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. Some of them add in an element representing four or five single units. 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.
Origins of numerals
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 numeral 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 for 1–4 in Unicode text.
𐩀 𐩁 𐩂 𐩃
The first three of these are very similar to our modern numerals, except that the 2 and 3 have been turned sideways during their migrations. The numeral for 4 has been given a smaller turn, and a stroke added.
Kharosthi numerals exist for 5–9, but these are not of the visual type. It was also possible to write numbers in Kharosthi in something like the manner of Roman numerals, so that 2 would be 𐩀𐩀. However, Kharosthi indicates a custom of counting on four fingers but not the thumb, so it uses a numeral for 4 rather than 5 as in Roman numerals. The Kharosthi numeral for 4 is very similar to X, so 7 in Kharosthi can be approximated in ASCII (right-to-left, again) )))X.
Naturally, we can teach a Turtle how to write this second form of Kharosthi numerals.
The details of this program are on another page.
The font for Kharosthi in Ubuntu Linux is in the package ttf-mph-2b-damase, with the name damase.ttf. Package names in other distributions may vary.
Visual numerals in Unicode
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.
𐄇 𐄈 𐄉 𐄊 𐄋 𐄌 𐄍 𐄎 𐄏
𐄐 𐄑 𐄒 𐄓 𐄔 𐄕 𐄖 𐄗 𐄘
𐄙 𐄚 𐄛 𐄜 𐄝 𐄞 𐄟 𐄠 𐄡
Counting Rod Numerals
Egyptian Hieroglyphics Heqat Measure
𓃉 𓃊 𓃋 𓃌 𓃍 𓃎 𓃏 𓃐 𓃑
🀙 🀚 🀛 🀜 🀝 🀞 🀟 🀠 🀡
Dice are also visual, but go up only to six, with no 0.
⚀ ⚁ ⚂ ⚃ ⚄ ⚅
Partitioning by fours and fives
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.
I II III IIII V VI VII VIII VIIII
with the later abbreviations IV for IIII, and IX for VIIII.
Counting Rod Numerals and Roman numerals are of the same structure as the Chinese and Japanese abacuses (abaci?) and the Roman counting board with its little pebbles (Latin, calculus/calculi, whence calculation and Differential and Integral Calculus).
Teaching the Turtle Visual Numerals
Here are the second form of Counting Rod Numerals drawn by a Turtle.
I leave the others as a programming challenge. You need to determine how to draw the elements, and how to specify the layout of elements in each numeral. For example, a program for the cuneiform wedge could go like this.
Mayan base 20 numerals
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. The following is a schematic rendering of the Mayan numerals.
The actual numerals carved in stone are sideways from these, and somewhat more elaborate.
We can teach the Turtle to write numbers in Mayan style in base 20. I am using an empty circle for 0 with these simplified forms rather than the actual Mayan shell glyph.
Visual Numerals in Turtle Art
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,
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.
Unicode character ranges for visual numerals
The original Unicode code space ran from U+0000 to U+FFFF, a space of 65,636 code points. Since then 16 more spaces of that size have been added, for a total of more than a million code points. Several writing systems for dead languages have been put into this extra space, and so have five-digit Unicode hex code point numbers.
The fonts listed here are available in Ubuntu Linux. Other distributions may vary. There may be other fonts for these ranges for Windows or Mac OS, either commercially or as freeware.
- Aegean Numerals
- Units U+10107–U+1010F
- Tens U+10110–U+10118
- Hundreds U+10119–U+10121
- Thousands U+10122–U+1012A
- Tens of thousands (myriad) U+1012B–U+10133
- Counting Rod Numerals U+1D360–U+1D368; U+1D369–U+1D371
- Cuneiform U+12415–U+1241D
- Egyptian Hieroglyphics heqat measure U+130C9–U+130D1. Other hieroglyphic numerals U+13386–U+1338E; U+131BC–U+131C4
- Mahjong Tiles U+1F019–U+1F021
- Dice U+2680–U+2685
Fonts for Visual Numerals
- Counting Rod
- Egyptian Hieroglyphic
- ttf-ancient-scripts/Aegyptus.otf Private Use Area
- Mahjong Tiles
- Dice (Miscellaneous Symbols)