Positive and negative numbers: definition, examples. Start in science

Identifying Positive and Negative Numbers

To determine positive and negative numbers, we use the coordinate line, which is located horizontally and directed from left to right.

Note 1

The origin on the coordinate line corresponds to the number zero, which is neither a positive nor a negative number.

Definition 1

The numbers corresponding to the points of the coordinate line that lie to the right of the origin are called positive.

Definition 2

The numbers corresponding to the points of the coordinate line that lie to the left of the origin are called negative.

From these definitions it follows that the set of all negative numbers is opposite to the set of all positive numbers.

Negative numbers are always written with a “–” (minus) sign.

Example 2

Examples of negative numbers:

Rational numbers $-\frac(9)(17)$, $-4 \frac(11)(23)$, $–5.25$, $–4,(79)$.
Irrational numbers$ -\sqrt(2)$, infinite non-periodic decimal fraction $–103.1012341981…$

To simplify writing, the “+” (plus) sign is often not written before positive numbers, and the “–” sign is always written before negative numbers. In such cases, it is necessary to remember that the entry “$17.4$” is equivalent to the entry “$+17.4$”, the entry “$\sqrt(5)$” is equivalent to the entry “$+\sqrt(5)$”, etc. d.

So the following definition of positive and negative numbers can be used:

Definition 3

Numbers written with a “+” sign are called positive, and with the sign “–” – negative.

The definition of positive and negative numbers is used, which is based on comparison of numbers:

Definition 4

Positive numbers are numbers Above zero, A negative numbers– numbers less than zero.

Note 3

Thus, the number zero separates positive and negative numbers.

Rules for reading positive and negative numbers

Note 4

When reading a number with a sign in front of it, read its sign first, and then the number itself.

Example 3

For example, “$+17$” is read “plus seventeen”,

“$-3 \frac(4)(11)$” read “minus three point four elevens.”

Note 5

It is worth noting that the names of the plus and minus signs are not declined, while numbers can be declined.

Example 4

Interpretation of positive and negative numbers

Positive numbers are used to denote an increase in some value, arrival, increase, increase in value, etc.

Negative numbers are used for opposite concepts - to indicate a decrease in some value, expense, deficiency, debt, decrease in value, etc.

Let's look at examples.

A reader borrowed $4$ books from the library. Positive value the number $4$ shows the number of books the reader has. If he needs to check out $2$ of books to the library, he can use a negative value of $–2$, which will indicate a decrease in the number of books the reader has.

Positive and negative numbers are often used to describe the values of various quantities in measuring instruments. For example, a thermometer for measuring temperature has a scale on which positive and negative values are marked.

Cooling outside by $3$ degrees, i.e. a decrease in temperature can be indicated by a value of $–3$, and an increase in temperature by $5$ degrees can be indicated by a value of $+5$.

It is customary to depict negative numbers in blue, which symbolizes cold, low temperature, and positive numbers in red, which symbolizes warmth, high temperature. Indicating positive and negative numbers using red and of blue color used in different situations to highlight the sign of numbers.

We know that if you add two or more natural numbers, then the result is a natural number. If you multiply natural numbers with each other, the result is always natural numbers. What numbers will be the result if you subtract another natural number from one natural number? If you subtract a smaller number from a larger natural number, the result will also be a natural number. What number will be if you subtract the larger number from the smaller number? For example, if we subtract 7 from 5. The result of such an action will no longer be a natural number, but will be a number less than zero, which we will write as a natural number, but with a minus sign, the so-called negative natural number. In this lesson we will learn about negative numbers. Therefore, we expand the set of natural numbers by adding “0” and negative integers to it. The new extended set will consist of numbers:

…-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6…

These numbers are called integers. Therefore, the result of our example 5 -7 = -2 will be an integer.

Definition. Integers are natural numbers, negative natural numbers and the number “0”.

We see an image of this set on a thermometer for measuring outdoor temperature.

The temperature may be “minus”, i.e. negative, maybe with a “plus” i.e. positive. A temperature of 0 degrees is neither positive nor negative, the number 0 is the boundary that separates positive numbers from negative ones.

Let's plot the integers on the number line.

Axis drawing

We see that there is an infinite number of numbers on the number line. Positive and negative numbers are separated by zero. Negative integers, such as -1, are read as "minus one" or "negative one".

Positive integers, for example “+3” are read as positive 3 or simply “three”, that is, for positive (natural) numbers the “+” sign is not written and the word “positive” is not pronounced.

Examples: mark +5, +6, -7, -3, -1, 0, etc. on the number line.

When you move to the right along the number axis, the numbers increase, and when you move to the left, they decrease. If we want to increase a number by 2, we move to the right along the coordinate axis by 2 units. Example: 0+2=2; 2+2=4; 4+2=6, etc. Conversely, if we want to decrease the number by 3, we will move to the left by 3 units. For example: 6-3=3; 3-3=0; 0-3=-3; etc.

1. Try to increase the number (-4) in 3 steps, increasing by 2 units each time.

Moving along the number axis as shown in the figure, we get 2 as a result.

2. Decrease the number 6 in six steps, decreasing it by 2 units for each step.

3. Increase the number (-1) in three steps, increasing it by 4 units at each step.

Using the coordinate line, it is easy to compare integers: of two numbers, the greater is the one that is located to the right on the coordinate line, and the smaller is the one that is to the left.

4. Compare numbers using > or< , для удобства сравнения изобрази их на координатной прямой:

3 and 2; 0 and -5; -34 and -67; -72 and 0, etc.

5. Remember how we marked points with natural coordinates on the coordinate ray. Dots are usually called in capital Latin letters. Draw a coordinate line, and taking a convenient unit segment, draw points with coordinates:

A) A(10),B(20),C(30),M(-10),N(-20)
B) C (100), B (200), K (300), F (-100)
B) U(1000),E(2000),R(-3000)

6. Write down all the integers located between -8 and 5, between -15 and -7, between -1 and 1.

When comparing numbers, we must be able to answer by how many units one number is greater or less than another.

