Velmyakina Kristina and Nikolaeva Evgenia
This research work is aimed at studying the use of positive and negative numbers in human life.
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MBOU "Gymnasium No. 1" of the Kovylkinsky municipal district
The use of positive and negative numbers in human life
Research
Completed:
6th class students
Velmyakina Kristina and Nikolaeva Evgenia
Head: teacher of mathematics and computer science
Sokolova Natalya Sergeevna
Kovylkino 2015
Introduction 2
1. The history of the emergence of positive and negative numbers 4
2.Using positive and negative numbers 6
Conclusion 13
List of used literature 14
Introduction
The introduction of positive and negative numbers was associated with the need to develop mathematics as a science that gives common ways solving arithmetic problems, regardless of the specific content and initial numerical data.
By examining the positive and negative numbers in mathematics lessons, we decided to find out where else, besides mathematics, these numbers are used. And it turned out that positive and negative numbers have quite wide application.
This research is aimed at studying the use of positive and negative numbers in human life.
The relevance of this topic lies in the study of the use of positive and negative numbers.
Objective: To study the use of positive and negative numbers in human life.
Object of study:Areas of application of positive and negative numbers in human life.
Subject of study:Positive and negative numbers.
Research method:reading and analysis of the literature used and observations.
To achieve the goal of the study, the following tasks were set:
1. Study the literature on this topic.
2. Understand the essence of positive and negative numbers in human life.
3. Explore the application of positive and negative numbers in various fields.
4. Draw conclusions.
- The history of positive and negative numbers
Positive and negative numbers first appeared in Ancient China already about 2100 years ago.
In the II century. BC e. The Chinese scholar Zhang Can wrote Arithmetic in Nine Chapters. From the contents of the book it is clear that this is not a completely independent work, but a revision of other books written long before Zhang Can. In this book, for the first time in science, negative quantities are encountered. They are understood by them differently than we understand and apply them. He does not have a complete and clear understanding of the nature of negative and positive quantities and the rules for working with them. He understood every negative number as a debt, and every positive number as property. He performed operations with negative numbers not in the same way as we do, but using reasoning about duty. For example, if we add another debt to one debt, then the result is debt, not property (t, that is, according to our (- a) + (- a) \u003d - 2a. 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 numbers expressing wealth (positive).Positive numbers in Chinese mathematics were called "chen" and depicted in red, and negative ones were called "fu" and depicted in black.This way of depicting was used in China until the middle of the 12th century, when Li Ye proposed a more convenient notation for negative numbers - the numbers that depicted negative numbers were crossed out with a dash obliquely from right to left.Although Chinese scholars explained negative quantities as debt, and positive ones as property, they still avoided a wide their use, since these numbers seemed incomprehensible, 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 it oge the decision was positive. In the 5th-6th centuries, negative numbers appear and are very widely distributed in Indian mathematics. Unlike China, in India, the rules for multiplication and division were already known. In India, negative numbers were systematically used in much the same way as we do now. Already in the work of the outstanding Indian mathematician and astronomer Brahmagupta (598 - about 660) we read: “property and property are 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 property from debt, and debt from property, then they take their amount.
The "+" and "-" signs were widely used in trading. Winemakers put a “-” sign on empty barrels, meaning a decline. If the barrel was filled, then the sign was crossed out and a “+” sign was received, meaning profit. These signs were introduced as mathematical ones by Jan Widmann in XV.
In European science, negative and positive numbers finally came into use only from the time of the French mathematician R. Descartes (1596 - 1650), who gave a geometric interpretation of positive and negative numbers as directed segments. In 1637 he introduced the "coordinate line".
In 1831, Gauss fully substantiated that negative numbers are absolutely equivalent in terms of rights with positive ones, and the fact that they can not be applied in all cases does not matter.
The history of the emergence of negative and positive numbers ends in the 19th century when William Hamilton and Hermann Grassmann created a complete theory of positive and negative numbers. From this moment begins the history of the development of this mathematical concept.
