% ConTeXT file
% filename: kaktovik-numerals-intro.tex
% W. Egger
% 19-04-2023

\input meta-imp-kaktovik.mkxl

\definefontfeature
  [default]
  [default]
  [metapost=kaktovik]

\setupbodyfont[dejavu]

\setuppagenumbering[location=footer]

\setuplayout
 [header=0pt]


\starttext
\starttitle[title=Kaktovik-numerals]

\startsubject[title=Intro]
On 18-04-2023 Haraban Ramm posted on the mailing-list about the Kaktovik numerals.
\blank
https://mathstodon.xyz/@johncarlosbaez/110215432175491555
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The Inuit have a counting system based on the base 20. Back in 1994 in the local school at Kaktovik in Alaska students developed a counting system for the local language on the base 20. There is a subunit in the form of 5.

The counting system with base 20 is known in different cultures. e.g. The Mayans (Mexico) used a system with dots and horizontal bars.

Even in modern languages there are traces of the base 20 counting system. e.g. French express 80 as \quote{quatre-vingt} and 90 as \quote{quatre-vingt-dix}...

\stopsubject

\startsubject[title=Hans Hagen's implementation]

A couple of hours after Hraban's post, Hans came already up with an implementation of the Kaktovik-numeral-shapes in \METAFUN.

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meta-imp-kaktovik.mkxl
\stopsubject

\startsubject[title=Further information on the Kaktovik-numerals-system]

https://www.youtube.com/watch?v=EyS6FfczH0Q&ab_channel=Artifexian

https://www.youtube.com/watch?v=fIZB4bRwxqI&ab_channel=Dave%27sMathVideos

https://www.youtube.com/watch?v=ObRFHiU_r9I&ab_channel=TheFerret

\stopsubject

\startsubject[title=The key to writing numbers in Kaktovik system]

In order to understand how numbers are composed hereunder is the way in the decimal and in the Kaktovik system.



\bTABLE
  \bTR
	  \bTD[nx=4,align=center] Decimal \eTD
		\bTD \strut \eTD
		\bTD[nx=4,align=center] Kaktovik \eTD
	\eTR
	\bTR
	  \bTD $10^3$ \eTD
		\bTD $10^2$ \eTD
		\bTD $10^1$ \eTD
		\bTD $10^0$ \eTD
		\bTD \strut \eTD
	  \bTD $20^3$ \eTD
		\bTD $20^2$ \eTD
		\bTD $20^1$ \eTD
		\bTD $20^0$ \eTD
  \eTR
	\bTR
	  \bTD 1 \eTD
		\bTD 0 \eTD
		\bTD 0 \eTD
		\bTD 0 \eTD
		\bTD \strut \eTD
	  \bTD 8 \eTD
		\bTD 0 \eTD
		\bTD 0 \eTD
		\bTD 0 \eTD	
  \eTR
\eTABLE

The following table shows the relation between the vertical bars (1 to 4) and the horizontal bars tuplets of 5

