Numbers

We explain everything about the numbers, what types exist and the characteristics of each one. Also, what are sets of numbers.

numbers
Numbers are vital for scientific thought and for the development of civilizations.

What are the numbers?

Numbers are abstractions, ideas or concepts created by the human being to represent, mainly, quantities and magnitudes. Numbers are one of humanity’s first inventions and played an important role in the creation of writing; since then they have been vital for scientific thought and for the daily life of civilizations.

At the same time, numbers and the relationships between them and with reality constitute a vast field of study, a fundamental part of the discipline of mathematics. Thus, there are various types and categories of numbers, and also different ways of representing them, possible operations and relationships between them, and even philosophical questions about what a number really is.

The word “number”, for its part, comes from the Latin numbersmade up of an ancient Indo-European root (nem-), which means “to distribute” or “to distribute”, and the suffix –it’swhich would then become –was. The ancestral word for “number”, thus, would have been namesalso related to other terms such as “norm” or “numismatics”.

See also: Algebraic language

Brief history of numbers

numbers history
The first numbers served for the invention of cuneiform writing.

Although it is not easy to find the origin of numbers, that is, of the concept of number itself, it is known that responded to the need to count in the ancestral societies of prehistory. Of these civilizations, bones with notches and sets of carvings have been found, which constitutes a clear sign of the primitive need of the human being to establish a system of registration of things or the passage of time.

The first such systems, however, are believed They were based on the use of the fingers and toes. For this reason, most number systems have a base decimal (10) or vigesimal (20).

However, the proper appearance of a written number, that is, of a symbol directly associated with a fixed amount, is a characteristic of more complex societies, such as those that arose in the Ancient Age, with great capacities for accumulating wealth and tax calculation needs, for trade, or to compose complex calendars.

It is estimated that the first written numbers appeared 5,000 years ago in Mesopotamia, on clay tablets which also served for the invention of cuneiform writing. In the following centuries, many other ancient cultures created their own methods and systems:

  • Additives, accumulating symbols to express greater value.
  • Positional, in which the order of the symbols expressed a greater or lesser value.
  • Hybrids, which combined the other two tendencies.
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Among them, the Egyptian systems (3000 BC approx.), Babylonian (2000 BC approx.), Maya (1000 BC approx.), Chinese (300 BC approx.), among others.

importance of numbers

numbers importance
Numbers are part of all kinds of operations that we carry out in our daily lives.

The creation of numbers is a central milestone of human civilization, which not only allowed ancestral people to count and compare sets of things to find out which had more ingredients (for example, which herd has more cows), but also allowed them to record what was told (for example, how many cows were in the herd yesterday). This may seem like a small thing today, but it forms the foundation of almost 10,000 years of study and use of numbers, which has spawned new and more complex systems and operations in which to apply them.

So the numbers are today an inseparable part of civilization, since they are part of the scientific, logistical, religious and all kinds of operations that we carry out in our daily lives. Without them, there would be no calendars, there would be no computer systems, and humanity would be unable to carry out the complex mathematical calculations that we have been capable of throughout history.

Roman numerals and Arabic numerals

Roman and Arabic numerals
Roman numerals use letters of the alphabet to represent exact values.

Since the numbers did not have a single common origin, but were simultaneously created by different cultures (each of which developed its method, its signs and its own registration rules), many of these number systems became extinct with the passage of time. time and were replaced by those of the great dominant powers. Hence, today two main sets of numbers are handled in the West, that is, two formats of numerical representation: Roman numerals and Arabic numerals.

  • roman numerals. Created and developed in Ancient Rome (around the 8th century BC) and used throughout its imperial times, this numbering system used letters of the Roman alphabet to represent exact values, and made up the figures depending on the location of each letter. Thus, for example, the letter I represented one, V five, X ten, L fifty and C Likewise, II represented two, VI six and XV fifteen, since the values ​​of the letters they accumulated; except if a letter preceded another of greater value, since in that case they were subtracted: IV represented four, IX nine and XC ninety. Roman numerals survive today for very specific uses, such as book chapters, century numbers, and other particular uses.
  • Arabic numerals. Created in India (and therefore actually called indo-arabians) and transmitted to the Islamic world, this decimal-based numbering system reached the West thanks to the Muslim invasion of southern Europe, and the establishment there of al-Andalus in the Iberian Peninsula. In this system the numbers are represented from one to ten by means of specific glyphs, which changed over time until they became the signs that are used today in almost the entire planet, the well-known 1, 2, 3, 4, 5, 6 , 7, 8, 9 and 0. The logic of these symbols, according to popular opinion, would lie in the total number of angles that each sign has, something that however historians deny. In any case, the construction of digits greater than ten is done by adding numbers to the right, thus going from units to tens and later to hundreds and so on (10, 100, 1000, etc.) always accumulating the value of the written numbers.
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Cardinal numbers and ordinal numbers

One of the main distinctions that exist between the numbers currently used has to do with what they denote:

  • Cardinal numbers: indicate quantities.
  • Ordinal numbers: indicate positions.