Let's draw a coordinate line. Let's draw points on it with coordinates from -5 to 5. The number 3 is two units less than 5, one less than 4, and 3 units more than zero. The number -1 is one less than zero, and 2 units more than -3.

7. Answer how many units:

3 is less than 4; -2 is less than 3; -5 is less than -4; 2 is greater than -1; 0 more than -5; 4 over -1

8. Draw a coordinate line. Write down 7 numbers, each of which is 2 units less than the previous one, starting with 6. What is this series last number? How many such numbers can there be if the number of numbers written down is not limited?

9. Write down 10 numbers, each of which is 3 units more than the previous one, starting with (-6). How many such numbers can exist if the series is not limited to ten?

Opposite numbers.

On the number line, for every positive number (or natural number), there is a negative number located to the left of zero at the same distance. For example: 3 and -3; 7 and -7; 11 and -11.

They say that the number -3 is the opposite of the number 3, and vice versa, -3 is the opposite of 3.

Definition: Two numbers that differ from each other only in sign are called opposite.

We know that if we multiply a number by +1, the number will not change. And if the number is multiplied by (-1), what happens? This number will change sign. For example, if 7 is multiplied by (-1) or negative one, the result is (-7), the number becomes negative. If (-10) is multiplied by (-1), we get (+10), i.e. we already get a positive number. Thus we see that the opposite numbers are obtained simple multiplication the original number by (-1). We see on the number axis that for each number there is only one opposite number. For example, for (4) the opposite will be (-4), for the number (-10) the opposite will be (+10). Let's try to find the opposite number of zero. He's gone. Those. 0 is the opposite of itself.

Now let's look at the number axis, what happens if you add 2 opposite numbers. We get that amount opposite numbers equals 0.

1. Game: Let the playing field be divided in half into two fields: left and right. There is a dividing line between them. There are numbers on the field. Passing through the line means multiplying by (-1), otherwise when passing through the dividing line, the number becomes the opposite.

Let the left field contain the number (5). What number will (5) turn into if the five crosses the dividing line once? 2 times? 3 times?

2. Fill out the following table:

3. From a variety of pairs, choose opposite pairs. How many pairs of these have you received?

9 ; -100; 1009; -63; -7; -9; 3; -33; 25; -1009; -2; 1; 0; 100; 27; 345; -56; -345; 33; 7.

Adding and subtracting integers.

Addition (or the "+" sign) means moving to the right on a number line.

1+3 = 4

-1 + 4 = 3
-3 + 2 = -1

Subtraction (or sign "-") means moving to the left on a number line

3 – 2 = 1
2 – 4 = -2
3 – 6 = -3
-3 + 5 = 2
-2 – 5 = -7
-1 + 6 = 5
1 – 4 = -3

Solve the following examples using the number line:

-3+1=
2)-4-1=
-5-1=
-2-7=
-1+3=
-1-4=
-6+7=

IN Ancient China when drawing up equations, the coefficients of minuends and subtrahends were written in numbers different color. Profits were indicated in red, and losses – in blue. Example, we sold 3 bulls and bought 2 horses. Let's consider another example: the housewife brought potatoes to the market and sold them for 300 rubles, we will add this money to the housewife's property and write it as +300 (red), and then she spent 100 rubles (we will write this money as (-100)( blue). Thus, it turned out that the housewife returned from the market with a profit of 200 rubles (or +200). Otherwise, numbers written in red paint were always added, and those written in blue paint were subtracted. By analogy, we will use blue paint to denote negative numbers.

Thus, we can consider all positive numbers as winnings, and negative numbers as losses or debts or losses.

Example: -4 + 9 = +5 The result (+5) can be considered as a win in any game; after first losing 4 points and then winning 9 points, the result will be a win of 5 points. Solve the following problems:

11. In the lotto game, Petya first won 6 points, then lost 3 points, then again won 2 points, then lost 5 points. What is the result of Petya's game?

12 (*). Mom put sweets in a vase. Masha ate 4 candies, Misha ate 5 candies, Olya ate 3 candies. Mom put 10 more candies in the vase, and there were 12 candies in the vase. How many candies were in the bowl at first?

13. In the house, one staircase leads from the basement to the second floor. The staircase consists of two flights of 15 steps each (one from the basement to the first floor, and the second from the first floor to the second). Petya was on the first floor. First he climbed the stairs 7 steps up, and then went down 13 steps. Where was Petya?

14. The grasshopper jumps along the number axis. One grasshopper jump is 3 divisions on the axis. The grasshopper first makes 3 jumps to the right, and then 5 jumps to the left. Where will the grasshopper end up after these jumps, if initially he was in 1) “+1”; 2) “-6”; 3) “0”; 4) “+5”; 5) “-2”; 6) “+ 3";7) "-1".

Until now, we have become accustomed to the fact that the numbers in question answered the question “how much.” But negative numbers cannot be the answer to the question “how much.” In an everyday sense, negative numbers are associated with debt, loss, with such actions as underdoing, under-jumping, underweight, etc. In all these cases we simply subtract the debt, the loss, the underweight. For example,

To the question “What is “a thousand without 100”?”, we must subtract 100 from 1000 and get 900.
The expression “3 hours to a quarter” means that we must subtract 15 minutes from 3 hours. Thus we get 2 hours 45 minutes.

Now solve the following problems:

15. Sasha bought 200g. oil, but the unscrupulous seller underweighted 5 grams. How much butter did Sasha buy?

16. At a running distance of 5 km. Volodya left the race before reaching the finish line of 200m. How far did Volodya run?

17. Filling three liter jar Mom didn’t add 100 ml of juice. How much juice was in the jar?

18. The movie should start at twenty minutes to eight. how many minutes What time and what time should the movie start?

19. Tanya had 200 rubles. and she owes Petya 50 rubles. After she paid off the debt, how much money did Tanya have left?

20. Petya and Vanya went to the store. Petya wanted to buy a book for 5 rubles. But he only had 3 rubles, so he borrowed 2 rubles from Vanya and bought a book. How much money did you have after purchasing from Petya?

3 - 5 = -2 (from what he had before the purchase, subtract the purchase price, we get -2 rubles, that is, two rubles of debt).

