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The world of numbers is very mysterious and interesting. Numbers are very important in our world. I want to learn as much as possible about the origin of numbers and their meaning in our lives. How to use them and what role do they play in our lives?
Last year in mathematics lessons we began to study the topic “Positive and Negative Numbers”. I had a question: when did negative numbers appear, in which country, which scientists studied this issue. I read on Wikipedia that a negative number is an element of the set of negative numbers, which (together with zero) appeared in mathematics when the set was expanded natural numbers. The purpose of the extension is to allow the subtraction operation to be performed on any number. As a result of the expansion, a set (ring) of integers is obtained, consisting of positive (natural) numbers, negative numbers and zero.
As a result, I decided to explore the history of negative numbers.
The purpose of this work is to study the history of the emergence of negative and positive numbers.
The object of study is negative numbers and positive numbers
History of positive and negative numbers
It took a long time for people to get used to negative numbers. Negative numbers seemed incomprehensible to them, they did not use them, they simply did not see much meaning in them. These numbers appeared much later than natural numbers and ordinary fractions.
The first information about negative numbers was found by Chinese mathematicians in the 2nd century. BC e. and even then, only the rules for adding and subtracting positive and negative numbers were known; the rules of multiplication and division did not apply.
In Chinese mathematics, positive quantities were called “chen”, negative quantities were called “fu”; they were portrayed different colors: “chen” - red, “fu” - black. This can be seen in the book “Arithmetic in Nine Chapters” (Author Zhang Can). This method of depiction was used in China until the middle of the 12th century, until Li Ye proposed a more convenient designation for negative numbers - the numbers that depicted negative numbers were crossed out with a line diagonally from right to left.
Only in the 7th century. Indian mathematicians began to widely use negative numbers, but treated them with some mistrust. Bhaskhara directly wrote: “People do not approve of abstract negative numbers...”. This is how the Indian mathematician Brahmagupta set out the rules of addition and subtraction: “property and property is property, the sum of two debts is debt; the sum of property and zero is property; the sum of two zeros is zero... Debt, which is subtracted from zero, becomes property, and property becomes debt. If it is necessary to take away property from debt, and debt from property, then they take their sum.” “The sum of two properties is property.”
(+x) + (+y) = +(x + y) (-x) + (-y) = - (x + y)
(-x) + (+y) = - (x - y) (-x) + (+y) = +(y - x)
0 - (-x) = +x 0 - (+x) = -x
The Indians called positive numbers "dhana" or "sva" (property), and negative numbers "rina" or "kshaya" (debt). Indian scientists, trying to find examples of such subtraction in life, came to interpret it from the point of view of trade calculations. If a merchant has 5000 rubles. and buys goods for 3000 rubles, he has 5000 - 3000 = 2000 rubles left. If he has 3,000 rubles, but buys for 5,000 rubles, then he remains in debt for 2,000 rubles. In accordance with this, it was believed that here a subtraction of 3000 - 5000 was performed, the result being the number 2000 with a dot at the top, meaning “two thousand debt.” This interpretation was artificial; the merchant never found the amount of debt by subtracting 3000 - 5000, but always subtracted 5000 - 3000.
A little later in Ancient India and China, they guessed that instead of the words “debt of 10 yuan” they should write simply “10 yuan”, but draw these hieroglyphs with black ink. And in ancient times there were no signs “+” and “-” either for numbers or for actions.
The Greeks also did not use signs at first. The ancient Greek scientist Diophantus did not recognize negative numbers at all, and if, when solving an equation, he got negative root, then it discarded it as "unavailable". And Diophantus tried to formulate problems and compose equations in such a way as to avoid negative roots, but soon Diophantus of Alexandria began to denote subtraction with a sign.
Rules for dealing with positive and negative numbers were proposed already in the 3rd century in Egypt. The introduction of negative quantities first occurred with Diophantus. He even used a special character for them. At the same time, Diophantus uses such figures of speech as “Let us add a negative to both sides,” and even formulates the rule of signs: “A negative multiplied by a negative gives a positive, while a negative multiplied by a positive gives a negative.”
In Europe, negative numbers began to be used from the 12th-13th centuries, but not until the 16th century. most scientists considered them “false”, “imaginary” or “absurd”, in contrast to positive numbers - “true”. Positive numbers were also interpreted as “property”, and negative numbers as “debt”, “shortage”. Even the famous mathematician Blaise Pascal argued that 0 − 4 = 0, since nothing can be less than nothing. In Europe, Leonardo Fibonacci of Pisa came quite close to the idea of negative quantity at the beginning of the 13th century. At a problem-solving competition with the court mathematicians of Frederick II, Leonardo of Pisa was asked to solve a problem: it was necessary to find the capital of several individuals. Fibonacci received negative meaning. “This case,” said Fibonacci, “is impossible, unless we accept that one had not capital, but debt.” However, negative numbers were first used explicitly at the end of the 15th century by the French mathematician Chuquet. Author of a handwritten treatise on arithmetic and algebra “The Science of Numbers in three parts" The symbolism of Shuque is close to modern.
