The history of negative numbers presentation. The concept of a negative number in modern mathematics and in history
A few thousand years ago, the need for measurement led to an expansion of the set of natural numbers, which until then were used by people. New, fractional numbers were introduced, with the help of which it became possible to make measurements (lengths, areas, weights, etc.) with any degree of accuracy allowed by the instruments.
This was not the case with negative numbers. In the practical activities of people, there was no need to introduce negative numbers, and they firmly entered mathematics and were applied only in the 17th century.
But in mathematics itself, the need to expand the numerical set by introducing new, negative numbers had been felt for a long time, and as the mathematical science developed, this need became more and more urgent.
So, back in the III century, the Greek mathematician Diophantus, when performing some transformations, for example
in fact, he already used the rule of multiplying negative numbers, which he expressed as follows: “Subtracted, multiplied by the added, results in the subtracted. The subtracted, multiplied by the subtracted, results in the added. "
From this formulation it is clear that Diophantus did not yet recognize the independent existence of negative numbers; for him they were the former numbers, "subtracted" from some other number. Therefore, if, for example, when solving an equation, a negative root was obtained, Diophantus simply rejected it as "unacceptable."
But already the Indian scientist Bramagupta (VII century) in his calculations freely used negative numbers and gave them a visual interpretation. He designated property with positive numbers and debt with negative numbers.
In this visual form, he also gave rules for actions with rational ones. numbers, for example: “The sum of two properties is property. The sum of the two debts is debt. The sum of property and debt is equal to their difference, and if they are equal, then zero, "and so on.
The Indian mathematician Bhaskara (XII century) uses the degree of a negative number. In his work "The Crown of the System" it says:
“The square of both positive and negative numbers gives a positive number, for example:
In Europe, mathematicians of the 16th century, although they sometimes used negative numbers, nevertheless called them complex and unclear, less than nothing, etc.
Only the Dutch mathematician Girard (XVI-XVII centuries) uses negative numbers along with positive ones. So, solving the equation
he cites three of its roots:
The rapid development of natural science and technology in the 17th century made increased demands on mathematics as well, demanded its further development and improvement of the mathematical apparatus. The non-use of negative numbers created unnecessary difficulties in mathematical calculations and transformations. Since the 17th century, negative numbers have become firmly established in mathematics and found practical applications. French philosopher and mathematician Descartes gives a visual interpretation of numbers using the points of the number axis. He uses negative numbers to graphically represent various processes and algebraic expressions.
ANDhistory suggests that people could not get used to negative numbers for a long time. Negative numbers seemed incomprehensible to them, they did not use them, they simply did not see any special meaning in them. Positive numbers have long been interpreted as "profit", and negative numbers - as "debt", "loss". Only in Ancient India and China did they guess instead of the words "debt of 10 yuan" to write simply "10 yuan", but draw these hieroglyphs with black ink. And the signs "+" and "-", which we talked about, in ancient times were not for numbers or for actions.V Ancient China knew only the rules for adding and subtracting positive and negative numbers; the rules of multiplication and division were not applied. In India, negative numbers were treated with some distrust, considering them peculiar, not entirely real. Bhashara directly wrote: "People do not approve of abstract negative numbers ..."European mathematicians did not approve of them for a long time, because the interpretation of "property-debt" aroused bewilderment and doubt. Indeed, it is possible to "add" or "subtract" assets and debts, but what real meaning can there be in "multiplying" or "dividing" assets by debt?
The G of the river was also not used at first, until in the 3rd century Diophantus of Alexandria began to denote subtraction with a sign.
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 emergence of modern signs "+", "-" is not entirely clear. The "+" sign probably comes from the abbreviated notation et, i.e. "and". However, it may have arisen from trade practice: the sold measures of wine were marked on the barrel “-”, and when the stock was restored, they were crossed out, and a “+” sign was obtained. And in Italy, the usurers, 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, crossed it out, it turned out something like our plus. The plus can be considered a crossed out minus!
Modern signs "+" and appeared in Germany in the last decade of the 15th century. in Widman's book, which was a guide to account for merchants (1489). Czech Jan Widman already wrote "+" and "-" for addition and subtraction. And a little later, the German scientist Michel Stiefel wrote "Complete Arithmetic", which was published in 1544, it was printed, and not written by hand. It contains the following entries for numbers: 0-2; 0 + 2; 0-5; 0 + 7. The numbers of the first kind, he called "less than nothing" or "lower than nothing." The numbers of the second kind are called "more than nothing" or "higher than nothing." You, of course, understand these names, because "nothing" is 0. О +
These numbers have always been talked about in academic circles. Other designations were proposed, images were invented.
The combined signs are first found in Girard (1626) in the form.
Such an entry has been supplanted by the iconsand . Secondary combinedinvented by the Portuguese da Cunha (1790), from whom they looked like this: and.
