The Stoneboard or Abacus

The abacus or stoneboard represents a common device which many number systems might be described.

The process of counting is to count a measure, say ten stones, and place one stone to the right for these ten stones. Counting continues until we are left with a string of remainders, each less than ten, as tens of stones have been replaced for one to the right.

The algorithm for converting from base ten to some other base follows this. We suppose 118 stones in the units column, which we remove dozens to leave 10 stones left. The dozens represents another number to count, we have nine, not a full dozen. So 118 decimal is 9 dozen 10.

The mathematical approach is to suppose that every different count of remainder has a different symbol, or ''digit. However, the historical record does not show this, and we shall use column-rune'' as a different word to digit. A good amount of creative skill is given over to populating a table with hundreds of different symbols, when even by 20, people have started composing symbols from parts, or using several digits to a column.

The view that one ought learn addition and multiplication tables is more in keeping with the use of different symbols for each column-value, this too can be avoided with row arithmetic of one kind or another.

Since the intention is to show how these things work, rather than to replicate the different numerals and characters, the discussion shall use western symbols.

Column Tokens
The number system used by the ancient Egyptians and others simply is to use a symbol for each column used. We will use the latin symbols M, C, X, I to represent the columns of units (I) and the values to the left of it.

A number like 234 might be written as CCXXXIIII. There is no need for a general zero, since 102 can be written as CII. The columns are named: the absence of decades in CII means there is no X in the name. Addition and subtraction is simply a matter of emptying two bags together, and carrying the count as needed. So while there is no need for an empty column, there is still a use for the empty bag. In the time of the medeavel era, a letter o was used to denote the absence of things to count. The Egyptians used a character of a centre-line to represent an empty board.

Whatever the system used, the units are on the right, because people are right-handed. The columns to the left of this are increasing value, this holds true whether the running script is left-to-right or right-to-left. A number is essentially a character-picture of the stones on the table.

Abacus Rows
A number like 10 is too much for the eye to count. If it were grouped into groups of three (such as the sumerian cuneform numbers, which are three rows of three), or the middle three on a sliding rail highlighted, it is very hard to pick that seven or eight is being used.

The abacus row overcomes this. A row is placed above the units, and units are counted to multiples of five, these fives are counted by pairs to tens. Such allows access to representing numbers clearly past the ability of the eye to count. As we see later, it overcomes also the need to create dozens of symbols to represent every number of a column separately. We shall also see that the row idiom allows access to larger bases.

The row of fives is placed further away from the operator. This is still reflected in calculators and numeric entry pads, where the values ascend as one moves higher up the keyboard. Note that access devices, such as telephones and key-pads, write the order of numbers from top to bottom. The symbols appear in the same order as they are read.

From Demotic to Alphabetic
Demotic numbers are an Egyptian invention, where in place of writing 234 as nine separate symbols, one uses a symbol for 200, 30 and 4. We can use upper and lower case letters here, a=1, b=2, etc. So 234 becomes Bc4. Again zero is not required, we see that a2 is XII (12) but A2 is CII (102)

Without numerals or letter-case, the letters of the alphabet are pressed into service, old letters and alternate character-forms used to create the 27 symbols. The greek form of 996 consists of three letters that have dropped out of use: vaw (or digamma), whence our F, qoppa, whence our Q, and tsampi (not transferred to the Roman alphabet).

From Alphabetic to Digital
In the West, the transfer to the column-rune system appears to have happened around 400 AD, when in order to simplify the addition and multiplication, the demotic forms were cascaded onto the units, so 1, a, and A are written in the same symbol. The effect of this meant that empty columns now needed a symbol, and the first letter to which there is no cascade is used.

So the first ten greek numbers (including vaw), come to represent 1-9 and then a=0, We see also the dreaded tables appear. Up to this point, it suffices to know that X times C is M, and this was repeated for every combination of X and C stone. Now we learn 2*3=6.

The greek letters were replaced eventually by the assorted hindu characters, and it was this system that Fibonacci introduced to Europe in 1200.

Digit-and-Column
Even before the Western transfer to the column system, in at least 300 BC, India were one of the first to embrace the use of 0, with the Sanskrit shunya being the word for 0. Indians also ditched using seperate symbols to represent powers of 10, so “2884” would not be “(2,000)(800)(80)(4)”, but “2884”; the places are infered by the reader. Additionally, “509” cannot be confused with “59” due to zero being introduced.

The chinese style of writing numbers is to write digit-and-column, such as 2C 3X 4, or 1C 2. According to Butterworth's The Mathematical Brain, this gives some form of advantage to the Chinese, insofar as they don't make the mistake of writing 20034 for two hundred and thirtyfour. This sort of error happens when there are already zeros in the number: a hundred and eight might be 1008 rather than 108.

A second method used by the Chinese when reading numbers, is state the digit-and-column, and then read the digits off. So 540 is not 5C 4 0 but simply 5C 4. The zero is only used medially, ie 5 C 0 4 is 504, and C 0 2 is 102. However, the Chinese also transferred to the previously-mentioned Indian system in about 690.