2 3 4 5 6 7 8 9 A B C D E F
U+1D2Cx 𝋀 𝋁 𝋂 𝋃 𝋄 𝋅 𝋆 𝋇 𝋈 𝋉 𝋊 𝋋 𝋌 𝋍 𝋎 𝋏
U+1D2Dx 𝋐 𝋑 𝋒 𝋓
Notes
1.^ As of Unicode version 15.0
2.^ Grey areas indicate non-assigned code points
Graphical display of Kaktovik Numerals
Official Unicode Consortium code chart (PDF)
  0 1 2 3 4 5 6 7 8 9 A B C D E F
U+1D2Cx 𝋀 𝋁 𝋂 𝋃 𝋄 𝋅 𝋆 𝋇 𝋈 𝋉 𝋊 𝋋 𝋌 𝋍 𝋎 𝋏
U+1D2Dx 𝋐 𝋑 𝋒 𝋓

Source: "Kaktovik numerals", Wikipedia, Wikimedia Foundation, (2023, January 27th), https://en.wikipedia.org/wiki/Kaktovik_numerals.

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See also
  • Maya numerals, a penta-vigesimal system from another American culture
References
  1. ^ Edna Ahgeak MacLean (2012) Iñupiatun Uqaluit Taniktun Sivunniuġutiŋit: North Slope Iñupiaq to English Dictionary
  2. ^ MacLean (2014) Iñupiatun Uqaluit Taniktun Sivuninit / Iñupiaq to English Dictionary, p. 840 ff.
  3. ^ a b MacLean (2014) Iñupiatun Uqaluit Taniktun Sivuninit / Iñupiaq to English Dictionary, p. 832
  4. ^ a b c d 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.
  5. ^ a b c d e f g h i j k l m n Bartley, William Clark (2002). "Counting on tradition: Iñupiaq numbers in the school setting". In Hankes, Judith Elaine; Fast, Gerald R. (eds.). Perspectives on Indigenous People of North America. Changing the Faces of Mathematics. Reston, Virginia: National Council of Teachers of Mathematics. pp. 225–236. ISBN 978-0873535069.
  6. ^ "Regarding Kaktovik Numerals. Resolution 89-09. Inuit Circumpolar Council. 1998". Archived from the original on February 2, 2017. Retrieved February 2, 2017.
  7. ^ Engblom-Bradley, Claudette (2009). "Seeing mathematics with Indian eyes". In Williams, Maria Sháa Tláa (ed.). The Alaska Native Reader: History, Culture, Politics. Duke University Press. pp. 237–245. ISBN 9780822390831. See in particular p. 244.
External links
  • free Kaktovik font, based on Bartley (1997)
  • Grunewald, Edgar (December 30, 2019). "Why These Are The Best Numbers!". YouTube. Archived from the original on December 20, 2021. Retrieved December 30, 2019. The video demonstrates how long division is easier with visually intuitive digits like the Kaktovik ones; the illustrated problems were chosen to work out easily, as the problems in a child's introduction to arithmetic would be.
  • Silva, Eduardo Marín; Miller, Kirk; Strand, Catherine (April 29, 2021). "Unicode request for Kaktovik numerals (L2/21-058R)" (PDF). Unicode Technical Committee Document Registry. Retrieved April 30, 2021.

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