Kaktovik numerals

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,

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. 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 did not 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.

Examples of long division

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:

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

* = ( * ) + ( * ) =

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

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.

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.

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 a college or university. Non-native students can see a practical example of a different world view, a part of ethnomathematics.

In Unicode

The Kaktovik numerals were added to the Unicode Standard in September, 2022, with the release of version 15.0. Several fonts support this block.