Literature. N.V. Alexandrova. Mathematical terms. Directory. Moscow "Higher School" 1978
Print material for the stand in the office, Word document [
Consider what negative numbers are. They come in many natural numbers and are used in mathematics to make subtraction as complete as addition. That is, thanks to the introduction of negative numbers, it became possible not only to subtract the smaller from the larger, but also vice versa. All negative numbers are less than zero and any positive number. They are located on the usual coordinate axis to the left of zero. With negative numbers, you can perform all the same arithmetic operations as with positive ones.
Features of actions with negative numbers:
- the product of a negative number by a negative number will be positive;
- the product of positive and negative will be negative;
- when dividing with a remainder of negative numbers (or negative and positive numbers), the quotient can be negative or positive, the remainder is always positive.
From the history of negative numbers
In the ancient world (Ancient Egypt, Greece, Babylon), negative numbers were not used and were rejected as impossible. They were first used in India and China from the 7th century AD to denote debts or shortages in trade. But actions with negative numbers were not sequenced. The Indian mathematician Brahmagupta began to consider the actions of multiplication and division with them a little later.
An example of using a negative number:
The merchant had 10,000 rubles. He bought goods for 8000. The remainder is 2000. If he buys goods for 12000, then he owes 2000. And in his accounting records this amount will be reflected as a negative number -2000.
In Europe, they began to be used in 1202. Mathematicians Leonard Pisansky, Bombelli, Girard considered them suitable for denoting a lack of something, debt. But the famous Pascal denied them even in the 17th century, and until the end of his life he continued to assert: "Nothing can be less than nothing (that is, zero)." Finally, the theory of negative numbers was formed in the 19th century by William Hamilton.
Known negative numbers:
- - 273.15 ° C Absolute zero temperature on the Kelvin scale;
- - 1.602 176 565.10 −19 Cl. The magnitude of the electron charge;
- - 270.85 ° C Space temperature.
Writing negative numbers
Until now, in mathematics, there is no separate sign for denoting a negative number. The traditionally used "minus" is also a subtraction sign. And this is algebraically wrong and sometimes misleading. How was it before? For example, in China for negative numbers there were special counting sticks in black and for positive numbers - red. In India, negative numbers were marked with a red horizontal line directly above the number itself.
Man invented number in order to somehow designate for himself and others the results of counting and measuring. Apparently, the first concepts of number in humans appeared in the Paleolithic era, but developed already in the Neolithic. The first step in the appearance of numbers, apparently, was the realization of the division of the measure into "one" and "many".
In the ancient world, for the first time, special signs began to be used to designate numbers: their images were preserved on clay tablets of Mesopotamia, on Egyptian papyri, and so on.
Mathematics developed further. And in different countries, their own special, authentic and noticeably different number systems began to form. Even a schoolchild now knows how Roman numerals and Arabic spellings differed. The numbers were passed from country to country, from culture to culture, as an important and valuable invention and heritage. Modern numbers, on which both Slavic and Western civilizations are built, are Arabic numbers, but borrowed from India. Many numbers, familiar to everyone now, were invented in India, for example, the number "0".
The division of numbers into positive and negative refers to the development of mathematicians of the Middle Ages. Again, negative numbers were first used in India. So it was easier for merchants to calculate losses and debts. At that time, arithmetic was already a highly developed applied field, and algebra was entering its development. With the introduction of Cartesian geometry, his coordinate systems of negative numbers have become firmly established. From here they do not leave to this day.
Complex numbers are a 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. At the time of Descartes, complex numbers, like negative ones, became firmly established in mathematical use.
Very old and long. Since negative numbers are something ephemeral, not real, people did not recognize their existence for a long time.
It all started in China, around II 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 they called them "property." The record that exists now did not exist then, and negative numbers were written in black, and positive numbers in red.
The first mention of negative numbers we find in the book "Mathematics in Nine Chapters" by the Chinese scientist Zhang Tsan.
Further, in V-VI For centuries, negative numbers began to be used quite widely in China and India. True, in China, they were still treated with caution, they tried to minimize their use, while in India, on the contrary, they were used very widely. There, calculations were made with them and negative numbers did not seem to be something incomprehensible.
Famous Indian scientists Brahmagupta Bhaskara ( VII-VIII century), who in their teachings left detailed explanations for 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 turned out to be a negative number, it was considered that there was no solution.
So in Europe, negative numbers were not recognized for a very long time. They were considered “imaginary” and “absurd”. No action was taken with them, but simply discarded if the answer was negative. It was 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 book "The Book of Abacus" in 1202.
Later, in 1544, Mikhail Shtifel, in his book "Complete Arithmetic", first introduced the concept of negative numbers and described in detail the actions with them. "Zero is between absurd and true numbers."
And in the XVII century mathematician René Descartes proposed putting negative numbers on the digital axis to the left of zero.
Since that time, negative numbers began to be widely used and recognized, although for a long time many scientists denied them.
In 1831 Gauss called negative numbers absolutely equivalent to positive ones. And the fact that not all actions can be performed with them was not considered something terrible, with fractions, for example, not all actions can be done either.
And in the XIX century Willman Hamilton and Hermann Grassmann created a complete and complete theory of negative numbers. Since that time, negative numbers have acquired their rights and now no one doubts their reality.