The recognition of negative numbers was facilitated by the work of the French mathematician, physicist and philosopher René Descartes. He proposed a geometric interpretation of positive and negative numbers - he introduced the coordinate line. (1637).
Positive numbers are represented on the number axis by points lying to the right of the beginning 0, negative numbers - to the left. The geometric interpretation of positive and negative numbers contributed to their recognition.
In 1544, the German mathematician Michael Stiefel first considered negative numbers as numbers less than zero (i.e. "less than nothing"). From this point on, negative numbers are no longer viewed as a debt, but in a completely new way. Stiefel himself wrote: “Zero is between true and absurd numbers...”
Almost simultaneously with Stiefel, the idea of negative numbers was defended by Bombelli Raffaele (about 1530-1572), an Italian mathematician and engineer who rediscovered the work of Diophantus.
Likewise, Girard considered negative numbers to be completely acceptable and useful, in particular, to indicate the lack of something.
Every physicist constantly deals with numbers: he always measures, calculates, calculates something. Everywhere in his papers there are numbers, numbers and numbers. If you look closely at the physicist’s notes, you will find that when writing numbers, he often uses the signs “+” and “-”. (For example: thermometer, depth and height scale)
Only in early XIX V. the theory of negative numbers completed its development, and “absurd numbers” received universal recognition.
Definition of the concept of number
IN modern world people constantly use numbers without even thinking about their origin. Without knowledge of the past it is impossible to understand the present. Number is one of the basic concepts of mathematics. The concept of number developed in close connection with the study of quantities; this connection continues to this day. In all branches of modern mathematics we have to consider different quantities and use numbers. Number is an abstraction used to quantitative characteristics objects. Having appeared back in primitive society From the needs of counting, the concept of number changed and enriched and turned into the most important mathematical concept.
Exists a large number of definitions of the concept “number”.
The first scientific definition of number was given by Euclid in his Elements, which he apparently inherited from his compatriot Eudoxus of Cnidus (about 408 - about 355 BC): “A unit is that in accordance with which each of the existing things are called one. A number is a set made up of units.” This is how the Russian mathematician Magnitsky defined the concept of number in his “Arithmetic” (1703). Even earlier than Euclid, Aristotle gave the following definition: “A number is a set that is measured using units.” In his “General Arithmetic” (1707), the great English physicist, mechanic, astronomer and mathematician Isaac Newton writes: “By number we mean not so much a set of units as the abstract relation of a quantity to another quantity of the same kind, taken as a unit.” . There are three types of numbers: integer, fractional and irrational. A whole number is something that is measured by one; fractional is a multiple of one, irrational is a number that is not commensurate with one.”
Mariupol mathematician S.F. Klyuykov also contributed to the definition of the concept of number: “Numbers are mathematical models real world invented by man for his knowledge.” He also introduced the so-called “functional numbers” into the traditional classification of numbers, meaning what is usually called functions all over the world.
Natural numbers arose when counting objects. I learned about this in 5th grade. Then I learned that the human need to measure quantities is not always expressed in whole numbers. After expanding the set of natural numbers to fractions, it became possible to divide any integer by another integer (with the exception of division by zero). Appeared fractional numbers. Subtract an integer from another integer when the subtracted is greater than the minuend, for a long time seemed impossible. What was interesting to me was the fact that for a long time many mathematicians did not recognize negative numbers, believing that they did not correspond to any real phenomena.
Origin of the words "plus" and "minus"
The terms come from the words plus - “more”, minus - “less”. At first, actions were denoted by the first letters p; m. Many mathematicians preferred or The origin of modern signs “+” and “-” is not entirely clear. The “+” sign probably comes from the abbreviation et, i.e. "And". However, it may have arisen from trade practice: sold measures of wine were marked “-” on the barrel, and when the stock was restored, they were crossed out, resulting in a “+” sign.
In Italy, moneylenders, when lending money, put the amount of the debt and a dash in front of the debtor’s name, like our minus, and when the debtor returned the money, they crossed it out, it turned out something like our plus.
Modern "+" signs appeared in Germany in last decade XV century in the book of Widmann, which was a guide to counting for merchants (1489). Czech Jan Widman already wrote “+” and “-” for addition and subtraction.
