We explain what a numbering system is and study the characteristics of each type of system through examples from different cultures.
What is a numbering system?
A numbering system is a set of symbols and rules through which the quantity of objects in a set can be expressed; that is, through which all valid numbers can be represented. This means that every numbering system contains a specific and finite set of symbols, as well as a specific and finite set of rules by which they can be combined.
Number systems were one of the main human inventions in ancient times, and each ancient civilization had its own system, related to its way of seeing the world, that is, its culture. You must read about Social Mobility once.
Broadly speaking, number systems can be classified into three different types:
- Non-positional systems: These are those in which each symbol has a fixed value, regardless of its position within the number (whether it appears first, to the side, or last).
- Semi-positional systems: These are those in which the value of a symbol tends to be fixed, but can be modified in specific situations (although these tend to be exceptions). This is understood as an intermediate system between positional and non-positional systems.
- Positional or weighted systems: These are those in which the value of a symbol is determined both by its own expression and by its place within the number. It can have more or less value or express different values depending on where it is located.
Numeral systems can also be classified based on the base digit they use for their calculations. For example, the current Western system is decimal (its base is 10), while the Sumerian numeral system was sexagesimal (its base was 60). Maybe you should definitely read about Social Classes once.
Non-positional numeral systems
Non-positional numeral systems were the first to exist and had the most primitive bases: fingers, knots in a string, or other recording methods for coordinating sets of numbers. For example, if you count with the fingers on one hand, you can later count with whole hands.
In these systems, digits have their own value, regardless of their location in the chain of symbols, and to form new symbols, the values of the symbols must be added (which is why they are also known as additive systems). These systems were simple and easy to learn, but required numerous symbols to express large quantities, so they were not entirely efficient.
Examples of this type of system include:
- The Egyptian numeral system: Emerging around the 3rd millennium BC, it was based on the ten (10) and used different hieroglyphs for each order of units: one for the unit, one for the ten, one for the hundred, and so on up to a million.
- The Aztec numeral system: Typical of the Aztec empire, it was based on the twenty (20) and used specific objects as symbols: a flag was equivalent to 20 units, a feather or a hair was equivalent to 400, a bag or sack was equivalent to 8000, among others.
- The Greek numeral system: Specifically the Ionic system, it was invented and spread in the eastern Mediterranean beginning in the 4th century BC, replacing the preexisting acrophonic system. It was an alphabetic system that used letters to represent numbers, matching the letter with its cardinal position in the alphabet (A=1, B=2). Thus, each number from 1 to 9 was assigned a letter, each ten another specific letter, each hundred another, until 27 letters were used: the 24 of the Greek alphabet and three special characters.
Semi-positional Number Systems
Semi-positional number systems combine the notion of a fixed value for each symbol with certain positioning rules, so they can be understood as a hybrid or mixed system between positional and non-positional systems. They are easy to represent large numbers, handling the order of numbers and formal procedures such as multiplication, thus representing a step forward in complexity compared to non-positional systems.
To a large extent, the emergence of semi-positional systems can be understood as a transition toward a more efficient numbering model that could meet the more complex needs of a more developed economy, such as that of the great empires of classical antiquity.
Examples of this numbering model include:
The Roman numeral system:
Created in Roman antiquity, it survives to this day. In this system, numerals were constructed using certain capital letters of the Latin alphabet (I = 1, V = 5, X = 10, L = 50, etc.), whose value was fixed and operated on the basis of addition and subtraction, depending on the location of the symbol. If the symbol was to the left of a symbol of equal or lesser value (as in II = 2 or XI = 11), the total values had to be added; while if the symbol was to the left of a symbol of greater value (as in IX = 9 or IV = 4), they had to be subtracted.
The Classical Chinese Numeral System:
Its origins date back to approximately 1500 BC and it is a very strict system of vertical representation of numbers through its own symbols, combining two distinct systems: one for colloquial, everyday writing, and another for commercial or financial records. It was a decimal system that featured nine different symbols that could be placed next to each other to add their values, sometimes inserting a special symbol or alternating the placement of the symbols to indicate a specific operation.
Positive Number Systems
Positive number systems are the most complex and efficient of the three types of number systems that exist. The combination of the symbols’ proper value and the value assigned by their position allows them to construct very high numbers with very few characters, adding and/or multiplying the value of each one, which makes them more versatile and modern systems.
Generally, positional systems use a fixed set of symbols, and through their combinations, the rest of the possible numbers are produced, ad infinitum, without the need to create new symbols, but rather by inaugurating new columns of symbols. Of course, this implies that an error in the chain also alters the total value of the number.
The first examples of such systems arose within the great empires or the most culturally and commercially demanding ancient cultures, such as the Babylonian Empire of the 2nd millennium BC. Examples of this type of numbering system include:
The modern decimal system
With just the digits 0 to 9, any possible number can be constructed by adding columns whose value is added as you move to the right, using the ten (10) as the base. Thus, by adding symbols to 1, we can construct 10, 195, 1958, or 19589. It is important to clarify that the symbols it uses come from Hindu-Arabic numerals.
The Hindu-Arabic numeral system
Invented by the ancient sages of India and later inherited by Muslim Arabs, it reached the West through Al-Andalus and eventually replaced traditional Roman numerals. In this system, similar to the modern decimal system, units from 0 to 9 are represented by specific glyphs, which represented the value of each one through lines and angles. The operating system of this system is basically the same as the modern Western decimal system.
The Mayan numeral system
It was created to measure time, rather than for mathematical transactions, and its base was vigesimal (20), and its symbols correspond to the calendar of this pre-Columbian civilization. Numbers, grouped by 20, are represented with basic symbols (dashes, dots, and snails or shells); and to move to the next twenty, a dot is added at the next level of writing. Furthermore, the Mayans were among the first to use the number zero.
References
All the information we offer is supported by authoritative and up-to-date bibliographic sources, ensuring reliable content in line with our editorial principles.
- “Number system” on Wikipedia.
- “Positive notation” on Wikipedia.
- “Other numbering systems” on the Andalusian Regional Government (Spain).
- “Numeration Systems” on Encyclopedia.com.
- “Numeral system (Mathematics)” on The Encyclopaedia Britannica.
