# Changes

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The numerals used in various languages, living and extinct, show their origins in stroke marks or scores (cuts), like the Chinese 一二三, and early forms of some Hindu-Arabic-European numerals. The following are Kharosthi numerals, the earliest forms known from India. Kharosthi was written from right to left, like the source for its writing, Aramaic.
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There are several historical examples of systems for writing numbers with numerals that show 1-9 copies of a basic visual unit, which can be a line, a circle, a wedge-shaped cuneiform indentation, or other shapes. An advantage of any of these systems for us to consider is that pre-literate preschoolers can deal with them by counting before they have memorized the conventional digits. This makes arithmetic particularly easy to demonstrate.
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==Origins of numerals==
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The numerals used in various languages, living and extinct, show their origins in stroke marks or scores (cuts), like the Chinese 一二三, and the earliest forms of Hindu-Arabic-European numerals. The following are Kharosthi numerals, among the earliest forms known from India. Kharosthi was written from right to left, like the source for its writing, Aramaic, and the source for all alphabetic writing, Phoenician. First we have an image, so that you don't need a Kharosthi font to view it, followed by the numerals in Unicode text.

[[File:KharosthiNumerals1-3.png]]

[[File:KharosthiNumerals1-3.png]]
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𐩀 𐩁 𐩂

𐩀 𐩁 𐩂
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Counting Rod Numerals and Mayan numerals (below) show clear indications of counting on fingers up to five, then with a whole hand for five plus fingers of the second hand, as in Roman numerals. Kharosthi numerals indicate counting on four fingers but not the thumb. The Kharosthi numeral for 4 is very similar to X, so 7 in Kharosthi could be written (right-to-left, again) )))X.
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Kharosthi numerals indicate counting on four fingers but not the thumb. The Kharosthi numeral for 4 is very similar to X, so 7 in Kharosthi could be written (right-to-left, again) )))X.

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i ii iii iiii v vi vii viii viiii
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or
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I II III IIII V VI VII VIII VIIII
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==Visual numerals in Unicode==
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with the later abbreviations iv and IV for iiii and IIII, and ix/IX for viiii/VIIII.
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Here are many of the visual numerals that have made it into Unicode as characters, first as a graphic, so that you can see them even if you do not have all of the fonts needed to display them. Then I have provided them as text so that you can test which fonts you lack.

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These number systems are of the same structure as the Chinese and Japanese [[Activities/TurtleArt/Tutorials/Abacus|abacuses]] (abaci?) and the Roman counting board with its little pebbles (Latin, ''calculus''/''calculi'', whence calculation and Differential and Integral [[Activities/TurtleArt/Tutorials/Kindergarten_Calculus|Calculus]]).
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Mayan numerals go up to 20, which is often interpreted as counting on fingers and toes, with groupings of five for hands and feet, although we have no other evidence for how they began. Here are the visual numerals that have made it into Unicode as characters, first as a graphic, so that you can see them even if you do not have all of the fonts needed to display them. Then I have provided them as text so that you can test which fonts you lack.

[[File:UnicodeNumerals.png]]

[[File:UnicodeNumerals.png]]
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🀙 🀚 🀛 🀜 🀝 🀞 🀟 🀠 🀡

🀙 🀚 🀛 🀜 🀝 🀞 🀟 🀠 🀡
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It should be obvious that having a Turtle write these numbers is fairly simple to do. We can create TA stacks for each digit in one of these systems, and display whatever sort of arithmetic we want. For example,
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Dice are also visual, but go up only to six, with no 0.
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⚀ ⚁ ⚂ ⚃ ⚄ ⚅
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==Partitioning by fours and fives==
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Counting Rod Numerals show clear indications of counting on fingers up to five, then with a whole hand for five plus fingers of the second hand, as in Roman numerals.
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I II III IIII V VI VII VIII VIIII
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[[File:TAMayanNumerals.png]]
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with the later abbreviations IV for IIII, and IX for VIIII.
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We can also put these numerals on Turtle Art tiles as text, thus.
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Counting Rod Numerals and Roman numerals are of the same structure as the Chinese and Japanese [[Activities/TurtleArt/Tutorials/Abacus|abacuses]] (abaci?) and the Roman counting board with its little pebbles (Latin, ''calculus''/''calculi'', whence calculation and Differential and Integral [[Activities/TurtleArt/Tutorials/Kindergarten_Calculus|Calculus]]).
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[[File:TACountingRodTiles.png]]
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==Mayan base 20 numerals==
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Mayan is also a candidate in part because it is the only one of these number systems with a zero, but Mayan has not yet been added to Unicode.
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Mayan is also a candidate, notable because it is the only one of these visual numeral systems with a numeral for zero, but Mayan has not yet been added to Unicode. Mayan numerals go up to 20. This is often interpreted as counting on fingers and toes, with groupings of five for hands and feet, although we have no other evidence for how they began.

[[File:MayanNumerals.svg]]

[[File:MayanNumerals.svg]]
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No matter. We can still teach the Turtle to write numbers in Mayan glyphs in base 20.Only the shell glyph for 0 requires any real thought.
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We can teach the Turtle to write numbers in Mayan glyphs in base 20. Only the shell glyph for 0 requires any real thought.

[[File:TAMayanNumerals.png]]

[[File:TAMayanNumerals.png]]
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An advantage of any of these systems is that pre-literate preschoolers can deal with them by counting before they have memorized the conventional digits. We can actually animate arithmetic in these numerals, by taking them apart into their constituents and reassembling them. Here are a few frames.
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==Visual Numerals in Turtle Art==
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It should be obvious that having a Turtle write most of these numerals is fairly simple to do. We can create TA stacks for each digit in one of these systems, and display whatever sort of arithmetic we want. For example,
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[[File:TAVisualArithmetic.png]]
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We can also put these numerals on Turtle Art tiles as text. In the following illustration, each Hieroglyphic heqat measure numeral is used as a variable name with value the number it represents.
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[[File:TAHieroglyphicNumeralTiles.png]]