A little later, the German scientist Michel Stiefel wrote "Complete Arithmetic", which was published in 1544. It contains the following entries for numbers: 0-2; 0+2; 0-5; 0+7. He called numbers of the first type “less than nothing” or “lower than nothing.” He called numbers of the second type “more than nothing” or “higher than nothing.” Of course, you understand these names, because “nothing” is 0.
Negative numbers in Egypt
However, despite such doubts, rules for operating with positive and negative numbers were proposed already in the 3rd century in Egypt. The introduction of negative quantities first occurred with Diophantus. He even used a special symbol for them (nowadays we use the minus sign for this purpose). True, scientists argue whether Diophantus’ symbol denoted a negative number or simply a subtraction operation, because in Diophantus negative numbers do not occur in isolation, but only in the form of positive differences; and he considers only rational positive numbers as answers to problems. But at the same time, Diophantus uses such figures of speech as “Let us add a negative to both sides,” and even formulates the rule of signs: “A negative multiplied by a negative gives a positive, while a negative multiplied by a positive gives a negative” (that is, which is now usually formulated: “Minus by minus gives a plus, minus by plus gives a minus”).
(-) (-) = (+), (-) (+) = (-).
Negative numbers in Ancient Asia
In Chinese mathematics, positive quantities were called “chen”, negative quantities were called “fu”; they were depicted in different colors: “chen” - red, “fu” - black. This method of depiction was used in China until the middle of the 12th century, until Li Ye proposed a more convenient designation for negative numbers - the numbers that depicted negative numbers were crossed out with a line diagonally from right to left. Indian scientists, trying to find examples of such subtraction in life, came to interpret it from the point of view of trade calculations.
If a merchant has 5000 rubles. and buys goods for 3000 rubles, he has 5000 - 3000 = 2000 rubles left. If he has 3,000 rubles, but buys for 5,000 rubles, then he remains in debt for 2,000 rubles. In accordance with this, it was believed that here a subtraction of 3000 - 5000 was performed, the result being the number 2000 with a dot at the top, meaning “two thousand debt.”
This interpretation was artificial; the merchant never found the amount of debt by subtracting 3000 - 5000, but always subtracted 5000 - 3000. In addition, on this basis, it was only possible to explain with a stretch the rules for adding and subtracting “numbers with dots,” but it was impossible was to explain the rules of multiplication or division.
In the 5th-6th centuries, negative numbers appeared and became very widespread in Indian mathematics. In India, negative numbers were used systematically, much as we do now. Indian mathematicians have been using negative numbers since the 7th century. n. e.: Brahmagupta formulated the rules for arithmetic operations with them. In his work we read: “property and property are property, the sum of two debts is debt; the sum of property and zero is property; the sum of two zeros is zero... Debt, which is subtracted from zero, becomes property, and property becomes debt. If it is necessary to take away property from debt, and debt from property, then they take their sum.”
The Indians called positive numbers "dhana" or "sva" (property), and negative numbers "rina" or "kshaya" (debt). However, in India there were problems with understanding and accepting negative numbers.
Negative numbers in Europe
European mathematicians did not approve of them for a long time, because the interpretation of “property-debt” caused bewilderment and doubt. In fact, how can one “add” or “subtract” property and debts, what real meaning can “multiplying” or “dividing” property by debt have? (G.I. Glazer, History of mathematics in school grades IV-VI. Moscow, Prosveshchenie, 1981)
That is why negative numbers have gained a place in mathematics with great difficulty. In Europe, Leonardo Fibonacci of Pisa came quite close to the idea of a negative quantity at the beginning of the 13th century, but negative numbers were first used explicitly at the end of the 15th century by the French mathematician Chuquet. Author of a handwritten treatise on arithmetic and algebra, “The Science of Numbers in Three Parts.” The symbolism of Shuquet is approaching the modern one (Mathematical encyclopedic Dictionary. M., Sov. encyclopedia, 1988)
Modern interpretation of negative numbers
In 1544, the German mathematician Michael Stiefel first considered negative numbers as numbers less than zero (i.e. "less than nothing"). From this point on, negative numbers are no longer viewed as a debt, but in a completely new way. Stiefel himself wrote: “Zero is between true and absurd numbers...” (G.I. Glazer, History of mathematics in school grades IV-VI. Moscow, Prosveshchenie, 1981)
After this, Stiefel devoted his work entirely to mathematics, in which he was a self-taught genius. One of the first in Europe after Nicola Chuquet began to operate with negative numbers.
The famous French mathematician René Descartes in “Geometry” (1637) describes the geometric interpretation of positive and negative numbers; positive numbers are represented on the number axis by points lying to the right of the beginning 0, negative numbers - to the left. The geometric interpretation of positive and negative numbers led to a clearer understanding of the nature of negative numbers and contributed to their recognition.