21. During the day the air temperature was 3°C or +3°, and at night 4°F or -4°. By how many degrees did the temperature drop? And how many degrees lower is the night temperature than the day temperature?

22. Tanya agreed to meet Volodya at a quarter to seven. What time and what time did they agree to meet?

23. Tim and a friend went to the store to buy a book that cost 97 rubles. But when they came to the store, it turned out that the book had risen in price and began to cost 105 rubles. Tim borrowed the missing amount from a friend and still bought the book. How much money did Tim owe his friend?

In this material we will explain what positive and negative numbers are. After the definitions have been formulated, we will show with examples what they are and reveal the basic meaning of these concepts.

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What are positive and negative numbers

In order to explain the basic definitions, we need a coordinate line. It will be positioned horizontally and directed from left to right: this will be easier to understand.

Definition 1

Positive numbers- these are the numbers that correspond to points in that part of the coordinate line that is located to the right of the origin.

Negative numbers- these are the numbers that correspond to points in the part of the coordinate line located on the left side of the origin (zero).

Zero, from which we choose directions, in itself does not belong to either negative or positive numbers.

From the definitions given above it follows that positive and negative numbers form certain sets that are opposite to each other (positive are opposed to negative, and vice versa). We have already mentioned this earlier in the article on opposite numbers.

Definition 2

We always write negative numbers with a minus.

Once we have introduced the basic definitions, we can easily give examples. Thus, any natural numbers are positive - 1, 9, 134,345, etc. Positive rational numbers are, for example, 7 9, 76 2 3, 4, 65 and 0, (13) = 0, 126712 ... and etc. To the positive irrational numbers refers to the number π, number e, 9 5, 809, 030030003... (this is the so-called infinite non-periodic decimal fraction).

Let's give examples of negative numbers. These are - 2 3 , − 16 , − 57 , 58 − 3 , (4) . Irrational negative numbers are, for example, minus pi, minus e, etc.

Can we immediately say that the value of the numerical expression log 3 4 - 5 is a negative number? The answer is not obvious. We will have to express this value as a decimal fraction and then look (for more information, see the material on comparing real numbers).

In order to clarify that a number is positive, they sometimes put a plus in front of it, just as they put a minus in front of a negative number, but most often it is omitted. Don't forget that + 5 = 5, + 1 2 3 = 1 2 3, + 17 = 17 and so on. In fact, these are different designations for the same number.

In the literature you can also find definitions of positive and negative numbers based on the presence of one or another sign.

Definition 3

Positive number is a number with a plus sign, and negative– having a minus sign.

There are also definitions based on the position of a given number relative to zero (remember that large numbers are located on the right side of the coordinate line, and smaller ones on the left).

Definition 4

Positive numbers– these are all numbers whose value is greater than zero. Negative numbers– these are all numbers less than zero.

It turns out that zero is a kind of separator: it separates negative numbers from positive ones.

We will separately focus on how to correctly read the records of positive and negative numbers, although, as a rule, there are no special problems with this. For negative numbers we always pronounce the minus, i.e. - 1 2 5 is “minus one point two fifths.”

In the case of positive numbers, we voice the plus only when it is explicitly indicated in the entry, i.e. + 7 is “plus seven”. It is incorrect to decline the names of mathematical symbols by case. For example, it would be correct to read the phrase a = - 5 as “a equals minus five,” rather than “minus five.”

Basic meaning of positive and negative numbers

We have already given basic definitions, but in order to make correct calculations, it is necessary to understand the very meaning of the positivity or negativity of a number. We will try to help you do this.

We consider positive numbers, that is, those that are greater than 0, as profit, gain, increase in the quantity of something, and negative numbers as deficiency, loss, expense, debt. Here are some examples:

We have 5 any items, for example, apples. The number 5 is positive, it indicates that we have something, we possess a certain amount of real-life objects. How then should we consider 5? It could, for example, mean that we must give someone five apples that we do not currently have.

The easiest way to understand this is by the example of money: if we have 6, 75 thousand rubles, then our income is positive: we were given money, and we have it. At the same time, at the cash register these expenses are indicated as - 6, 75, that is, for them it is a loss.

On a thermometer, an increase in temperature by 4.5 values can be described as + 4.5, and a decrease, in turn, as - 4.5. Instruments designed to measure often use positive and negative numbers because they are useful for displaying changes in quantities. For example, in a thermometer, negative numbers are indicated in blue - this means falling, cold, decreasing heat; positive ones are marked in red - this is the color of fire, growth, increase in warmth. These colors are very often used to write such numbers, because... they are very visual - with their help you can always clearly identify income and expenses, profit and loss.

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NUMBER, one of the basic concepts of mathematics; originated in ancient times and gradually expanded and generalized. In connection with the counting of individual objects, the concept of positive integer (natural) numbers arose, and then the idea of the limitlessness of the natural series of numbers: 1, 2, 3, 4. Problems of measuring lengths, areas, etc., as well as isolating shares of named quantities led to the concept of a rational (fractional) number. The concept of negative numbers arose among Indians in the 6th-11th centuries.

For the first time negative numbers are found in one of the books of the ancient Chinese treatise “Mathematics in Nine Chapters” (Jan Can - 1st century BC). A negative number was understood as debt, and a positive number as property. Addition and subtraction of negative numbers was done based on reasoning about debt. For example, the addition rule was formulated as follows: “If you add another debt to one debt, the result is debt, not property.” There was no minus sign then, and in order to distinguish between positive and negative numbers, Can Can wrote them with ink of different colors.

The idea of negative numbers had a hard time gaining a place in mathematics. These numbers seemed incomprehensible and even false to the mathematicians of antiquity, and actions with them were unclear and had no real meaning.

Use of negative numbers by Indian mathematicians.

In the 6th and 7th centuries AD, Indian mathematicians already systematically used negative numbers, still understanding them as a debt. Since the 7th century, Indian mathematicians have used negative numbers. They called positive numbers “dhana” or “sva” (“property”), and negative numbers “rina” or “kshaya” (“debt”). For the first time, all four arithmetic operations with negative numbers were given by the Indian mathematician and astronomer Brahmagupta (598 - 660).

