Numerals

Numerals, signs or symbols for graphic representation of numbers. The earliest forms of numerical notation were simply groups of straight lines, either vertical or horizontal, each line corresponding to the number 1. Such a system is inconvenient when dealing with large numbers, and as early as 3400 bc in Egypt and 3000 bc in Mesopotamia a special symbol was adopted for the number 10. The addition of this second symbol made it possible to express the number 11 with two instead of 11 individual symbols and the number 99 with 18 instead of 99 individual symbols. Later numerical systems introduced extra symbols for a number between 1 and 10, usually either 4 or 5, and additional symbols for numbers greater than 10. In Babylonian cuneiform notation the numeral used for 1 was also used for 60 and for powers of 60; the value of the numeral was indicated by its context. This was a logical arrangement from the mathematical point of view because 60 ⁰ = 1, 60¹ = 60, and 60² = 3600. The Egyptian hieroglyphic system used special symbols for 10, 100, 1,000, and 10,000.

The ancient Greeks had two parallel systems of numerals. The earlier of these was based on the initial letters of the names of numbers: the number 5 was indicated by the letter pi; 10 by the letter delta; 100 by the antique form of the letter H; 1,000 by the letter chi; and 10,000 by the letter mu. The later system, which was first introduced about the 3rd century bc, employed all the letters of the Greek alphabet plus three letters borrowed from the Phoenician alphabet as number symbols. The first nine letters of the alphabet were used for the numbers 1 to 9, the second nine letters for the tens from 10 to 90, and the last nine letters for the hundreds from 100 to 900. Thousands were indicated by placing a bar to the left of the appropriate numeral, and tens of thousands by placing the appropriate letter over the letter M. The later Greek system had the advantage that large numbers could be expressed with a minimum of symbols, but it had the disadvantage of requiring the user to memorize a total of 27 symbols.

The system of number symbols created by the Romans had the merit of expressing all numbers from 1 to 1,000,000 with a total of seven symbols: I for 1, V for 5, X for 10, L for 50, C for 100, D for 500, and M for 1,000. Roman numerals are read from left to right. The symbols representing the largest quantities are placed at the left; immediately to the right of those are the symbols representing the next largest quantities, and so on. The symbols are usually added together. For example, LX = 60 and MMCIII = 2,103.  represents 1,000,000—a small bar placed over the numeral multiplies the numeral by 1,000. Thus, theoretically, it is possible, by using an infinite number of bars, to express the numbers from 1 to infinity. In practice, however, one bar is usually used; two are rarely used, and more than two are almost never used. Roman numerals are still used today, more than 2,000 years after their introduction. The Roman system's one drawback, however, is that it is not suitable for rapid written calculations.

The common system of numerical notation in use in most parts of the world today is the Arabic system. This system was first developed by the Hindus and was in use in India in the 3rd century bc. At that time the numerals 1, 4, and 6 were written in substantially the same form used today. The Hindu numeral system was probably introduced into the Arab world about the 7th or 8th century ad. The first recorded use of the system in Europe was in ad 976.

The important innovation in the Arabic system was the use of positional notation, in which individual number symbols assume different values according to their position in the written numeral. Positional notation is made possible by the use of a symbol for zero. The symbol 0 makes it possible to differentiate between 11, 101, and 1,001 without the use of additional symbols, and all numbers can be expressed in terms of ten symbols, the numerals from 1 to 9 plus 0. Positional notation also greatly simplifies all forms of written numerical calculation.