- Applying Positive and Negative Numbers
- The medicine
Nearsightedness and farsightedness
Negative numbers express the pathology of the eye. Nearsightedness (myopia) is manifested by a decrease in visual acuity. In order for the eye to see clearly distant objects in myopia, diffusing (negative) lenses are used.Myopia (-), farsightedness (+).
Farsightedness (hypermetropia) is a type of refraction of the eye, in which the image of an object is focused not on a specific area of the retina, but in a plane behind it. This state of the visual system leads to the fuzziness of the image that the retina perceives.
The cause of farsightedness may be a shortened eyeball, or a weak refractive power of the optical media of the eye. By increasing it, it is possible to ensure that the rays will be focused where they are focused in normal vision.
With age, vision, especially near vision, deteriorates more and more due to a decrease in the accommodative ability of the eye due to age-related changes in the lens - the elasticity of the lens decreases, the muscles that hold it weaken, and as a result, vision decreases. That's whyage-related farsightedness (presbyopia ) is present in almost all people after 40-50 years.
With small degrees of farsightedness, high vision is usually maintained both in the distance and near, but there may be complaints of fatigue, headache, dizziness. With an average degree of hypermetropia, distance vision remains good, but close vision is difficult. With high farsightedness - poor eyesight both far and near, since all the possibilities of the eye to focus on the retina an image of even distant objects have been exhausted.
Farsightedness, including age-related, can only be detected with a thorough examination.diagnostic examination (with medical dilation of the pupil, the lens relaxes and the true refraction of the eye appears).
Myopia - This is an eye disease in which a person sees poorly objects located far away, but sees well those objects that are close. Nearsightedness is also called myopia.
It is believed that about eight hundred million people suffer from myopia. Everyone can suffer from myopia: both adults and children.
Our eyes have a cornea and a lens. These component eyes are able to transmit rays, refracting them. And an image appears on the retina. Then this image becomes nerve impulses and is transmitted along the optic nerve to the brain.
If the cornea and lens refract the rays so that the focus is on the retina, then the image will be clear. Therefore, people without any eye diseases will see well.
With myopia, the image is blurry and fuzzy. This may happen for the following reasons:
- if the eye is greatly elongated, then the retina moves away from a stable focus location. With myopia in humans, the eye reaches thirty millimeters. And in a normal healthy person, the size of the eye is twenty-three - twenty-four millimeters - if the lens and cornea refract the rays of light too much.
According to statistics, every third person on earth suffers from myopia, that is, myopia. It is difficult for such people to see objects that are far from them. But at the same time, if a book or notebook is close to the eyes of a person who suffers from myopia, then he will see these objects well..
2) Thermometers
Let's look at the scale of a conventional outdoor thermometer.
It has the form shown on scale 1. Only positive numbers are marked on it, and therefore, when indicating the numerical value of the temperature, it is necessary to additionally explain 20 degrees of heat (above zero). This is inconvenient for physicists - you can't substitute words into a formula! Therefore, in physics, a scale with negative numbers (scale 2) is used.
3) Phone balance
When checking the balance on your phone or tablet, you can see a number with a sign (-), which means that this subscriber has a debt and cannot make a call until he replenishes his account, while a number without a sign (-) means that you can call or make some or any other function.
- Sea level
Let's look at physical map peace. Land plots on it are painted various shades green and brown, and the seas and oceans are painted 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) this or that color means, for example, this:
Depth and height scale in meters
Deeper 5000 2000 200 0 200 1000 2000 4000 higher
On this scale we see only positive numbers and zero. Zero is the height (and depth too) at which the surface of the water in the World Ocean is located. The use of only non-negative numbers in this scale is inconvenient for a mathematician or physicist. The physicist gets such a scale.
Altitude scale in meters
Less than -5000 -2000 -200 0 200 1000 2000 4000 more
Using such a scale, it is enough to indicate the number without any additional words: positive numbers correspond to various places on land that are above the surface of the sea; negative numbers correspond to points under the sea surface.
In the scale of heights considered by us, 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.
5) Qualities of a person
Each person is individual and unique! However, we do not always think about what character traits define us as a person, what attracts people in us and what repels us. Highlight the positive and negative qualities of a person. For example, positive traits activity, nobility, dynamism, courage, enterprise, determination, independence, courage, honesty, vigor, negative, aggressiveness, irascibility, competitiveness, criticality, stubbornness, selfishness.
