Numeral Systems

Every civilization has known in some way about the universal concept of numbers, and every civilization created a numeral system to represent them. Humans count things, and early numeral systems were all about counting. Most of us probably take our familiar base-10 numeral system for granted, but looking at early numeral systems can help you appreciate the power of good notation (a theme I think runs throughout the history of math).

Certainly early humans had to count things, whether is was deer or logs for the fire, but early civilizations with their reliance on agriculture really highlighted the need to keep track of counts. In many ways, commerce drove the development of numeral systems. It was the needs of merchants to keep track of their money that made Europeans open to the use of Hindu-Arabic numerals described in the book Liber Abaci by Leonardo of Pisa (better known to us as Fibonacci). Going back even further in history, it was the needs of early civilizations to keep track of counts larger than 10 or 20 that led to the development of different numeral systems.

This will be a necessarily shallow introduction to numeral systems. Books can and have been written about Babylonian, Egyptian, and Greek mathematics, as well as less famous civilizations such as the ancient Cretans. Someday, I'd like to read some of those books, so I can know more about them than what is here.

Any numeral system should have at a minimum two characteristics: it should represent a useful set of numbers (all the natural numbers at least), and it should have a unique representation for each number. There are two types of numeral systems—additive systems and positional, though Archaeologist and anthropologists, i.e. people that know that they are doing, may classify ancient numeral systems differently.

Additive systems are the simplest number systems, in which a number . The simplest additive system is a unary system, where every natural number is represented by a corresponding number of symbols. The Cretans, for example, from 1700-1200 BCE, used a unary system, which consisted essentially of tally marks.

Additive systems can be improved by employing special symbols for the grouping of units. The ancient Egyptians employed a base 10 numeral system, with special symbols for powers of ten. The Greeks had a slightly more complicated system, an extended decimal system with special symbols for powers of 10, five, and multiples of five and powers of 10. The symbols have the same meaning regardless of their position. More familiar are Roman numerals. When I was a kid we had to learn about Roman numerals, though I've never figured out why it was important to learn about them. Additive systems are simple, and for purposes of representing natural numbers they work; larger numbers get a bit cumbersome, but every natural number can be represented, and there is a unique representation for each number. Where additive systems really fall apart is with arithmetic. Try multiplying 2459671 and 45689 in the additive numeral system of your choice and you can see the problems.

Positional numeral systems are a bit more complicated than additive systems; the location of the symbol matters. In a base-b positional numeral system, we write the number

as:

For example, in the base 10 numeral system with which we are familiar, when we write 1231, we mean the number:



The Babylonians had a sexagesimal, or base-60 numeral system, which really works quite well; other than the (to us) unusual base and symbolism, the Babylonians system seems quite modern, especially when compared to additive systems such as the Romans. One notable difference between this ancient systems and our own is the lack of a symbol for zero.

The number zero was a surprisingly long time coming. Many cultures had trouble with the idea of 0 as a number. Zero was synonymous with nothing, and how could nothing be a number? The Babylonians dealt with the lack of a positional value by leaving the place blank. This works, but you can see how sloppy penmanship could lead to numerical confusion. The Mesoamerican Long Count Calendar, a non-repeating base-20 and base-18 calendar dating to 36 BCE and used by several ancient Central American civilizations including the Maya, makes use of several glyphs for zero as a placeholder. The Mayans base-20 numeral system settled on a shell glyph for zero, again as a placeholder. 

Zero appears sporadically as a placeholder in Old World mathematics, but the first instance of zero as a number withing a positional numeral system is generally credited to Indian mathematicians, who by the 9^th century AD were carrying out calculations with zero as a number. The Hindu-Arabic number system, with its zero, was first described by the renowned Persian mathematician Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī, whose book on Arithmetic, written in 825 AD, is considered one of the most influential ancient texts. One of al-Khwārizmī's later fans, Leonardo of Pisa, introduced European mathematicians to the Hindu-Arabic numeral system through his book Liber Abaci. While the symbols have evolved somewhat, they are remarkably similar to the original numeral system described by al-Khwārizmī.

With the introduction of the number zero, the positional numeral system was complete. You can pick your base—decimal, hexadecimal, binary, whatever—the essential qualities are the same. Arithmetical computation is easy. The history of numeral systems is fascinating; it took a very long time to get to something that we take for granted.

The evolution of numeral systems didn't stop with the Hindu-Arabic numeral system. Complex numbers, with the imaginary unit i, were a later discovery, but by that time mathematicians were thinking in the decimal numeral system we use today, and with a good system of notation, the possibilities are endless.