For example, he formulated the division rule as follows: “A positive divided by a positive, or a negative divided by a negative becomes positive. But positive divided by negative, and negative divided by positive, remains negative."

(Brahmagupta (598 - 660) is an Indian mathematician and astronomer. Brahmagupta’s work “Revision of the Brahma System” (628) has reached us, a significant part of which is devoted to arithmetic and algebra. The most important here is the doctrine of arithmetic progression and solution quadratic equations, which Brahmagupta dealt with in all cases where they had valid solutions. Brahmagupta allowed and considered the use of zero in all arithmetic operations. In addition, Brahmagupta solved some indefinite equations in integers; he gave the rule for composing right triangles with rational sides, etc. Brahmagupta knew the inverse triple rule; he found the approximation P, the earliest interpolation formula of the 2nd order. His interpolation rule for sine and inverse sine at equal intervals is a special case of the Newton–Stirling interpolation formula. In a later work, Brahmagupta gives an interpolation rule for unequal intervals. His works were translated into Arabic in the 8th century.)

Understanding negative numbers by Leonard Fibonacci of Pisa.

Independently of the Indians, the Italian mathematician Leonardo Fibonacci of Pisa (13th century) came to understand negative numbers as the opposite of positive numbers. But it took about 400 more years before “absurd” (meaningless) negative numbers received full recognition by mathematicians, and negative decisions tasks were no longer discarded as impossible.

(Leonardo Fibonacci of Pisa (c. 1170 - after 1228) - Italian mathematician. Born in Pisa (Italy). Elementary education received in Bush (Algeria) under the guidance of a local teacher. Here he mastered the arithmetic and algebra of the Arabs. He visited many countries in Europe and the East and expanded his knowledge of mathematics everywhere.

He published two books: “The Book of Abacus” (1202), where the abacus was considered not so much as an instrument, but as calculus in general, and “Practical Geometry” (1220). Based on the first book, many generations of European mathematicians studied the Indian positional number system. The presentation of the material in it was original and elegant. The scientist also made his own discoveries, in particular, he initiated the development of issues related to T.N. Fibonacci numbers, and gave original technique extracting the cube root. His works only became widespread at the end of the 15th century, when Luca Pacioli revised them and published them in his book Summa.

Consideration of negative numbers by Mikhail Stifel in a new way.

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. (Mikhail Stiefel (19.04.1487 - 19.06.1567) - famous German mathematician. Michael Stiefel studied in a Catholic monastery, then became interested in the ideas of Luther and became a rural Protestant pastor. While studying the Bible, he tried to find a mathematical interpretation in it. As a result His research predicted the end of the world on October 19, 1533, which, of course, did not happen, and Michael Stiefel was imprisoned in Württemberg prison, from which Luther himself rescued him.

After this, Stiefel devoted his work entirely to mathematics, in which he was a self-taught genius. One of the first in Europe after N. Schuke began to operate with negative numbers; introduced fractional and zero exponents, as well as the term “exponent”; in the work “Complete Arithmetic” (1544) he gave the rule for dividing by a fraction as multiplying by the reciprocal of the divisor; took the first step in the development of techniques that simplify calculations with large numbers, for which he compared two progressions: geometric and arithmetic. Later this helped I. Bürgi and J. Napier create logarithmic tables and develop logarithmic calculations.)

Modern interpretation of negative numbers by Girard and Rene Descartes.

The modern interpretation of negative numbers, based on plotting unit segments on the number line to the left of zero, was given in the 17th century, mainly in the works of the Dutch mathematician Girard (1595–1634) and the famous French mathematician and philosopher René Descartes (1596–1650). ) (Girard Albert (1595 - 1632) - Belgian mathematician. Girard was born in France, but fled to Holland from persecution catholic church because he was a Protestant. Albert Girard made major contributions to the development of algebra. His main work was the book A New Discovery in Algebra. For the first time he expressed the fundamental theorem of algebra about the presence of a root in an algebraic equation with one unknown. Although Gauss was the first to give a rigorous proof. Girard is responsible for the derivation of the formula for the area of a spherical triangle.) Since 1629 in the Netherlands. He laid the foundations of analytical geometry, gave the concepts of variable quantities and functions, and introduced many algebraic notations. He expressed the law of conservation of momentum and gave the concept of impulse of force. Author of a theory that explains the formation and movement of celestial bodies by the vortex motion of matter particles (Descartes vortices). Introduced the concept of reflex (Descartes arc). The basis of Descartes' philosophy is the dualism of soul and body, “thinking” and “extended” substance. He identified matter with extension (or space), and reduced movement to the movement of bodies. The general cause of motion, according to Descartes, is God, who created matter, motion and rest. Man is a connection between a lifeless bodily mechanism and a soul with thinking and will. The unconditional foundation of all knowledge, according to Descartes, is the immediate certainty of consciousness (“I think, therefore I exist”). The existence of God was considered as a source of objective significance of human thinking. In the doctrine of knowledge, Descartes is the founder of rationalism and a supporter of the doctrine of innate ideas. Main works: “Geometry” (1637), “Discourse on the Method. "(1637), "Principles of Philosophy" (1644).

DESCARTES (Descartes) Rene (Latinized - Cartesius; Cartesius) (March 31, 1596, Lae, Touraine, France - February 11, 1650, Stockholm), French philosopher, mathematician, physicist and physiologist, founder of modern European rationalism and one of the most influential metaphysicians of the New Age.

Life and writings

Born into a noble family, Descartes received a good education. In 1606, his father sent him to the Jesuit college of La Flèche. Considering not very good health Descartes, he was given some relaxations in the strict regime of this educational institution, eg. , were allowed to get up later than others. Having acquired a lot of knowledge at the college, Descartes at the same time became imbued with antipathy towards scholastic philosophy, which he retained throughout his life.

After graduating from college, Descartes continued his education. In 1616, at the University of Poitiers, he received a bachelor's degree in law. In 1617, Descartes enlisted in the army and traveled extensively throughout Europe.

