CHAPTER 1 History of Numbers

The Greek numbering system is another example of an additive numbering system. Around the fifth century B.C., the Greeks of Ionia developed this additive numbering system that had a more extensive set of symbols to memorize than the Egyptians. In the Ionic Greek numbering system, the ordinary Greek alphabet of 24 letters was used and 3 Phoenician letters were added to come up with the following numerals (see Figure 1.3.1).

Since the Ionic system is additive, all numbers between 1 and 999 could be represented by at most three symbols. For example,

For larger numbers an accent mark (,) was placed to the left and below the appropriate unit letter to indicate multiplication by 1,000. In this way ,β was 2 × 1,000 = 2,000.

Another method for making even larger numbers employed the use of the letter M (derived from the word myriad, meaning 10,000). This was placed next to, or even below, the symbols indicating multiplication by 10,000.

For example, βM was 2 × 10,000 = 20,000. A double MM meant to multiply by 10,000² = 100,000,000. This is similar to a concept used in the Roman numbering system that we will examine in the next section.

Example 1.3.1

Translate the following Ionic Greek numbers into our own Hindu-Arabic numbers.

. . . to part (a) Matching the given symbols with those found in Figure 1.3.1 we find that

. . . to part (b) Part (b) requires you understand the accent mark concept already discussed. It means to multiply the unit symbol it is in front of by 1,000. Doing this and matching symbols from Figure 1.3.1 we will find

. . . to part (c) Part (c) requires your understanding of using the symbol M which means to multiply everything in front of it by 10,000. We will find that

You will also want to be able to write our own numbers in the Ionic Greek form. Example 1.3.2 illustrates how this is to be done.

Example 1.3.2

Write the following numbers using the Ionic Greek alpha-numeric symbols.

(a) 666

(b) 8128

(c) 357,084

. . . to part (a) Referring to Figure 1.3.1 we will look for the symbols that make up the number 666. Since 666 = 600 + 60 + 6 these are the symbols we are looking for. The solution is

. . . to part (b) Likewise for 8128 we need to locate the symbols from Figure 1.3.1 to make this number. But we also need to remember that 8,000 requires us to use the accent mark. The solution is

. . . to part (c) This part requires us to understand how the given number, 357,084 is broken down. We should first note that 357,084 = 350,000 + 7,000 + 80 + 4. Key here was the break down of 350,000 since 35 × 10,000 = 350,000. Using Figure 1.3.1 then we should find that

Numerically speaking, this system of numeration afforded much economy of writing. Where the Greek alphabetic numeral for 800 was a single symbol (omega), the Egyptian hieroglyphics would have had to use eight symbols of the coil of rope. A later use of hieratic script writing would afford the Egyptians the same economy of writing as the Ionic Greeks.