Historical Development and Limitations of Early Number Systems

Key Insight

Numbers initially served to count things like possessions or transactions, evolving from individual drawings of items (e.g., four ducks) to a single drawing with a numerical indicator, and eventually to systematic numbering to manage larger quantities, such as 27 ducks. The pervasive use of a base-ten, or decimal, number system in most civilizations stems from the anatomical convenience of counting on ten fingers or toes, a link exemplified by the word 'digit' referring to both. This foundational choice is arbitrary; if humans had eight or twelve fingers, counting systems would likely differ. Yet, base-ten numbers are endowed with special nomenclature, like 'decade' for ten years, 'century' for one hundred years, and 'millennium' for one thousand years, and are the basis for powers of ten such as 10^6 (million) and 10^9 (billion).

Of the early number systems, Roman numerals remain in common use, appearing on clocks, monument dates (e.g., MCMLIII for 1953), book page numbers, and movie copyright notices. The system represents numbers like 27 ducks as 'XXVII', where each symbol denotes a specific value. The principal symbols still used are I (1, possibly from a scratch mark or single finger), V (5, likely a hand symbol), X (10, two V's), L (50), C (100, from Latin 'centum'), D (500), and M (1000, from Latin 'mille').

For a considerable period, Roman numerals were favored in European bookkeeping due to their perceived ease in addition and subtraction. These operations involved simply combining and then simplifying symbols according to rules, such as 'five I's make a V' or 'two V's make an X'. However, this system proved cumbersome for more sophisticated operations like multiplication and division. Many other ancient number systems, including that of the Greeks, shared similar inadequacies for complex arithmetic, which explains why the Greeks, despite their extraordinary advancements in geometry, are not noted for their contributions to algebra.