The year 1619 turned out to be a key year for Descartes scientifically. It was at this time, as he himself wrote in his diary, that the foundations of a new “most amazing science” were revealed to him. Most likely, Descartes had in mind the discovery of the universal scientific method, which he subsequently fruitfully applied in a variety of disciplines.

In the 1620s, Descartes met the mathematician M. Mersenne, through whom he long years“kept in touch” with the entire European scientific community.

In 1628, Descartes settled in the Netherlands for more than 15 years, but did not settle in any one place, but changed his place of residence about two dozen times.

In 1633, having learned about the condemnation of Galileo by the church, Descartes refused to publish his natural philosophical work “The World,” which outlined the ideas of the natural origin of the universe according to the mechanical laws of matter.

In 1637, Descartes’ work “Discourse on Method” was published in French, with which, as many believe, modern European philosophy began.

In 1641 the main thing appears philosophical essay Descartes "Reflections on First Philosophy" (in Latin), and in 1644 "Principles of Philosophy", a work conceived by Descartes as a compendium summarizing the most important metaphysical and natural philosophical theories of the author.

Descartes's last philosophical work, The Passions of the Soul, published in 1649, also had a great influence on European thought. In the same year, at the invitation of the Swedish Queen Christina, Descartes went to Sweden. The harsh climate and unusual regime (the queen forced Descartes to get up at 5 a.m. to give her lessons and carry out other assignments) undermined Descartes' health, and, having caught a cold, he died of pneumonia.

The philosophy of Descartes clearly illustrates the desire of European culture to liberate itself from old dogmas and build a new science and life itself “from scratch.” The criterion of truth, Descartes believes, can only be the “natural light” of our mind. Descartes does not deny the cognitive value of experience, but he sees its function exclusively in coming to the aid of reason where own strength the latter is not enough for knowledge. Reflecting on the conditions for achieving reliable knowledge, Descartes formulates the “rules of method” with the help of which one can arrive at the truth. Initially thought by Descartes to be very numerous, in the “Discourse on Method”, he reduces them to four main provisions that constitute the “quintessence” of European rationalism: 1) start with the undoubted and self-evident, i.e. with that which cannot be thought to be the opposite, 2) divide any problem into as many parts as necessary to solve it effective solution, 3) start with the simple and gradually move towards the complex, 4) constantly recheck the correctness of the conclusions. The self-evident is grasped by the mind in intellectual intuition, which cannot be confused with sensory observation and which gives us a “clear and distinct” comprehension of the truth. Dividing a problem into parts makes it possible to identify “absolute” elements in it, that is, self-evident elements from which subsequent deductions can be based. Descartes calls deduction the “movement of thought” in which the cohesion of intuitive truths occurs. The weakness of human intelligence requires checking the correctness of the steps taken to ensure that there are no gaps in reasoning. Descartes calls this verification “enumeration” or “induction.” The result of consistent and ramified deduction should be the construction of a system of universal knowledge, “universal science.” Descartes compares this science to a tree. Its root is metaphysics, its trunk is physics, and its fruitful branches are formed by concrete sciences, ethics, medicine and mechanics, which bring direct benefit. From this diagram it is clear that the key to the effectiveness of all these sciences is correct metaphysics.

What distinguishes Descartes from the method of discovering truths is the method of presenting already developed material. It can be presented “analytically” and “synthetically”. The analytical method is problematic, it is less systematic but more conducive to understanding. Synthetic, as if “geometrizing” material, is more strict. Descartes still prefers the analytical method.

Doubt and certainty

The initial problem of metaphysics as a science about the most general kinds of being is, as in any other disciplines, the question of self-evident foundations. Metaphysics must begin with the undoubted statement of some existence. Descartes “tests” the theses about the existence of the world, God and our “I” for self-evidence. The world can be imagined as non-existent if we imagine that our life is a long dream. One can also doubt the existence of God. But our “I,” Descartes believes, cannot be questioned, since doubt itself in its existence proves the existence of doubt, and therefore of the doubting I. “I doubt, therefore I exist” - this is how Descartes formulates this most important truth, denoting the subjectivist turn of European philosophy New time. In more general view this thesis sounds like this: “I think, therefore I exist” - cogito, ergo sum. Doubt is only one of the “modes of thinking,” along with desire, rational comprehension, imagination, memory, and even sensation. The basis of thinking is consciousness. Therefore, Descartes denies the existence of unconscious ideas. Thinking is an integral property of the soul. The soul cannot help but think; it is a “thinking thing,” res cogitans. Recognizing the thesis of one's own existence as undoubted does not mean, however, that Descartes considers the non-existence of the soul generally impossible: it cannot but exist only as long as it thinks. Otherwise, the soul is a random thing, that is, it can either be or not be, because it is imperfect. All random things derive their existence from the outside. Descartes states that the soul is maintained in its existence every second by God. Nevertheless, it can be called a substance, since it can exist separately from the body. However, in reality, the soul and body interact closely. However, the fundamental independence of the soul from the body is for Descartes the guarantee of the probable immortality of the soul.

Doctrine of God

From philosophical psychology, Descartes moves on to the doctrine of God. He gives several proofs of the existence of a supreme being. The most famous is the so-called “ontological argument”: God is an all-perfect being, therefore the concept of him cannot lack the predicate of external existence, which means it is impossible to deny the existence of God without falling into contradiction. Another proof offered by Descartes is more original (the first was well known in medieval philosophy): in our mind there is an idea of God, this idea must have a cause, but the cause can only be God himself, since otherwise the idea supreme reality would be generated by something that does not possess this reality, that is, there would be more reality in the action than in the cause, which is absurd. The third argument is based on the necessity of God's existence to sustain human existence. Descartes believed that God, while not in himself bound by the laws of human truth, is nevertheless the source of man’s “innate knowledge,” which includes the very idea of God, as well as logical and mathematical axioms. Descartes believes that our belief in the existence of external things comes from God. material world. God cannot be a deceiver, and therefore this faith is true, and the material world really exists.