\blank

\bTABLE
  \setupTABLE[r][each][frame=off,align=flushright]
	\setupTABLE[r][even][style=\tfxx]
	\setupTABLE[r][odd][style=\tfd]
	\setupTABLE[c][1][frame=on]
	\setupTABLE[r][1][frame=on]
	\bTR
	 \bTD[frame=off] \kaktoviknumerals{0}\eTD
	 \bTD \kaktoviknumerals{1}\eTD
	 \bTD \kaktoviknumerals{2}\eTD
	 \bTD \kaktoviknumerals{3}\eTD
	 \bTD \kaktoviknumerals{4}\eTD
	\eTR
	\bTR
	 \bTD 0\eTD
	 \bTD 1\eTD
	 \bTD 2\eTD
	 \bTD 3\eTD
	 \bTD 4\eTD
	\eTR
	\bTR
	 \bTD \kaktoviknumerals{5}\eTD
	 \bTD \kaktoviknumerals{6}\eTD
	 \bTD \kaktoviknumerals{7}\eTD
	 \bTD \kaktoviknumerals{8}\eTD
	 \bTD \kaktoviknumerals{9}\eTD
	\eTR
	\bTR
	 \bTD 5\eTD
	 \bTD 6\eTD
	 \bTD 7\eTD
	 \bTD 8\eTD
	 \bTD 9\eTD
	\eTR
	\bTR
	 \bTD \kaktoviknumerals{10}\eTD
	 \bTD \kaktoviknumerals{11}\eTD
	 \bTD \kaktoviknumerals{12}\eTD
	 \bTD \kaktoviknumerals{13}\eTD
	 \bTD \kaktoviknumerals{14}\eTD
	\eTR
	\bTR
	 \bTD 10\eTD
	 \bTD 11\eTD
	 \bTD 12\eTD
	 \bTD 13\eTD
	 \bTD 14\eTD
	\eTR
	\bTR
	 \bTD \kaktoviknumerals{15}\eTD
	 \bTD \kaktoviknumerals{16}\eTD
	 \bTD \kaktoviknumerals{17}\eTD
	 \bTD \kaktoviknumerals{18}\eTD
	 \bTD \kaktoviknumerals{19}\eTD
	\eTR
	\bTR
	 \bTD 15\eTD
	 \bTD 16\eTD
	 \bTD 17\eTD
	 \bTD 18\eTD
	 \bTD 19\eTD
	\eTR	
\eTABLE

\stopsubject

\startsubject[title=Examples]

\starttabulate[|r|Mr|r|]
 \NC 20   \NC 1 \times 20^1 + 0 \times 20^0\NC \kaktoviknumerals{1}\kaktoviknumerals{0} \NC\NR
 \NC 30   \NC 1 \times 20^1 + 10 \times 20^0\NC \kaktoviknumerals{1}\kaktoviknumerals{10} \NC\NR
 \NC 40   \NC 2 \times 20^1  + 0 \times 20^0 \NC \kaktoviknumerals{2}\kaktoviknumerals{0} \NC\NR
 \NC 100  \NC 5 \times 20^1 + 0 \times 20^0\NC \kaktoviknumerals{5}\kaktoviknumerals{0} \NC\NR
 \NC 1000 \NC 2 \times 20^2 + 10 \times 20^1 + 0 \times 20^0 
 \NC \kaktoviknumerals{2}\kaktoviknumerals{10}\kaktoviknumerals{0}\NC\NR
 \NC 2023 \NC 5 \times 20^2 + 0 \times 20^1 + 3 \times 20^0  
 \NC \kaktoviknumerals{5}
 \kaktoviknumerals{0}\kaktoviknumerals{1}\kaktoviknumerals{3}\NC\NR
\stoptabulate

\blank
So the date of today would be:
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19 - 4 - 2023
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\kaktoviknumerals{19} -- \kaktoviknumerals{4} --
\kaktoviknumerals{5}\kaktoviknumerals{0}\kaktoviknumerals{1}\kaktoviknumerals{3}

\blank

Frappant is the possibility to solve arithmetic visually with these Kaktovik-numerals. The approach is to look how patterns fit in the left and right part of the equation.


\starttabulate[|Ml|l|][EQ={$\rightarrow$}]
  \NC \rm Decimal \NC Kaktovik \NC\NR
  \NC 2 + 2 = 4 
	\EQ
\kaktoviknumerals{2} $+$ \kaktoviknumerals{2} $=$ \kaktoviknumerals{4}
	\NC\NR
	\NC 17 - 7 = 10 
	\EQ
	\kaktoviknumerals{17} $-$ \kaktoviknumerals{7} $=$ \kaktoviknumerals{10}
	\NC\NR
	\NC 1503 / 3 
	\EQ
	\kaktoviknumerals{3}\kaktoviknumerals{15}\kaktoviknumerals{3} $/$ \kaktoviknumerals{3} $=$ \kaktoviknumerals{1}\kaktoviknumerals{5}\kaktoviknumerals{1}
	\NC\NR
	\NC 364 / 3
	\EQ
	\kaktoviknumerals{18}\kaktoviknumerals{4} $/$ \kaktoviknumerals{3} $=$ \kaktoviknumerals{6}\kaktoviknumerals{1} remainder \kaktoviknumerals{1}
\stoptabulate

\stopsubject
\stoptitle
\stoptext