Almost simultaneously with Stiefel, the idea of negative numbers was defended by R. Bombelli Raffaele (about 1530-1572), an Italian mathematician and engineer who rediscovered the work of Diophantus.
Bombelli and Girard, on the contrary, considered negative numbers to be quite acceptable and useful, in particular for indicating the lack of something. The modern designation for positive and negative numbers with the signs “+” and “-” was used by the German mathematician Widmann. The expression “lower than nothing” shows that Stiefel and some others mentally imagined positive and negative numbers as points on a vertical scale (like a thermometer scale). Then developed by the mathematician A. Girard, the idea of negative numbers as points on a certain line, located on the other side of zero than positive ones, turned out to be decisive in providing these numbers with citizenship rights, especially as a result of the development of the coordinate method by P. Fermat and R. Descartes .
Conclusion
In my work, I investigated the history of the emergence of negative numbers. During the research, I concluded:
Modern science encounters quantities of such a complex nature that to study them it is necessary to invent new types of numbers.
When introducing new numbers great importance have two circumstances:
a) the rules of action over them must be fully defined and not lead to contradictions;
b) new number systems should help either solve new problems or improve already known solutions.
Currently, time has seven generally accepted levels of generalization of numbers: natural, rational, real, complex, vector, matrix and transfinite numbers. Some scientists propose to consider functions as functional numbers and expand the degree of generalization of numbers to twelve levels.
I will try to study all these sets of numbers.
Application
POEM
"Adding negative numbers and numbers with different signs»
If you really want to fold
The numbers are negative, there is no need to bother:
We need to quickly find out the sum of the modules,
Then take and add a minus sign to it.
If numbers with different signs are given,
To find their sum, we are all right there.
We can quickly select a larger module.
From it we subtract the smaller one.
The most important thing is not to forget the sign!
Which one will you put? - we want to ask
We’ll tell you a secret, it couldn’t be simpler,
Write down the sign where the module is greater in your answer.
Rules for adding positive and negative numbers
Add minus to minus,
You can get a minus.
If you add up minus, plus,
Will it turn out to be an embarrassment?!
You choose the sign of the number
Which is stronger, don't yawn!
Take them away from the modules
Make peace with all the numbers!
The rules of multiplication can be interpreted this way:
“My friend’s friend is my friend”: + ∙ + = + .
“The enemy of my enemy is my friend”: ─ ∙ ─ = +.
“The friend of my enemy is my enemy”: + ∙ ─ = ─.
“The enemy of my friend is my enemy”: ─ ∙ + = ─.
The multiplication sign is a dot, it has three signs:
Cover two of them, the third will give the answer.
For example.
How to determine the sign of the product 2∙(-3)?
Let's cover the plus and minus signs with our hands. There remains a minus sign
Bibliography
"Story ancient world", 5th grade. Kolpakov, Selunskaya.
“History of mathematics in antiquity”, E. Kolman.
"Student's Handbook." Publishing house "VES", St. Petersburg. 2003
Great mathematical encyclopedia. Yakusheva G.M. and etc.
Vigasin A.A., Goder G.I., “History of the Ancient World,” 5th grade textbook, 2001.
Wikipedia. Free encyclopedia.
Emergence and development mathematical science: Book. For the teacher. - M.: Education, 1987.
Gelfman E.G. "Positive and Negative Numbers" tutorial in mathematics for the 6th grade, 2001.
Head. ed. M. D. Aksyonova. - M.: Avanta+, 1998.
Glazer G. I. "History of mathematics at school", Moscow, "Prosveshchenie", 1981
Children's encyclopedia "I know the world", Moscow, "Enlightenment", 1995.
History of mathematics in school, grades IV-VI. G.I. Glazer, Moscow, Education, 1981.
M.: Philol. LLC "WORD": OLMA-PRESS, 2005.
Malygin K.A.
Mathematical encyclopedic dictionary. M., Sov. encyclopedia, 1988.
Nurk E.R., Telgmaa A.E. "Mathematics 6th grade", Moscow, "Enlightenment", 1989
Textbook 5th grade. Vilenkin, Zhokhov, Chesnokov, Shvartsburd.
Friedman L.M.. "Studying Mathematics", educational publication, 1994.
E.G. Gelfman et al., Positive and negative numbers in the Buratino theater. Mathematics textbook for 6th grade. 3rd edition, revised, - Tomsk: Publishing House Tomsk University, 1998
Encyclopedia for children. T.11. Mathematics
The history of the emergence of negative numbers is very old and long. Since negative numbers are something ephemeral, unreal, people for a long time did not recognize their existence.
