January 01, 2008

Alphabetic Numerals

In pointing out that English letters and numbers resemble each other to an unfortunate degree, Geoff mentions that some systems are even worse. The very worst in this respect are the writing systems that do not distinguish letters from numbers at all.

This unfortunate idea begins, I'm afraid, with us Jews. In pre-modern Hebrew the letters of the alphabet were assigned numerical values according to their position in the alphabet. Thus, aleph א, the first letter of the alphabet, represents 1, bet ב, the second letter, represents 2, gimel ג, the third letter, represents 3, and so forth. The system is not place-based: 10 is represented by yod י ‭, 20 by kaf   כ ‭, and so forth. 312 is shin yod bet   ניב ‭= 300 + 10 + 2. This system is still used for some purposes in modern Hebrew but for the most part has been replaced by the "Arabic" numerals (which actually come from India) used as in English.

The ancient Greeks used a similar system, with alpha α representing 1, beta β 2, and so on. 312 in ancient Greek was written τιβ. From Greek this system spread to the two writing systems used by the Slavs, Cyrillic (312 = ТБІ), and Glagolitic (312 = ⰕⰁⰉ), and to Armenian (312 = ՅԺԲ). In Cyrillic when letters were used as numerals they were usually written with a sort of zigzag line over the top known as a titlo. This was not, however, a reliable indicator that the letters in question represented a number as the titlo was also used to mark abbreviations.

A somewhat different approach was introduced in the early sixth century CE by the Indian mathematician Āryabhata (आर्यभट), who assigned a numerical value to each CV syllable formed by taking the Cartesian product of the Sanskrit consonants and vowels (including syllabic l and r) in their usual order. Thus, the first vowel letter /a/ अ represents 1, the second /i/ इ 100, the third /u/ उ 10,000, and so on through the isolated vowels. The first consonant letter /k/ क represents the coefficient 1, the second /kh/ ख 2, the third /g/ ग 3. Thus, the syllable /ku/ कु represents 3 * 10,000 = 30,000, and the syllable /ghi/ घि represents 4 * 100 = 400. This never became the usual way of writing numbers in India.

Alphabetic numeral systems are problematic not only because letters may be confused with numbers and because, not being place-based, it can only represent a limited range of numbers, but because it is not well suited to doing arithmetic. As a result, these systems have all been replaced for most purposes by the "Arabic" numerals, although sometimes retained for limited purposes, such as printing page numbers in books.

Roman numerals (312 = cccxii) in the form in which most of us are familiar with them are a system of the same type, but they actually have a different origin. The Romans borrowed their numerals, like many other things, from the Etruscans, whose numerals were not, in general, the same as their letters. In Etruscan 1 was I and 10 was X as in Latin, but 5 was Λ, 50 was ⋔, 100 was 8, and 1000 was ⊕. Only gradually did the Romans modify this system into the version we still occasionally use today in which the numerals are all the same as letters.

Posted by Bill Poser at January 1, 2008 05:03 PM