Thus, suppose that we have a certain number of candies in a bag, which we take out one by one and deposit on the table. We can use the cardinal numbers to find out how many candies there are in all (1, 2, 3, 4, and 5 candies in all) or we can use the ordinal numbers to find out in what order they come out of the bag (1st or first, 2nd or second, 3rd or third, 4th or fourth, and 5th or fifth).

Cardinal numbers, as we have just seen, are written as usual, while cardinal numbers require the appearance of an order symbol (°), or they are transcribed into letters using a combination of prefixes, roots, and suffixes. The ordinal numbers are also necessary to compose the names of the fractions: one fourth (¼), two fifths (⅖), etc.

More in: Ordinal numbers

Prime numbers and composite numbers

Prime numbers are a certain type of particular numbers, greater than 1 and that cannot be divided except by themselves and by unity.. This means that they cannot be decomposed into whole numbers, such as 2, 3, 5, 7, 11, 13, 17 or 19.

prime numbers they are infinite and they appear when counting with a frequency that many mathematicians have found intriguing, so they have wanted to find the exact pattern that determines when a prime number appears. Between the number 1 and the number 1000, for example, there are 168 prime numbers.

Numbers that are not prime are known as composite numbers.. These numbers can be divided by other numbers without giving fractional results. Examples of composite numbers are: 4, 6, 10, 15, 18, 22, etc.

More in: Prime numbers

sets of numbers

sets of numbers
Sets are infinite collections of numbers that share fundamental properties.

Numbers are studied by Number Theory, a discipline at the service of mathematics, and are often organized in sets, that is, in infinite groupings of numbers that share fundamental properties. These numerical sets are:

  • Natural numbers (N). Also called counting numbers, are those that we use daily and that are used to count, they begin with 0, 1, 2 and end at infinity. Their name is due to the fact that they obey the natural logic of the universe, that is, the things that exist and can be counted, such as how many fingers we have in our hand, or how many windows a building has. The natural numbers are classified into cousins Y compounds.
  • Integers (Z). It is a set formed by the natural numbers and their negative counterparts, that is, numbers preceded by the minus sign (-) and imaginarily located below (or to the left) of zero: -1, -2, -3… -999. Integers, thus, are an infinite set of positive (greater than 0) and negative (less than 0) numbers, as long as they are not fractional (hence the name of integers). This set is traditionally represented by the letter Z, from the German zahlen (“numbers”).
  • Rational numbers (Q). Rational numbers are both integers and fractional numbers, since this set is understood as the totality of numbers that can be represented as the quotient between an integer and a positive natural number. The set is represented by the letter Q (from quotient, “quotient” in various European languages). Examples of rational numbers are: 1, -1, ½, ¼, etc.
  • Irrational numbers (I). They are numbers whose decimal expression is neither exact nor periodic, that is, they do not comply with the quotient rule to be rational numbers. Numbers with infinite and aperiodic decimals, such as √ 7 or 3.1415918… belong to the irrational numbers, represented as a set by the letter I.
  • Real numbers (R). It is a set that includes both rational and irrational numbers, that is, any number that can be represented on the number line between minus infinity (negative infinity) and plus infinity (positive infinity) is a real number, regardless of the rest. of its properties. These numbers are represented by the letter R and any number that we can think of serves as an example of them.
  • Complex numbers (C). They are a prolongation or extension of the real numbers, which constitute an algebraically closed field that can be represented as the sum of a real number and an imaginary number. These are numbers that do not “exist” in nature, but must be sought and propitiated by students of pure mathematics through complex equations and calculations applied to other disciplines, such as physics, electronics and engineering.
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Continue with: Mathematical Thinking

References

  • “Number” on Wikipedia.
  • “Number Etymology” in the Online Spanish Etymological Dictionary.
  • “Number Sets” at NROC Project.
  • “The history of numbers” (video) at the Catholic University of Loja (Ecuador).
  • “Number (mathematics)”in The Encyclopaedia Britannica.