• What is a number system?
• Non-positional number systems
• Semi-positional number systems
• Positional number systems

We explain what a numbering system is and we study the characteristics of each type of system, through examples from different cultures.

Every number system contains a certain and finite set of symbols.

What is a number system?

A number system is a set of symbols and rules by which the number of objects in a number can be expressed. set, that is, through which all valid numbers can be represented. This means that every number system contains a given and finite set of symbols, plus a given 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 ancient civilizations had its own system, related to its way of seeing the world, that is, with 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 number (if it appears first, to one 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 rather 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 number, being able to 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 number 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 numbering system was sexagesimal (its base was 60).

Non-positional number systems

Non-positional systems were easy to learn but required numerous symbols.

Non-positional number systems were the first to exist and had the most primitive bases: fingers, knots on a rope, or other recording methods for coordinating number sets. For example, if you count on the fingers of one hand, then you can count on whole hands.

In these systems, the 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 (for this reason 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 these types of systems are:

• The Egyptian number system. Emerged around the third millennium BC. C., was based on the ten and used hieroglyphs different 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 20 as its base and used specific objects as symbols: a flag equaled 20 units, a feather or a few hairs equaled 400, a bag or sack equaled 8,000, among others.
• The Greek number system.Specifically the Ionian, was invented and spread in the eastern Mediterranean from the fourth century BC. C., replacing the pre-existing acrophonic system. It was an alphabetic system, which used letters to mean numbers, matching the letter with its cardinal place 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 systems responded to the needs of a more developed economy.

Semi-positional number 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 enjoy facilities to represent large numbers, managing the order of numbers and formal procedures such as multiplication, so they represent a step forward in complexity compared 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 the figures were built 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 where the symbol appears.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 ​​should be added; while if the symbol was to the left of a higher value symbol (as in IX = 9, or IV = 4), they had to be subtracted.
• The classical Chinese number system. Its origins date back to approximately 1500 BC. C. and is a very strict system of vertical representation of numbers through their 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 location of the signs to indicate a specific operation.

Positional number systems

The current numbering system comes from the Hindu-Arabic system.

Positional number systems are the most complex and efficient of the three types of number system that exist. The combination of the proper value of the symbols and the value assigned by their position allows them to build 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 combination the rest of the possible figures are produced, ad infinitum, without the need to create new signs, but rather by inaugurating new columns of symbols. Of course, this implies that an error in the string also alters the total value of the number.

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 second millennium BC. C. Examples of this type of numbering system are:

• The modern decimal system.With just the digits from 0 to 9, it allows you to build any number possible, adding columns whose value is added as you move to the right, having the ten as a base. Thus, adding symbols to 1 we can build 10, 195, 1958 or 19589. It is important to clarify that the symbols used come from Hindu-Arabic numerals.
• The Hindu-Arabic number system. Invented by the ancient sages of India and later inherited by the Muslim Arabs, it reached the West through Al-Andalus and ended up replacing the Roman numerals traditional. 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 by means of lines and angles. The system of operation of this system 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 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 to the next score, a point is added at the next level of writing. In addition, the Mayans they were among the first to use the number zero.