It all started in China, around the 2nd century BC. Perhaps they were known in China before, but the first mention dates back to that time. There they began to use negative numbers and considered them “debts,” while the positive ones were called “property.” The record that exists now did not exist then, and negative numbers were written in black, and positive numbers in red.
We find the first mention of negative numbers in the book “Mathematics in Nine Chapters” by the Chinese scientist Zhang Can.
Next, in V-VI centuries negative numbers began to be used quite widely in China and India. True, in China they were treated with caution and tried to minimize their use, but in India, on the contrary, they were used very widely. There calculations were made with them and negative numbers did not seem incomprehensible.
Indian scientists Brahmagupta Bhaskara (VII-VIII centuries) are famous, who in their teachings left detailed explanations of working with negative numbers.
And in Antiquity, for example, in Babylon and in Ancient Egypt, negative numbers were not used at all. And if the calculation resulted in a negative number, it was considered that there was no solution.
Likewise, in Europe negative numbers were not recognized for a very long time. They were considered “imaginary” and “absurd.” They did not perform any actions with them, but simply discarded them if the answer was negative. They believed that if you subtract any number from 0, then the answer will be 0, since nothing can be less than zero- emptiness.
For the first time in Europe, Leonardo of Pisa (Fibonacci) turned his attention to negative numbers. And he described them in his work “The Book of Abacus” in 1202.
Leonardo Fibonacci Leonardo Fibonacci
Later, in 1544, Mikhail Stiefel, in his book “Complete Arithmetic,” first introduced the concept of negative numbers and described in detail the operations with them. “Zero is between the absurd and the true numbers.”
And in the 17th century, mathematician Rene Descartes proposed putting negative numbers on the digital axis to the left of zero.
Rene Descartes Rene Descartes
From that time on, negative numbers began to be widely used and accepted, although for a long time many scientists denied them.
In 1831, Gauss called negative numbers absolutely equivalent to positive ones. And I didn’t consider the fact that not all actions can be performed with them to be something terrible; with fractions, for example, not all actions can be done either.
And in the 19th century, Wilman Hamilton and Hermann Grassmann created a complete theory of negative numbers. Since that time, negative numbers have gained their rights and now no one doubts their reality.
Man invented the number in order to somehow indicate for himself and others the results of counting and measurement. Apparently, the first concepts of number among people appeared in the Paleolithic era, but developed already in the Neolithic. The first step in the appearance of numbers, apparently, was the awareness of the division of measure into “one” and “many”.
In the Ancient world they first began to use special signs to designate numbers: their images were preserved on clay tablets of Mesopotamia, on Egyptian papyri, and so on.
Mathematics developed further. And in various countries their own special, authentic and noticeably different number systems began to form. Even a schoolchild now knows how the Roman and Arabic writing of numerals differed. Numbers have been passed down from country to country, culture to culture, as an important and valuable invention and heritage. Modern numbers, on which both Slavic and Western civilization is built, are Arabic numbers, but borrowed from India. Many numbers that are now familiar to everyone were invented in India, for example, the number “0”.
The division of numbers into positive and negative dates back to the developments of mathematicians of the Middle Ages. Again, negative numbers were first used in India. This made it easier for merchants to calculate losses and debts. At that time, arithmetic was already a highly developed applied field, and algebra was beginning to develop. With the introduction of Cartesian geometry, his coordinate systems, negative numbers firmly came into use. They haven't left here to this day.
Complex numbers are modern concept, such numbers are also called “imaginary numbers” and are derived from the formal solution of cubic and quadratic equations. Their “father” was the medieval mathematician Gerolamo Cardano. During the time of Descartes, complex numbers, like negative numbers, became firmly established in mathematical use.
History of Negative Numbers
It is known that natural numbers arose when counting objects. The human need to measure quantities and the fact that the result of a measurement is not always expressed as an integer led to the expansion of the set of natural numbers. Zero and fractional numbers were introduced.
Process historical development The concept of numbers didn't end there. However, the first impetus for expanding the concept of number was not always purely the practical needs of people. It also happened that the problems of mathematics itself required expanding the concept of number. This is exactly what happened with the emergence of negative numbers. Solving many problems, especially those involving equations, involved subtracting a larger number from a smaller number. This required the introduction of new numbers.
Negative numbers first appeared in Ancient China already about 2100 years ago. They also knew how to add and subtract positive and negative numbers; the rules of multiplication and division were not applied.
In the II century. BC e. Chinese scientist Zhang Can wrote the book Arithmetic in Nine Chapters. From the contents of the book it is clear that this is not a completely independent work, but a reworking of other books written long before Zhang Can. In this book, negative quantities are encountered for the first time in science. They are understood differently from the way we understand and apply them. He does not have a complete and clear understanding of the nature of negative quantities and the rules for operating with them. He understood every negative number as a debt, and every positive number as property. He performed operations with negative numbers not the same way as we do, but using reasoning about debt. For example, if you add another debt to one debt, then the result is debt, not property (i.e., according to us (- x) + (- x) = - 2x. The minus sign was not known then, therefore, in order to distinguish the numbers , expressing debt, Zhan Can wrote them in a different ink than the numbers expressing property (positive).