6) Physics and comb
Place a few small pieces of thin paper on the table. Take a clean, dry plastic comb and run it through your hair 2-3 times. When combing your hair, you should hear a slight crackle. Then slowly bring the comb to the scraps of paper. You will see that they are first attracted to the comb, and then repelled from it.
The same comb can attract water. Such an attraction is easy to observe if you bring the comb to a thin stream of water flowing calmly from the faucet. You will see that the trickle is noticeably curved.
Now roll out of thin paper (preferably tissue paper) two tubes 2-3 cm long. and 0.5 cm in diameter. Hang them side by side (so that they lightly touch each other) on silk threads. After combing your hair, touch the comb to the paper tubes - they will immediately disperse to the sides and remain in this position (that is, the threads will be rejected). We see that the tubes repel each other.
If you have a glass rod (or a tube, or a test tube) and a piece of silk cloth, then the experiments can be continued.
Rub the stick on the silk and bring it to the scraps of paper - they will begin to "jump" on the stick in the same way as on the comb, and then slide off it. A trickle of water is also deflected by a glass rod, and paper tubes that you touch with a stick repel each other.
Now take one stick, which you touched with a comb, and the second tube, and bring it to each other. You will see that they are attracted to each other. So, in these experiments, the forces of attraction and the forces of repulsion are manifested. In experiments, we have seen that charged objects (physicists say charged bodies) can attract each other, or they can repel each other. This is explained by the fact that there are two types, two types of electric charges, and charges of the same type repel each other, and charges different types are attracted.
7) Counting time
AT different countries differently. For example, in Ancient Egypt every time I started 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 came again, then the second, the third. The count of years used by the inhabitants of one of the oldest cities in the world, Rome, was different. The Romans considered the year of the foundation of their city the first, the next - the second, and so on.
The count of years that we use arose long ago and is associated with the veneration of Jesus Christ, the founder of the Christian religion. The count of 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. The time counted from the Nativity of Christ, we call OUR ERA (and we write NE for short). Our era has been going on for two thousand years. Consider the "time line" in the figure.
Founding Beginning The first mention of Moscow Birth of A. S. Pushkin
rome uprising
Spartacus
Conclusion
Working with various sources and exploring various phenomena and processes, we found out that negative and positive are used in medicine, physics, geography, history, in modern means communication, in the study of human qualities and other areas of human activity. This topic is relevant and is widely used and actively used by man.
This work can be used in mathematics lessons, motivating students to study positive and negative numbers.
Bibliography
- Vigasin A.A., Goder G.I., “History ancient world”, textbook 5th grade, 2001.
- Vygovskaya V.V. "Pourochnye development in Mathematics: Grade 6" - M.: VAKO, 2008.
- Newspaper "Mathematics" №4, 2010
- Gelfman E.G. "Positive and Negative Numbers" tutorial in mathematics for the 6th grade, 2001.
Now we will analyze positive and negative numbers. First, we give definitions, introduce notation, after which we give examples of positive and negative numbers. We will also dwell on the semantic load that positive and negative numbers carry.
Page navigation.
Positive and Negative Numbers - Definitions and Examples
To give determination of positive and negative numbers will help us. For convenience, we will assume that it is located horizontally and directed from left to right.
Definition.
The numbers that correspond to the points of the coordinate line lying to the right of the origin are called positive.
Definition.
The numbers that correspond to the points of the coordinate line lying to the left of the origin are called negative.
The number zero corresponding to the origin is neither positive nor negative.
From the definition of negative and positive numbers, it follows that the set of all negative numbers is the set of numbers that are opposite to all positive numbers (if necessary, see the article opposite numbers). Therefore, negative numbers are always written with a minus sign.
