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: Ī 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: The students could keep track of the strokes of the intermediate steps with colored pencils in an elaborated system of chunking. 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.

For example,Īnother advantage came in doing long division. It was even easier for subtraction: one could simply look at the number and remove the appropriate number of strokes to get the answer. Adding two digits together would look like their sum. 𝋆 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 (zero in the quotient), and into the remaining white digits once.Īn advantage the students discovered of their new system was that arithmetic was easier than with the Arabic numerals. This proved visually helpful in doing arithmetic. In this way, they created a iconic notation with the sub-base of 5 forming the upper part of the digit, and the remainder forming the lower part. 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. 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. In base-20 positional notation, the number twenty is written with the digit for 1 followed by the digit for 0.

Aesthetics: They should be pleasing to look at.

Distinctiveness: They should 'look very different from Arabic numerals,' so there would not be any confusion between notation in the two systems.

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.

Iconicity: There should be a 'clear relationship between the symbols and their meanings'.

Visual simplicity: The symbols should be 'easy to remember'.

Īfter brainstorming, the students came up with several qualities that an ideal system would have: Their teacher, William Bartley, guided them. 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. The students first addressed this lack by creating ten extra symbols, but found these were difficult to remember. In the early 1990s, during a math enrichment activity at Harold Kaveolook school in Kaktovik, Alaska, 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.

Map of Alaska highlighting North Slope Borough, part of Iñupiaq Nunauruat 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. 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: x24px|𝋌.

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). The Kaktovik digits graphically reflect the lexical structure of the Iñupiaq numbering system. Thus 78 is identified as three score fifteen-three. 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. Iñupiaq, like other Inuit languages, has a base-20 counting system with a sub-base of 5.

Subtracting babylonian numerals