In Chinese mathematics, positive quantities were called “chen” and were depicted in red, while negative quantities were called “fu” and were depicted in black. This method of depiction was used in China until the middle of the 12th century, until Li Ye proposed a more convenient designation for negative numbers - the numbers that depicted negative numbers were crossed out with a line diagonally from right to left. Although Chinese scientists explained negative quantities as debt, and positive quantities as property, they still avoided their widespread use, since these numbers seemed incomprehensible, and actions with them were unclear. If the problem led to a negative solution, then they tried to replace the condition (like the Greeks) so that in the end a positive solution would be obtained.
In the 5th-6th centuries, negative numbers appeared and became very widespread in Indian mathematics. For calculations, mathematicians of that time used a counting board, on which numbers were depicted using counting sticks. Since there were no signs + and – at that time, positive numbers were depicted with red sticks, and negative numbers were depicted with black sticks and were called “debt” and “shortage.” Positive numbers were interpreted as “property.” Unlike China, the rules of multiplication and division were already known in India. In India, negative numbers were used systematically, much as we do now. Already in the work of the outstanding Indian mathematician and astronomer Brahmagupta (598 - about 660) we read: “property and property is property, the sum of two debts is a debt; the sum of property and zero is property; the sum of two zeros is zero... Debt, which is subtracted from zero, becomes property, and property becomes debt. If it is necessary to take away property from debt, and debt from property, then they take their sum.”
Indian mathematicians used negative numbers when solving equations, and subtraction was replaced by addition with an equally opposite number.
Along with negative numbers, Indian mathematicians introduced the concept of zero, which allowed them to create a decimal number system. But for a long time, zero was not recognized as a number; “nullus” in Latin means no, the absence of a number. And only after 10 centuries, in the 17th century, with the introduction of a coordinate system, zero became a number.
The Greeks also did not use signs at first. The ancient Greek scientist Diophantus did not recognize negative numbers at all, and if, when solving an equation, a negative root was obtained, he discarded it as “inaccessible.” And Diophantus tried to formulate problems and compose equations in such a way as to avoid negative roots, but soon Diophantus of Alexandria began to denote subtraction with the sign .
Despite the fact that negative numbers have been used for a long time, they were treated with some distrust, considering them not entirely real, their interpretation as property-debt caused bewilderment: how can one “add” and “subtract” property and debts?
In Europe, recognition came a thousand years later. Leonardo of Pisa (Fibonacci) came quite close to the idea of a negative quantity at the beginning of the 13th century, who also introduced it to solve financial tasks with debts and came to the idea that negative quantities should be taken in the opposite sense of positive ones. In those years, the so-called mathematical duels were developed. At a problem-solving competition with the court mathematicians of Frederick II, Leonardo of Pisa (Fibonacci) was asked to solve a problem: it was necessary to find the capital of several individuals. Fibonacci received a negative value. “This case,” said Fibonacci, “is impossible, unless we assume that one had not capital, but debt.”
In 1202, he first used negative numbers to calculate his losses. However, negative numbers were used explicitly for the first time at the end of the 15th century by the French mathematician Chuquet.
Nevertheless, until the 17th century, negative numbers were “in the fold” and for a long time they were called “false”, “imaginary” or “absurd”. And even in the 17th century, the famous mathematician Blaise Pascal argued that 0-4 = 0 because there is no number that can be less than nothing, and until the 19th century, mathematicians often discarded negative numbers in their calculations, considering them meaningless...
Bombelli and Girard, on the contrary, considered negative numbers to be quite acceptable and useful, in particular for indicating the lack of something. An echo of those times is the fact that in modern arithmetic the operation of subtraction and the sign of negative numbers are denoted by the same symbol (minus), although algebraically these are completely different concepts.
In Italy, when lending money, moneylenders put the amount of the debt and a line in front of the debtor’s name, like our minus, and when the debtor returned the money, they crossed it out, so it looked like our plus. You can consider a plus as a crossed out minus!
Modern notation for positive and negative numbers with signs
“+” and “-” were used by the German mathematician Widmann.
The German mathematician Michael Stiefel, in his book “Complete Arithmetic” (1544), first introduced the concept of negative numbers as numbers less than zero (less than nothing). This was a very big step forward in justifying negative numbers. He made it possible to view negative numbers not as a debt, but in a completely different, new way. But Stiefel called negative numbers absurd; actions with them, in his words, “also go absurdly, topsy-turvy.”