Now, knowing the definitions of positive and negative numbers, we can easily write examples of positive and negative numbers. Examples of positive numbers are natural numbers 5 , 792 and 101 330 , and indeed any natural number is positive. Examples of positive rational numbers are numbers , 4.67 and 0,(12)=0.121212... , and negative ones are numbers , −11 , −51.51 and −3,(3) . As examples of positive irrational numbers, one can give the number pi, the number e, and the infinite non-periodic decimal fraction 809.030030003 ..., and examples of negative ir rational numbers are the numbers minus pi, minus e and the number equal to . It should be noted that in the last example it is by no means obvious that the value of the expression is a negative number. To find out for sure, you need to get the value of this expression in the form decimal fraction, and how this is done, we will tell in the article comparison of real numbers.
Sometimes positive numbers are preceded by a plus sign, just as negative numbers are preceded by a minus sign. In these cases, you should know that +5=5 . 
etc. That is, +5 and 5, etc. is the same number, but differently denoted. Moreover, you can find the definition of positive and negative numbers, based on the plus or minus sign.
Definition.
Numbers with a plus sign are called positive, and with a minus sign - negative.
There is another definition of positive and negative numbers based on comparing numbers. To give this definition, it is enough to remember that the point on the coordinate line corresponding to a larger number lies to the right of the point corresponding to a smaller number.
Definition.
positive numbers are numbers that are greater than zero, and negative numbers are numbers less than zero.
Thus, zero, as it were, separates positive numbers from negative ones.
Of course, we should also dwell on the rules for reading positive and negative numbers. If the number is written with a + or - sign, then the name of the sign is pronounced, after which the number is pronounced. For example, +8 is read as plus eight, and as minus one point two fifths. The names of the signs + and − are not declined by cases. An example correct pronunciation is the phrase "a equals minus three" (not minus three).
Interpretation of positive and negative numbers
We have been describing positive and negative numbers for quite some time now. However, it would be nice to know what meaning they carry in themselves? Let's deal with this issue.
Positive numbers can be interpreted as income, as an increase, as an increase in some value, and the like. Negative numbers, in turn, mean exactly the opposite - expense, lack, debt, decrease in some value, etc. Let's deal with this with examples.
We can say that we have 3 items. Here, the positive number 3 indicates the number of items we have. How can you interpret a negative number −3? For example, the number -3 could mean that we have to give someone 3 items that we don't even have in stock. Similarly, we can say that at the box office they gave us 3.45 thousand rubles. That is, the number 3.45 is associated with our arrival. In turn, a negative number -3.45 will indicate a decrease in money in the cash register that issued this money to us. That is, −3.45 is the expense. Another example: an increase in temperature by 17.3 degrees can be described as a positive number +17.3, and a decrease in temperature by 2.4 can be described using a negative number as a change in temperature by -2.4 degrees.
Positive and negative numbers are often used to describe the values of any quantities in various measuring instruments. The most accessible example is a device for measuring temperatures - a thermometer - with a scale on which both positive and negative numbers are written. Often negative numbers are depicted in blue (it symbolizes snow, ice, and at temperatures below zero degrees Celsius water begins to freeze), and positive numbers are written in red (the color of fire, the sun, at temperatures above zero degrees ice begins to melt). Writing positive and negative numbers in red and blue is also used in other cases when it is necessary to emphasize the sign of numbers.
Bibliography.
- Vilenkin N.Ya. etc. Mathematics. Grade 6: textbook for educational institutions.
Negative numbers are located to the left of zero. For them, as well as for positive numbers, an order relation is defined that allows you to compare one integer with another.
For every natural number n there is one and only one negative number, denoted by -n, which complements n to zero: n + (− n) = 0 . Both numbers are called opposite for each other. Subtraction of an integer a is equivalent to adding to its opposite: -a.
Properties of negative numbers
Negative numbers follow almost the same rules as natural numbers, but have some peculiarities.
Historical outline
Literature
- Vygodsky M. Ya. Reference book elementary mathematics. - M.: AST, 2003. - ISBN 5-17-009554-6
- Glazer G.I. History of mathematics in the school. - M.: Enlightenment, 1964. - 376 p.
Links
Wikimedia Foundation. 2010 .
