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 It is a set of symbols and rules through which the number of objects in a set can be expressed that is, through which all valid numbers can be represented. This means that every number system contains a certain and finite set of symbols, as well as a certain and finite set of rules by which to combine them.
numbering systems were one of the main human inventions in ancient times and each of the civilizations of yesteryear had its own system, related to its way of seeing the world, that is, to its culture.
Broadly speaking, numbering systems can be classified into three different types:
- Non-positional systems. They are those in which each symbol corresponds to a fixed value, regardless of the position it occupies within the figure (if it appears first, to the side or after).
- Semi-positional systems They are those in which the value of a symbol tends to be fixed, but can be modified in particular situations of appearance (although they tend to be exceptions). It is understood as an intermediate system between the positional and the non-positional.
- Positional or weighted systems. They are those in which the value of a symbol is determined both by its own expression and by the place it occupies within the figure, and can be worth more or less or express different values depending on where it is located.
It is also possible to classify numbering systems based on the figure they use as the basis for their calculations. Thus, for example, the current Western system is decimal (since its base is 10), while the Sumerian number system was sexagesimal (its base was 60).
Non-positional number systems
Non-positional number systems were the first to exist and they had the most primitive bases: the fingers of the hands, knots in a rope or other recording methods to coordinate numerical sets. For example, if you count on the fingers of one hand, you can then count on whole hands.
In these systems digits have a value of their own, regardless of their location in the symbol string 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, easy to learn, but required numerous symbols to express large quantities, so they were not entirely efficient.
Examples of this type of systems are the following:
- The Egyptian number system. Emerged around the 3rd millennium BC. C., 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 the million.
- The Aztec number system. Typical of the Mexica empire, it had twenty as a base (20) and used specific objects as symbols: a flag was equivalent to 20 units, a feather or a few hairs were equivalent to 400, a bag or sack was equivalent to 8,000, among others.
- The Greek number system. Specifically, Ionic was invented and spread in the eastern Mediterranean starting in the 4th century BC. C., replacing the pre-existing acrophonic system. It was an alphabetical system, which used letters to signify numbers, matching the letter with its cardinal place in the alphabet (A=1, B=2). Thus, a letter was assigned to each number from 1 to 9, to each ten another specific letter, to each hundred another, until 27 letters were used: the 24 of the Greek alphabet and three special characters.
Semi-positional numbering systems
Semi-positional numbering systems combine the notion of the fixed value of each symbol with certain positioning rules so they can be understood as a hybrid or mixed system between positional and non-positional. They have facilities for representing large figures, managing the order of numbers and formal procedures such as multiplication, so that they represent a step forward in complexity with respect to non-positional systems.
To a large extent, the emergence of semi-positional systems can be understood as the transition towards a more efficient numbering model that could satisfy the more complex needs of a more developed economy, such as that of the great empires of classical antiquity.
Examples of this numbering model are:
- The Roman numeral system. Created in Roman antiquity, it survives to this day. In this system, figures 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 based on addition and subtraction, depending on the place of appearance of the symbol. If the symbol was to the left of a symbol of equal or lesser value (as in Yo I = 2 or in x I = 11), the total values had to be added; while if the symbol was to the left of a higher value symbol (as in Yo X = 9, or IV = 4), had to be subtracted.
- The classical Chinese number system. Its origins date back to approximately 1500 BC. C and it is a very strict system of vertical representation of numbers through its own symbols, combining two different systems: one for colloquial and everyday writing, and another for commercial or financial records. It was a decimal system that had nine different signs that could be placed next to each other to add their values, sometimes inserting a special sign or alternating the placement of the signs to indicate a specific operation.
Positional number systems
Positional number systems are the most complex and efficient of the three types of number systems that exist. The combination of the own value of the symbols and the value assigned by their position allows them to construct very high figures 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 combinatorics the rest of the possible figures are produced to infinity, without the need to create new signs, but rather inaugurating new columns of symbols. Of course, this implies that an error in the chain also alters the total value of the figure.
The first examples of systems of this type arose within the great empires or the most demanding ancient cultures in cultural and commercial matters, such as the Babylonian Empire of the 2nd millennium BC. C. Examples of this type of numbering system are:
- The modern decimal system. With just the digits from 0 to 9, you can build any possible figure, adding columns whose value is added as you move to the right, based on the ten (10). Thus, by adding symbols to 1 we can construct 10, 195, 1958 or 19589. It is important to clarify that the symbols used come from Indo-Arabic numerals.
- The Indo-Arabic numeral system. Invented by the ancient wise men of India and later inherited by the Muslim Arabs, it reached the West through Al-Andalus and ended up replacing the traditional Roman numerals. In this system, similar to the modern decimal, the units from 0 to 9 are represented by specific glyphs, which represented the value of each one through lines and angles. The way this system works is basically the same as the modern Western decimal system.
- The Mayan number system. It was created to measure time, instead of to make mathematical transactions, and its base was vigesimal (20) and its symbols correspond to the calendar of this pre-Columbian civilization. The figures, grouped 20 by 20, are represented with basic signs (stripes, dots and snails or shells); and to move on to the next twenty, a point is added at the next level of writing. Furthermore, the Mayans were among the first to use the number zero.
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References
- “Numbering system” on Wikipedia.
- “Positional notation” on Wikipedia.
- “Other numbering systems” in the Junta de Andalucía (Spain).
- “Numeration Systems” at Encyclopedia.com.
- “Numeral system (Mathematics)” in The Encyclopaedia Britannica.