After Stiefel, scientists began to perform operations with negative numbers more confidently.
Negative solutions to problems were increasingly retained and interpreted.
In the 17th century The great French mathematician Rene Descartes proposed putting negative numbers on the number line to the left of zero. Now it all seems so simple and understandable to us, but to reach this idea, it took eighteen centuries of work of scientific thought from the Chinese scientist Zhang Can to Descartes.
In the works of Descartes, negative numbers received, as they say, a real interpretation. Descartes and his followers recognized them on an equal basis with positive ones. But in operations with negative numbers, not everything was clear (for example, multiplication by them), so many scientists did not want to recognize negative numbers as real numbers. A large and long dispute broke out among scientists about the essence of negative numbers and whether to recognize negative numbers as real numbers or not. This dispute after Descartes lasted about 200 years. During this period, mathematics as a science developed very greatly, and negative numbers were encountered at every step. Mathematics has become unthinkable, impossible without negative numbers. All more scientists it became clear that negative numbers are real numbers, just as real, in fact existing numbers, as positive numbers.
Negative numbers have hardly won their place in mathematics. No matter how hard scientists try to avoid them. However, they did not always succeed in this. Life presented science with new and new tasks, and more and more often these tasks led to negative solutions in China, India, and Europe. Only at the beginning of the 19th century. the theory of negative numbers completed its development, and “absurd numbers” received universal recognition.
Every physicist constantly deals with numbers: he always measures, calculates, calculates something. Everywhere in his papers there are numbers, numbers and numbers. If you look closely at the physicist’s notes, you will find that when writing numbers, he often uses the signs “+” and “-”.
How do positive, and especially negative numbers arise in physics?
A physicist deals with various physical quantities that describe the various properties of objects and phenomena around us. Building height, distance from school to home, mass and temperature human body, car speed, can volume, force electric current, refractive index of water, power nuclear explosion, the voltage between the electrodes, the duration of a lesson or recess, the electric charge of a metal ball - all these are examples physical quantities. A physical quantity can be measured.
One should not think that any characteristic of an object or natural phenomenon can be measured and, therefore, is a physical quantity. It's not like that at all. For example, we say: “Which beautiful mountains around! And what beautiful lake there below! And what a beautiful spruce tree over there on that rock! But we cannot measure the beauty of the mountains, the lake, or this lonely spruce!” This means that a characteristic such as beauty is not a physical quantity.
Measurements of physical quantities are carried out using measuring instruments such as a ruler, watch, scales, etc.
So, numbers in physics arise as a result of measuring physical quantities, and the numerical value of a physical quantity obtained as a result of measurement depends: on how this physical quantity is defined; from the units of measurement used.
Let's look at the scale of a regular street thermometer.
It has the form shown on scale 1. Only positive numbers are printed on it, and therefore, when indicating numerical value temperatures have to be further explained by 20 degrees Celsius (above zero). This is inconvenient for physicists - after all, you can’t put words into a formula! Therefore, in physics a scale with negative numbers is used.
Let's look at the physical map of the world. The land areas on it are painted in various shades of green and brown, and the seas and oceans are painted in blue and blue. Each color has its own height (for land) or depth (for seas and oceans). A scale of depths and heights is drawn on the map, which shows what height (depth) a particular color means,
Using such a scale, it is enough to indicate the number without any additional words: positive numbers answer various places on land above the surface of the sea; negative numbers correspond to points below the sea surface.
In the height scale we considered, the height of the water surface in the World Ocean is taken as zero. This scale is used in geodesy and cartography.
In contrast, in everyday life we usually take the height of the earth’s surface (in the place where we are) as zero height.
3.1 How were years counted in ancient times?
IN different countries differently. For example, in Ancient Egypt, every time he began to rule new king, the counting of years began anew. The first year of the king's reign was considered the first year, the second - the second, and so on. When this king died and a new one came to power, the first year began again, then the second, the third. The counting of years used by the inhabitants of one of the most ancient cities in the world, Rome, was different. The Romans considered the year the city was founded to be the first, the next year to be the second, and so on.
The counting of years that we use arose a long time ago and is associated with the veneration of Jesus Christ, the founder Christian religion. Counting the years from the birth of Jesus Christ was gradually adopted in different countries. In our country, it was introduced by Tsar Peter the Great three hundred years ago. We call the time calculated from the Nativity of Christ OUR ERA (and we write it in abbreviated form N.E.). Our era continues for two thousand years.
Conclusion
Most people know negative numbers, but there are some whose representation of negative numbers is incorrect.
Negative numbers are most common in the exact sciences, mathematics and physics.
