Figure (mathematics)

The ten digits used in Western culture. With them we can graphically represent the numbers.

A figure is a symbol or graphic character used to represent a number. For example, the characters «0», «1», «2», «3», «4», «5», «6», «7», «8» and «9» are digits of the Arabic numeral system, while the characters «I », «V», «X», «L”, “C”, “D » and «M» are figures from the Roman numeral system.

The numbers are also used as identifiers in: telephone numbers, road numbers; as order indicators in: serial numbers; such as codes (ISBN), etc.

Number and number

A numeral is a string of figures used to denote a number (not an identifying code). By way of example, the numerals «21», «2», «3», «4» and «500» represent in the Arabic system the same numbers as the respective numerals «XXI», «II», «III», «IV» and D» in the roman system.

Cipher and digit

A digit number is a number that can be expressed using a single digit numeral. By extension, a digit can be said to be each symbol or figure used to express a numeral or a number.

Small scales for the decimal system.

In the decimal system they are: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Thus, 157 is made up of the digits 1, 5 and 7. The name digit comes from the Latin digitus finger, because the 10 fingers correspond to the 10 digits in the common number system based on 10, that is, a decimal digit.

In mathematics and computer science, a numeric digit is a symbol, eg. «3», which used in combinations, v.gr. «37», represents numbers (whole or real) in positional numeral systems.

By tradition, at least since Ancient Egyptian times, the decimal system is used, due to the archaic use of ten fingers to help count, although there is no special reason why a numeral system should use the base ten.

In the decimal system, 10 digits are needed, although they have a different value depending on their position in the numeral, since their value varies from ten to ten, that is, units, tens (10¹), hundreds (10²), thousands (10³), and so on, such that a digit to the left has ten times the value of the given position and to the right right the tenth part of its value. To separate values less than unity, the decimal point is used (in Europe the comma). This method of positional notation comes from India and was transmitted to the West by Muslim mathematicians during the Middle Ages.

The simplest is the binary system, which only requires two digits, generally represented by 0 and 1; in the binary system they vary two by two: units, pairs (2¹), quatrains (2²), and so on. It is a system widely used in computing.

Examples of digits include any of the decimal characters "0" through "9", or the binary characters "0" or "1", and the digits "0"..."9", " A»,...,«F» used in the hexadecimal system. In a given numeral system, if the base (radical, in English en:radix) is an integer, the number of digits required, for the integer part, is equal to the next integer of the logarithm of the number to be represented divided by the logarithm of base. For the fractional part, the number of digits will depend on the precision needed to handle.

Graphic signs

In number systems, digits are combined to represent different numbers. If the value is determined by the position of the digit, it is called positional notation. If the digits have a fixed value, which does not depend on their position, it is called additive notation, such as Roman numerals.

Origin and evolution of the word cipher

The zero of the learned

When the 10th-century Arabs adopted numbering from India, they translated the word "sunya", which meant 'empty' or 'blank', by "sifr», 'empty' in Arabic. Later, the Indo-Arabic numeral system was introduced in Italy and the word “sifr” was Latinized as “zephirum”. The process began in the early 13th century and over time a succession of changes culminated in the Italian word “zero”.

Almost in parallel, a similar process developed in Germany. Jordanus Nemorarius changed the word "sifr" to "figure". For a time in Europe both words denoted zero. As one of the testimonies of this stage, the English word «cipher» currently has two meanings: 'cipher', in the modern sense, and 'zero' in its archaic form, according to its etymology..

The words «cifra», «chiffre», «cipher», «ziffer» and «zero >» represented zero for the learned.

The number of the masses

History does not contemplate the titles and honors of the learned. Social processes irremediably change some of the original concepts. When the mass adopts a use, all efforts to the contrary are useless.

In ancient times and in the Middle Ages, calculations were made by experts. Until the definitive adoption of the position system and zero, multiplication and division were carried out by duplications and mediations, respectively. For example, to multiply a number by 13, the multiplier was broken down into powers of 2, in this case, 8 + 4 + 1. The multiplicand was doubled two and three times. Then the triple doubling, the double doubling, and the original amount were added together. The division followed a similar but inverse process. The calculations required a lot of work time and the cost was high. A residue of this can be seen in the way old measurements such as the English inch are subdivided: halves, fourths, eighths, sixteenths, thirty-seconds.

The merchants of those times had to cover these expenses to have control and information about their businesses. When the news of the new numbering system reached them, they very readily saw the advantage it would give them. Calculations were easy to perform, and higher education was no longer required to master arithmetic operations. They would not have to pay for the service of an expert.

It's really remarkable that these people realized the critical role of zero in the new system. The mass identified the entire system with its most characteristic feature, the number, using, then, number with the sense of numerical sign that it has today in our civilization. This use was totally opposite to the meaning of the number of the learned.

The secret and the fight

Traders thought it prudent to reserve that use for themselves, as an advantage. The system was used secretly. In this way, the word "figure" was used as a secret sign. The words "decrypt" and "encryption" survive from that stage. An encrypted code is a text of meaning that is inaccessible if the key is not available. When the key is obtained the secret is revealed, the secret code is deciphered, "zeroed" or the secret.

For selfish reasons the merchants kept the system to themselves. On the other hand, there was a reaction from supporters of traditions and defenders of ancient philosophies, joined by those who lived on the difficult calculations of yesteryear. For these reasons, the system took a long time to catch on. The fighting lasted from the 11th century to the 15th century. In some places it was even banned. But by the beginning of the 16th century it was already firmly established and did not suffer any delay in its development.

Supporters of the position system were called "algorists" and defenders of the old system, "abacists", because they used the abacus in their calculations. In those times also "abaci" was synonymous with arithmetic.

Current use of the word

Once the new system was fully adopted, the use of the word "figure" in the sense of a number sign was so strongly entrenched that the effort of scholars to return to the original meaning of "zero" was futile. They had no choice but to leave "figure" with that sense and take "zero" to designate empty space until they reached the use it has now.

Other meanings

In astronomy, an astronomical digit is each of the equal parts into which the diameter of the lunar and solar disks is divided to express the importance of an eclipse. Thus, a lunar eclipse of 8 digits affects two thirds of the diameter of our planet (see magnitude of an eclipse).