Philosophy of nature

Having convinced himself of the existence of the material world, Descartes began to study its properties. The main property of material things is extension, which can appear in various modifications. Descartes denies the existence of empty space on the grounds that wherever there is extension, there is also an “extended thing,” res extensa. Other qualities of matter are vaguely conceived and, perhaps, Descartes believes, exist only in perception, and are absent in the objects themselves. Matter consists of the elements fire, air and earth, the only difference being their size. Elements are not indivisible and can transform into each other. Trying to reconcile the concept of discreteness of matter with the thesis about the absence of emptiness, Descartes puts forward a very interesting thesis about instability and absence a certain shape in the smallest particles of matter. Descartes recognizes collision as the only way to convey interactions between elements and things consisting of their mixture. It occurs according to the laws of constancy, arising from the unchanging essence of God. In the absence of external influences, things do not change their state and move in a straight line, which is a symbol of constancy. In addition, Descartes talks about the conservation of the original momentum in the world. Movement itself, however, is not initially inherent in matter, but is introduced into it by God. But just one initial push is enough for a correct and harmonious cosmos to gradually assemble independently from the chaos of matter.

Body and soul

Descartes devoted a lot of time to studying the laws of functioning of animal organisms. He considered them to be subtle machines capable of independently adapting to environment and respond appropriately to external influences. The experienced effect is transmitted to the brain, which is a reservoir of “animal spirits”, tiny particles, the entry of which into the muscles through the pores that open due to deviations of the brain “pineal gland” (which is the seat of the soul), leads to contractions of these muscles. The movement of the body is composed of a sequence of such contractions. Animals have no souls and do not need them. Descartes said that he was more surprised by the presence of a soul in humans than by its absence in animals. The presence of a soul in a person, however, is not useless, since the soul can correct the natural reactions of the body.

Descartes the physiologist

Descartes studied the structure of various organs in animals and examined the structure of embryos at various stages of development. His doctrine of “voluntary” and “involuntary” movements laid the foundations for the modern doctrine of reflexes. The works of Descartes presented schemes of reflex reactions with the centripetal and centrifugal parts of the reflex arc.

The significance of Descartes' works in mathematics and physics

Descartes' natural scientific achievements were born as a “by-product” of the unified method of a unified science he developed. Descartes is credited with creating modern systems notations: he introduced signs for variables (x, y, z.), coefficients (a, b, c.), notation for degrees (a2, x-1.).

Descartes is one of the authors of the theory of equations: he formulated the rule of signs for determining the number of positive and negative roots, raised the question of the boundaries of real roots and put forward the problem of reducibility, i.e., the representation of an integer rational function with rational coefficients in the form of a product of two functions of this kind. He pointed out that an equation of the 3rd degree is solvable in square radicals (and also indicated a solution using a compass and straightedge if the equation is reducible).

Descartes is one of the creators of analytical geometry (which he developed simultaneously with P. Fermat), which made it possible to algebraize this science using the coordinate method. The coordinate system he proposed received his name. In his work “Geometry” (1637), which opened the interpenetration of algebra and geometry, Descartes first introduced the concepts of a variable quantity and a function. He interprets a variable in two ways: as a segment of variable length and constant direction (the current coordinate of a point describing a curve with its movement) and as a continuous numerical variable running through a set of numbers expressing this segment. In the field of study of geometry, Descartes included "geometric" lines (later called algebraic by Leibniz) - lines described by hinged mechanisms in motion. He excluded transcendental curves (Descartes himself calls them “mechanical”) from his geometry. In connection with the study of lenses (see below), "Geometry" sets out methods for constructing normals and tangents to plane curves.

"Geometry" had a huge influence on the development of mathematics. IN Cartesian system coordinates received a real interpretation of negative numbers. Descartes actually interpreted real numbers as the ratio of any segment to a unit (although the formulation itself was given later by I. Newton). Descartes' correspondence also contains his other discoveries.

In optics, he discovered the law of refraction of light rays at the boundary of two different media (set out in Dioptrics, 1637). Descartes made a major contribution to physics by giving a clear formulation of the law of inertia.

Influence of Descartes

Descartes had a tremendous influence on subsequent science and philosophy. European thinkers adopted his calls for the creation of philosophy as an exact science (B. Spinoza) and for the construction of metaphysics on the basis of the doctrine of the soul (J. Locke, D. Hume). Descartes also intensified theological debate on the possibility of proving the existence of God. Descartes' discussion of the question of the interaction of soul and body, to which N. Malebranche, G. Leibniz and others responded, as well as his cosmogonic constructions had a huge resonance. Many thinkers made attempts to formalize Descartes' methodology (A. Arnauld, N. Nicole, B. Pascal). In the 20th century, Descartes' philosophy is often referred to by participants in numerous discussions on the problems of philosophy of mind and cognitive psychology.

In order to develop this approach, which is understandable and natural for us now, it took the efforts of many scientists over eighteen centuries from Can Tsang to Descartes.

Consisting of positive (natural) numbers, negative numbers and zero.

All negative numbers, and only them, are less than zero. On the number line, negative numbers are located to the left of zero. For them, as for positive numbers, an order relation is defined that allows one to compare one integer with another.

n -n, which complements n to zero: n + (− n) = 0 . Both numbers are called opposite for each other. Subtracting an Integer a is equivalent to adding it with its opposite: -a.

Properties of Negative Numbers

Negative numbers follow almost the same rules as natural numbers, but have some special features.

Historical sketch

Literature

Vygodsky M. Ya. Guide to elementary mathematics. - M.: AST, 2003. - ISBN 5-17-009554-6
Glazer G.I. History of mathematics at school. - M.: Education, 1964. - 376 p.

Links

Wikimedia Foundation. 2010.

