[Math Lair] Greek Ionic Numerals

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The Ionic system of numerals (also known as the Alexandrian system), gained popularity in Greco-Roman times, starting in around 100 B.C. By 50 A.D., this system had replaced the older Attic (or Herodianic) system almost completely. Ionic numbers were used in Europe until the tenth century, when they were supplanted by Roman numerals.

This system used the 24 letters in the Greek alphabet, as well as three other symbols: the digamma, which represented 6, the koppa, which represented 90, and the sampi, which represented 900. The first nine letters represented the numbers 1 to 9. The second nine represented the multiples of 10 from 10 to 90. The third nine represented the multiples of 100 from 100 to 900. To represent thousands, an accent mark was used on the letters from 1 to 9. For example, "A" represents 1, while ",A" represents 1,000.

For representing numbers larger than 10,000, there were several systems used. For example, putting M in front of a group of numerals multiplies them by 10,000, and MM in front of a group of numerals multiplies them by 100 million.

Fractions were either represented as a sum of unit fractions, or several different forms were used. For example, keeping in mind that γ represents 3 and η represents 8, 38 could have been represented in any of the following ways:

Some later works use fractions that look like modern fractions, except that the denominator is on the top and the numerator is on the bottom.

The advantages of the Ionic system was that numbers could be represented using only a few symbols. For example, every number between 1 and 9,999 could be represented using four or fewer symbols. This made this system handy for coins. Since the numerals could be mistaken for Greek words, they were often printed with a bar over them, or were bracketed with rows of dots.

The main disadvantage to this system was that it was harder to do arithmetic in Ionic numerals than it was in the Attic system or using Roman numerals. For example, it is not at all intuitive that ζ × μ = σπ. (7 × 40 = 280). Even with addition, it is possible to see in the other systems that, for example, | + || = ||| or I + II = III, but it is not so obvious that α + β = γ. While it was possible for addition and multiplication to be memorized by using the sounds of the numbers (i.e. letters), most work was done on a counting board and only written down in Ionic numerals. It was not until the Arabic number system was introduced into Europe that it was possible to do calculations on paper rather than on counting boards.

Sources used (see bibliography page for titles corresponding to numbers): 14.