In physics, negative numbers arise as a result of measurements and calculations of physical quantities. Negative number – shows the value electric charge. In other sciences, such as geography and history, a negative number can be replaced with words, for example, below sea level, and in history - 157 BC. e.
Literature
1. Great scientific encyclopedia, 2005.
2. Vigasin A. A., “History of the Ancient World,” 5th grade textbook, 2001.
3. Vygovskaya V.V. “Lesson-based developments in Mathematics: 6th grade” - M.: VAKO, 2008
4. “Positive and negative numbers”, textbook on mathematics for the 6th grade, 2001.
5. Children's encyclopedia “I know the world”, Moscow, “Enlightenment”, 1995.
6.. “Studying Mathematics”, educational publication, 1994.
7. “Elements of historicism in teaching mathematics in secondary school”, Moscow, “Prosveshchenie”, 1982
8. Nurk E.R., Telgmaa A.E. “Mathematics 6th grade”, Moscow, “Enlightenment”, 1989
9. “History of mathematics at school”, Moscow, “Prosveshchenie”, 1981.
In ancient times, a person who could count was considered a sorcerer. Not all literate people possessed such “witchcraft”. It was mainly scribes who knew how to count, and also, of course, merchants.
But even those who knew how to count, every now and then faced some kind of riddles and pitfalls. Addition, the simplest arithmetic operation, could be mastered with a certain amount of imagination. All you had to do was imagine that the same sticks, pebbles, and shells were once sheep, another time they were fruits, and the third time they were actually stars in the sky. And then it’s simple. Know yourself, add a stick to the stick and count the total. This is roughly how we were taught counting in first grade.
But problems were already beginning with subtraction. It was not always possible to subtract one number from another. Sometimes you take away, take away, and lo and behold, there’s nothing left. Nothing more to take away! So subtraction was a tricky operation and it was not always possible to do it.
True, you could get smart and take counting sticks of two colors, for example, black and white. Then one could subtract the white sticks, and then, when there is nothing left, start laying out the black sticks, as if in reserve. In this case, the subtraction could always be performed. True, the result expressed in black sticks would be difficult to interpret. Let's say two white sticks are two sheep. And two black sticks equal how many sheep?
But here merchants would come to the rescue. "All clear!" - they would say. - “Two black sticks are two sheep that you should give away, but have not given yet. This is a duty!
And the holy fathers, having thought about it, would have supported them. “Indeed,” they would say, “We are counting the years from the birth of Christ. But even before that there were people in the world. This means that the black sticks are the years that remained from some ancient event before the beginning of our chronology.”
In general, we came up with an interpretation of negative numbers in a minute. It took humanity more than a thousand years to do this. And in the thirteenth century they learned about negative numbers (and not only about them) in Europe. In 1202, a merchant (again a merchant, you can’t escape them, merchants!) Leonardo of Pisa (1170 - 1250) published a manual on arithmetic, in which he outlined what he had learned from mathematical books on Arabic which I read while visiting trade affairs in Egypt. Namely, the concept of zero (that is, a digit that denotes the absence of a number), the concept of positional notation of numbers (that is, how to write any number using just ten digits), and the rules of arithmetic operations with numbers written like this way. Among other things, Leonardo of Pisa also described the numbers obtained by subtracting a larger number from a smaller number, that is, negative numbers. Leonardo also showed that with the help of such numbers it is convenient to record losses or debts. He was great mathematician, Leonardo of Pisa. He was also known by the nickname Fibonacci (son of Bonacci). One of Fibonacci's discoveries was a special sequence of numbers, which at that time were considered a mathematical delight. And in our time, Fibonacci numbers are widely used not only in mathematics, but also in natural science and even in economics.
In general, problems similar to the problems described above with negative numbers arose with all “reverse” arithmetic operations. Two integers could be multiplied to produce a whole number. But the result of dividing two integers by an integer did not always turn out to be an integer. This also led to confusion. As in S. Marshak’s children’s poem: “And my answer was: two diggers and two thirds.” That is, in order for the result of division to always exist, it was necessary to introduce, master and understand, so to speak, the “physical meaning” of fractional numbers. Nowadays this is taught in the second grade. Humanity has been mastering fractional numbers for almost a thousand years. And again - thanks to the merchants! This is who mathematics owes its progress to!
Already in the 18th century, mathematicians came up with special numbers in order to obtain another “reverse” operation, extraction square root from negative numbers. These are the so-called “complex” numbers. It’s difficult to imagine them, but it’s possible to get used to them. And the benefits of using complex numbers big. The existence of these "strange" numbers greatly facilitated the calculation of complex AC electrical circuits, and also made it possible to calculate the profile of an aircraft wing.
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