See what “Negative numbers” are in other dictionaries:

Real numbers less than zero, such as 2; 0.5; π, etc. See Number... Great Soviet Encyclopedia

- (values). The result of successive additions or subtractions does not depend on the order in which these actions are performed. Eg. 10 5 + 2 = 10 +2 5. Not only the numbers 2 and 5 are rearranged here, but also the signs in front of these numbers. Agreed... ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron

numbers are negative- Numbers in accounting that are written in red pencil or red ink. Topics: accounting... Technical Translator's Guide

NEGATIVE NUMBERS- numbers in accounting that are written in red pencil or red ink... Great Accounting Dictionary

The set of integers is defined as the closure of the set of natural numbers with respect to the arithmetic operations of addition (+) and subtraction (). Thus, the sum, difference and product of two integers are again integers. It consists of... ... Wikipedia

Numbers that arise naturally when counting (both in the sense of enumeration and in the sense of calculus). There are two approaches to determining natural numbers; numbers used in: listing (numbering) objects (first, second, ... ... Wikipedia

Coefficients E n in the expansion The recurrent formula for the E. number has the form (in symbolic notation, (E + 1)n + (E 1)n=0, E0 =1. In this case, E 2n+1=0, E4n are positive, E4n+2 negative integers for all n=0, 1, ...; E2= 1, E4=5, E6=61, E8=1385 ... Mathematical Encyclopedia

A negative number is an element of the set of negative numbers, which (together with zero) appeared in mathematics when expanding the set of natural numbers. The purpose of the extension is to allow the subtraction operation to be performed on any number. As a result... ... Wikipedia

Arithmetic. Painting by Pinturicchio. Apartment Borgia. 1492 1495. Rome, Vatican Palaces ... Wikipedia

Hans Sebald Beham. Arithmetic. 16th century Arithmetic (ancient Greek ἀ ... Wikipedia

Books

Mathematics. 5th grade. Educational book and workshop. Positive and negative numbers. In 2 parts. Part 2. Federal State Educational Standards, Gelfman E.G.. The educational book and workshop for grade 5 are part of the teaching materials in mathematics for grades 5–6, developed by a team of authors led by E.G. Gelfman and M.A. Kholodnaya as part of the project...

Within the framework of natural numbers, you can only subtract a smaller number from a larger one, and the commutative law does not involve subtraction - for example, the expression 3 + 4 − 5 (\displaystyle 3+4-5) is valid, and an expression with rearranged operands 3 − 5 + 4 (\displaystyle 3-5+4) unacceptable...

Adding negative numbers and zero to natural numbers makes subtraction possible for any pair of natural numbers. As a result of this expansion, a set (ring) of “integers” is obtained. With further expansions of the set of numbers by rational or real numbers, corresponding negative values are obtained for them in the same way. For complex numbers, the ordering is undefined, and the concept of a “negative number” does not exist.

For every natural number n there is one and only one negative number, denoted -n, which complements n to zero:

n + (− n) = 0. (\displaystyle n+\left(-n\right)=0.)

Both numbers are called opposites of each other. Subtracting an Integer a from another integer b is equivalent to addition b with the opposite for a:

b − a = b + (− a) . (\displaystyle b-a=b+\left(-a\right).)

Example: 25 − 75 = − 50. (\displaystyle 25-75=-50.)

Properties of Negative Numbers

Negative numbers obey almost the same algebraic rules as natural numbers, but they have some special features.

If any set of positive numbers is bounded below, then any set of negative numbers is bounded above.
When multiplying integers, the following applies: rule of signs: the product of numbers with different signs is negative, with the same signs - positive.
When both sides of an inequality are multiplied by a negative number, the sign of the inequality is reversed. For example, multiplying the inequality 3 −10.

When dividing with a remainder, the quotient can have any sign, but the remainder, by convention, is always non-negative (otherwise it is not uniquely determined). For example, divide −24 by 5 with a remainder:

− 24 = 5 ⋅ (− 5) + 1 = 5 ⋅ (− 4) − 4 (\displaystyle -24=5\cdot (-5)+1=5\cdot (-4)-4).

Variations and generalizations

The concepts of positive and negative numbers can be defined in any ordered ring. Most often, these concepts refer to one of the following number systems:

The above properties 1-3 also hold in the general case. TO complex numbers the concepts “positive” and “negative” are not applicable.

Historical sketch

Ancient Egypt, Babylon and Ancient Greece did not use negative numbers, and if equations had negative roots (when subtracting), they were rejected as impossible. The exception was Diophantus, who in the 3rd century already knew rule of signs and knew how to multiply negative numbers. However, he considered them only as an intermediate step, useful for calculating the final, positive result.

For the first time, negative numbers were partially legalized in China, and then (from about the 7th century) in India, where they were interpreted as debts (shortages), or, like Diophantus, recognized as temporary values. Multiplication and division for negative numbers had not yet been defined. The usefulness and validity of negative numbers was gradually established. The Indian mathematician Brahmagupta (7th century) already considered them on a par with positive ones.

In Europe, recognition came a thousand years later, and even then for a long time negative numbers were called “false,” “imaginary,” or “absurd.” The first description of them in European literature appeared in the “Book of Abacus” by Leonard of Pisa (1202), who interpreted negative numbers as debt. Bombelli and Girard, in their writings, considered negative numbers to be quite acceptable and useful, in particular to indicate the lack of something. Even in the 17th century, Pascal believed that 0 − 4 = 0 (\displaystyle 0-4=0), since “nothing can be less than nothing.” 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 the 17th century, with the advent of analytical geometry, negative numbers received a visual geometric representation on the number axis. From this moment on, their full equality comes. Nevertheless, the theory of negative numbers was in its infancy for a long time. For example, the strange proportion 1: (− 1) = (− 1) : 1 (\displaystyle 1:(-1)=(-1):1)- in it the first term on the left is greater than the second, and on the right - vice versa, and it turns out that the greater is equal to the less (“paradox

1. Questions related to negative numbers are one of the difficult questions for students to master.

The history of the development of mathematics shows that negative numbers are much more difficult for humans, this is due to the fact that negative numbers are less connected with practical life.

Negative numbers arose from the need to perform with known numbers. The mathematicians of ancient Greece did not recognize negative numbers; they could not give them a specific interpretation. Only in the work of Diophantus (3rd century AD) are there transformations that lead to the need to perform operations on negative numbers.

Negative numbers appear only in rudimentary form. They received a fairly wide distribution in the works of Indian scientists. They called positive numbers real, and negative numbers not real, false. Negative numbers were considered as debt, and positive numbers as cash.

