Kaktovik numerals – Wikipedia
Inuit numeral system for a base-20 counting system
The Kaktovik numerals or Kaktovik Iñupiaq numerals[1] are a base-20 system of numerical digits created by Alaskan Iñupiat. They’re visually iconic, with shapes that point out the quantity being represented.
The Iñupiaq language has a base-20 numeral system, as do the opposite Eskimo–Aleut languages of Alaska and Canada (and previously Greenland). Arabic numerals, which had been designed for a base-10 system, are insufficient for Iñupiaq and different Inuit languages. To treatment this drawback, college students in Kaktovik, Alaska, invented a base-20 numeral notation in 1994, which has unfold among the many Alaskan Iñupiat and has been thought-about to be used in Canada.
The picture right here exhibits the Kaktovik digits 0 to 19. Bigger numbers are composed of those digits in a positional notation: Twenty is written as a one and a zero (
), forty as a two and a zero (
), 4 hundred as a one and two zeros (), eight hundred as a two and two zeros (), and so forth.
Iñupiaq, like different Inuit languages, has a base-20 counting system with a sub-base of 5. That’s, portions are counted in scores (as in Danish, Welsh and in some French numbers resembling quatre-vingts ‘eighty’), with intermediate numerals for five, 10, and 15. Thus 78 is recognized as three rating fifteen-three.[2]
The Kaktovik digits graphically replicate the lexical construction of the Iñupiaq numbering system. For instance, the quantity seven is known as tallimat malġuk in Iñupiaq (‘five-two’), and the Kaktovik digit for seven is a prime stroke (5) linked to 2 backside strokes (two): 
. Equally, twelve and seventeen are referred to as qulit malġuk (‘ten-two’) and akimiaq malġuk (‘fifteen-two’), and the Kaktovik digits are respectively two and three prime strokes (ten and fifteen) with two backside strokes: 
, 
.[3]
Values[edit]
Within the desk are the decimal values of the Kaktovik digits as much as three locations to the left and to the appropriate of the items’ place.[3]
| 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 |
Within the early Nineties, throughout a math enrichment exercise at Harold Kaveolook faculty in Kaktovik, Alaska,[4] college students famous that their language used a base 20 system and located that, after they tried to jot down numbers or do arithmetic with Arabic numerals, they didn’t have sufficient symbols to characterize the Iñupiaq numbers.[5]
The scholars first addressed this lack by creating ten further symbols, however discovered these had been tough to recollect. The center faculty within the small city had 9 college students, so it was attainable for your complete class to work collectively to create a base-20 notation. Their instructor, William Bartley, guided them.[5]
After brainstorming, the scholars got here up with a number of qualities that a super system would have:
- Visible simplicity: The symbols must be “simple to recollect”
- Iconicity: There must be a “clear relationship between the symbols and their meanings”
- Effectivity: It must be “simple to jot down” the symbols, and they need to be capable to be “written shortly” with out lifting the pencil from the paper
- Distinctiveness: They need to “look very totally different from Arabic numerals,” so there wouldn’t be any confusion between notation within the two techniques
- Aesthetics: They need to be pleasing to take a look at[5]
In base-20 positional notation, the quantity twenty is written with the digit for 1 adopted by the digit for 0. The Iñupiaq language doesn’t have a phrase for zero, and the scholars determined that the Kaktovik digit 0 ought to seem like crossed arms, that means that nothing was being counted.[5]
When the middle-school pupils started to show their new system to youthful college students within the faculty, the youthful college students tended to squeeze the numbers down to suit contained in the same-sized block. On this method, they created an iconic notation with the sub-base of 5 forming the higher a part of the digit, and the rest forming the decrease half. This proved visually useful in doing arithmetic.[5]
Computation[edit]
Abacus[edit]
The scholars constructed base-20 abacuses within the faculty workshop.[4][5] These had been initially meant to assist the conversion from decimal to base-20 and vice versa, however the college students discovered their design lent itself fairly naturally to arithmetic in base-20. The higher part of their abacus had three beads in every column for the values of the sub-base of 5, and the decrease part had 4 beads in every column for the remaining items.[5]
Arithmetic[edit]
.svg)
(black) goes into the primary two digits of the dividend (purple) one time, for a one within the quotient (purple). It suits into the following two digits (purple) as soon as if rotated, so the following digit within the quotient (purple) is a one rotated (a 5). The final two digits are matched as soon as for a last one within the quotient (blue)..svg)
goes into the primary three digits of the dividend twice (traced in purple and blue), for a two within the quotient (purple and blue), into the following three as soon as (inexperienced), doesn’t match into the following three digits (thus zero within the quotient), and suits into the remaining pink digits as soon as.A bonus the scholars found of their new system was that arithmetic was simpler than with the Arabic numerals.[5] Including two digits collectively would look like their sum. For instance,
- 2 + 2 = 4
is
+ =
It was even simpler for subtraction: one might merely take a look at the quantity and take away the suitable variety of strokes to get the reply.[5] For instance,
- 4 − 1 = 3
is
- − =
One other benefit got here in doing long division. The visible features and the sub-base of 5 made lengthy division with massive dividends nearly as simple as quick division, because it did not require writing in subtables for multiplying and subtracting the intermediate steps.[4] The scholars might preserve observe of the strokes of the intermediate steps with coloured pencils in an elaborated system of chunking.[5]
A simplified multiplication table might be made by first discovering the merchandise of every base digit, then the merchandise of the bases and the sub-bases, and at last the product of every sub-base:
These tables are functionally full for multiplication operations utilizing Kaktovik numerals, however for components with each bases and sub-bases it’s essential to first disassociate them:
6 * 3 = 18
is
* = ( * ) + ( * ) =
Within the above instance the issue (6) isn’t discovered within the desk, however its parts, (1) and (5), are.