The first rules of addition and subtraction belong to Indian scientists. And they are associated with the interpretation of these numbers as property and debt.

For a long time, scientists could not explain or interpret the product of two negative numbers. Why is the product of 2 debts property? Scientists such as Euler and Comey gave their explanations for the rule of the product of numbers, but they led to erroneous results.

The German scientist M. Stiefel was the first to define negative numbers as numbers less than zero in 1544.

The first mathematical interpretation was given by Rene Descartes in 1737 in his book “Analytical Geometry”. He considered negative numbers as independent numbers, located on the OX axis to the left of the origin. However, he called these numbers false. Negative numbers received general recognition in the first half of the 21st century, and negative numbers entered the history of mathematics.

2. Various techniques introducing negative numbers. In the educational literature, there are 3 ways to introduce negative numbers.

1) Cases are considered when the calculation on a set of positive numbers is false.

2) Consider vectors located on the same line; the need to characterize not only their length, but also their direction leads to the concept of positive and negative numbers.

3) Introducing negative numbers by placing changing quantities in opposite directions.

Method of introducing a negative number.

Before giving the concept of a negative number, it is necessary to show with specific examples that the known numbers are not enough to characterize the position of a point on a straight line to the origin.

Using a sufficient number of examples, it is necessary to show the inconvenience of the concept of drawing a number axis to the right or left, up or down. It is necessary to postpone the beginning of the counting and so that for the certainty of such scales, which are to the right with a plus sign, to the left with the opposite sign - minus.

The textbook discusses a sufficient number of examples showing the advisability of using certain signs to indicate the direction of opposite movement. For the concept of introducing a negative number, it is necessary to use a demonstration thermometer and other aids.

Familiarity with opposite numbers is facilitated by studying the center of symmetry.

The concept of opposite numbers is connected by symmetrical dots. At the same time, the introduction of this concept is based on the geometric interpretation of positive and negative numbers.

The section on opposite numbers introduces the definition of integers. Natural numbers, opposite numbers, zero are called integers. Modulus of a number - the concept of modulus of a number gives from the origin to the point the corresponding number. Students should pay attention to how to motivate the determination of the modulus of a number.

In textbooks, the concept of the modulus of a number is introduced by considering examples, and they explain how to find the modulus of a number. It is explained that the modulus of a number cannot be negative because the modulus of a number is a distance; attention is drawn to the fact that for a positive number the modulus is equal to the number itself. The modulus of a negative number is equal to the opposite number.

Comparison of numbers.

The relations of equality and inequality between positive and negative numbers are introduced by definition; they cannot be obtained by proof, and it is very important to show students the appropriateness of the definition by specific examples and geometric images.

Students should become so familiar with the arrangement of numbers on the number line that it can serve as the primary means of comparing numbers. Sometimes difficulties arise in comparing negative numbers; to overcome them, you need to consider them on a number line.

Operations on negative and positive numbers.

The main thing that the teacher needs to take into account when considering this material is that the operations of addition and subtraction on positive and negative numbers are introduced by definition, and the formulations of these definitions should include concepts previously known to students about these actions. Subtraction and division are defined as the inverses of addition and multiplication.

The textbook provides a separate definition of the action of adding numbers with different signs; the wording of these rules contains instructions for the following actions. The textbook spends a lot of time on how to approach the action of addition. The main attention is paid to the consideration of specific problems, while referring to the coordinate line.

No matter how the addition rule is introduced to students, it should be clear that nothing is proven when considering the following examples.

The examples are intended only to illustrate the appropriateness of the rules. Students must master the skills of adding 2 negative numbers with different signs, opposite numbers, zero with positive and negative numbers.

When considering the properties of actions, it is important to show students that with established definitions of the actions of adding and subtracting numbers, all the laws that apply to positive numbers are preserved.

Students are given the formulation of the commutative and associative laws and write each of them using letters.

Subtraction of negative numbers is defined as the inverse action of addition. Subtraction comes down to adding the opposite number.

Multiplication of positive and negative numbers presents the greatest difficulty, the difficulty lies in the fact that the student feels the need to prove the rules of signs when multiplying, and the teacher must convince the students that such a proof cannot be sought or demanded, thus the action of multiplication is introduced by a definition that can be introduced interpret the rule of signs in different ways and in different ways. Addition and multiplication have a lot in common, but interpreting the rules of multiplication is more difficult.

Let's consider an explanation of the rules of multiplication by considering specific problems, the solution of which requires calculation using the formula a in, for different a and b. The disadvantage of this method is that they prove the multiplication rule.

Many authors follow the path where at the beginning the formulation of the rules of multiplication is given, then it is explained with examples and problems. The student is convinced in concrete mathematics of the practical feasibility of the introduced definition. Typically, in textbooks, the formulation of the rules for multiplying numbers with different signs and the rules for multiplying natural numbers are presented with schedules of series of examples.

In this case, the provision is used that if you change the sign of one of the factors, the sign of the product will change.

The rule is formulated in a form convenient for use. It is necessary to draw students' attention to the conditions for a product to be equal to zero.

Division of positive and negative numbers is considered as the inverse action of multiplication. The student is told that dividing positive and negative numbers has the same meaning as dividing positive numbers. It is important to pay attention to the laws of calculation and multiplication of expressions.

Just as in the case of addition, the rule for adding and multiplying natural numbers can be derived from multiplying numbers. Assuming that the sign rule for the sum is known.

In the 6th grade, in the topic of rational numbers, negative numbers are introduced to memory, which can be written as a fraction. A lot of people are signing rational numbers You can confuse your attention that when it is feasible:, +, *, - to a number not equal to zero.

When subtracting or performing actions, the student receives numbers from the same set and this set has the property of being closed in relation to actions of the first and second degree. For addition, the commutative and associative laws are valid: there is a neutral element, there is an opposite element.

For multiplication, the first distributive and combinational laws are valid; there is a neutral element 1, the opposite element ().

Practical lesson No. 2

Subject: Study of function in ShKM

1. Methodology for introducing the concept of function.

2. Methodology for studying individual functions

3. Types of functions studied in basic school

Literature: , . Further reading I.