The Kaktovik numerals have gained broad use amongst Alaskan Iñupiat. They’ve been launched into language-immersion applications and have helped revive base-20 counting, which had been falling into disuse among the many Iñupiat as a result of prevalence of the base-10 system in English-medium colleges.[4][5]
When the Kaktovik center faculty college students who invented the system graduated to the highschool in Barrow, Alaska (now renamed Utqiaġvik), in 1995, they took their invention with them. They had been permitted to show it to college students on the native center faculty, and the area people Iḷisaġvik College added an Inuit arithmetic course to its catalog.[5]
In 1996, the Fee on Inuit Historical past Language and Tradition formally adopted the numerals,[5] and in 1998 the Inuit Circumpolar Council in Canada really helpful the event and use of the Kaktovik numerals in that nation.[6]
Significance[edit]
Scores on the California Achievement Test in arithmetic for the Kaktovik center faculty improved dramatically in 1997 in comparison with earlier years. Earlier than the introduction of the brand new numerals, the common rating had been within the twentieth percentile; after their introduction, scores rose to above the nationwide common. It’s theorized that having the ability to work in each base-10 and base-20 may need comparable benefits to these bilingual college students have from participating in two methods of desirous about the world.[5]
The event of an indigenous numeral system helps to show to Alaskan-native college students that math is embedded of their tradition and language slightly than being imparted by western tradition. It is a shift from a beforehand generally held view that arithmetic was merely a necessity to get into school/college. Non-native college students can see a sensible instance of a special world view, part of ethnomathematics.[7]
In Unicode[edit]
The Kaktovik numerals had been added to the Unicode Normal in September, 2022, with the discharge of model 15.0. A font is offered within the external links.
See additionally[edit]
- Maya numerals, a penta-vigesimal system from one other American tradition
References[edit]
- ^ Edna Ahgeak MacLean (2012) Iñupiatun Uqaluit Taniktun Sivunniuġutiŋit: North Slope Iñupiaq to English Dictionary
- ^ MacLean (2014) Iñupiatun Uqaluit Taniktun Sivuninit / Iñupiaq to English Dictionary, p. 840 ff.
- ^ a b MacLean (2014) Iñupiatun Uqaluit Taniktun Sivuninit / Iñupiaq to English Dictionary, p. 832
- ^ 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 unique on June 25, 2013. Retrieved February 27, 2017.
- ^ a b c d e f g h i j k l m n Bartley, William Clark (2002). “Relying on custom: Iñupiaq numbers within the faculty setting”. In Hankes, Judith Elaine; Quick, Gerald R. (eds.). Views on Indigenous Individuals of North America. Altering the Faces of Arithmetic. Reston, Virginia: Nationwide Council of Academics of Arithmetic. pp. 225–236. ISBN 978-0873535069.
- ^ “Regarding Kaktovik Numerals. Resolution 89-09. Inuit Circumpolar Council. 1998”. Archived from the original on February 2, 2017. Retrieved February 2, 2017.
- ^ Engblom-Bradley, Claudette (2009). “Seeing arithmetic with Indian eyes”. In Williams, Maria Sháa Tláa (ed.). The Alaska Native Reader: Historical past, Tradition, Politics. Duke College Press. pp. 237–245. ISBN 9780822390831. See specifically p. 244.
Exterior hyperlinks[edit]
- free Kaktovik font, based mostly on Bartley (1997)
- Grunewald, Edgar (December 30, 2019). “Why These Are The Best Numbers!”. YouTube. Archived from the unique on December 20, 2021. Retrieved December 30, 2019. The video demonstrates how lengthy division is simpler with visually intuitive digits just like the Kaktovik ones; the illustrated issues had been chosen to work out simply, as the issues in a toddler’s introduction to arithmetic can be.
- Silva, Eduardo Marín; Miller, Kirk; Strand, Catherine (April 29, 2021). “Unicode request for Kaktovik numerals (L2/21-058R)” (PDF). Unicode Technical Committee Doc Registry. Retrieved April 30, 2021.