- Negative landforms
- Negative and positive zero
See what "Negative numbers" are in other dictionaries:
Negative numbers- real numbers less than zero, for example 2; 0.5; π etc. See Number... Great Soviet Encyclopedia
Positive and negative numbers- (values). The result of successive additions or subtractions does not depend on the order in which these operations are performed. Eg. 10 5 + 2 \u003d 10 +2 5. Here, not only the numbers 2 and 5 are permuted, but also the signs in front of these numbers. Agreed... ... encyclopedic Dictionary F. Brockhaus and I.A. Efron
numbers are negative- Numbers in accounting that are written in red pencil or red ink. Accounting topics... Technical Translator's Handbook
NEGATIVE NUMBERS- numbers in accounting that are written in red pencil or red ink ... Big accounting dictionary
Whole numbers- 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
Integers- numbers that naturally arise during counting (both in the sense of enumeration and in the sense of calculus). There are two approaches to determining the natural numbers of the number used in: enumerating (numbering) objects (first, second, ... ... Wikipedia
EULER NUMBERS are the coefficients E n in the decomposition. The recursive formula for E. h. has the form , E4n+2 negative integers for all n=0, 1, . . .; E2= 1, E4=5, E6=61, E8=1385 … Mathematical Encyclopedia
A negative number- A negative number is an element of the set of negative numbers, which (together with zero) appeared in mathematics when the set of natural numbers was expanded. The purpose of the extension is to provide a subtraction operation for any numbers. As a result ... ... Wikipedia
History of arithmetic- Arithmetic. Painting by Pinturicchio. Borgia apartments. 1492 1495. Rome, Vatican Palaces ... Wikipedia
Arithmetic- Hans Sebald Beham. Arithmetic. XVI century Arithmetic (other Greek ἀ ... Wikipedia
Books
- Mathematics. Grade 5 Educational book and workshop. In 2 parts. Part 2. Positive and negative numbers, . The textbook and workshop for grade 5 are part of the teaching materials for mathematics for grades 5-6, developed by a team of authors led by E. G. Gelfman and M. A. Kholodnaya as part of ...
Chalina Irina
Presentation on the history of negative numbers.
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Negative numbers Chalina Irina
Mathematics - vivat! Glory, glory, glory! Don't serenades her, Don't shout bravo to her. Once upon a time there were 2 numbers, Lived, did not grieve. One is a minus, the other is a plus, We were friends cheerfully. Signs are different in everything, But you can put, To add up the number, Which should be. Plus by plus - we get a plus, Plus by minus - there will be a minus. Well, if we add (-20) (-8), then in the end we will get the number (-28).
Negative number A negative number is an element of the set of negative numbers, which (together with zero) appeared in mathematics when the set of natural numbers was expanded. The purpose of the extension is to provide a subtraction operation for any numbers. As a result of the expansion, a set (ring) of integers is obtained, consisting of positive (natural) numbers, negative numbers and zero. All negative numbers, and only they, are less than zero. On the number axis, negative numbers are located to the left of zero. For them, as well as for positive numbers, an order relation is defined that allows you to compare one integer with another.
Historical reference History says that people could not get used to negative numbers for a long time. Negative numbers seemed incomprehensible to them, they were not used, they simply did not see the meaning in them. Positive numbers were interpreted as "profit", and negative - as "debt", "loss". In Ancient Egypt, Babylon and Ancient Greece did not use negative numbers, and if negative roots of equations were obtained (when subtracted), they were rejected as impossible. 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 (shortage), or recognized as an intermediate stage, useful for calculating the final, positive result. But there were no + or - signs in ancient times either for numbers or for actions. True, multiplication and division for negative numbers had not yet been defined. The Greeks also did not use signs at first, until Diophantus of Alexandria in the 3rd century began to use the “-” sign when solving linear equations. The “+” sign appeared as a result of the opposite action to the “-” sign by crossing out the minus. It was very similar to the plus that we use now. He already knew the rule of signs and knew how to multiply negative numbers. However, he considered them only as temporary values.
The usefulness and legality of negative numbers were established gradually. 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”. Even Pascal thought that 0 − 4 = 0, since nothing can be less than nothing. Bombelli and Girard, on the contrary, considered negative numbers quite acceptable and useful, in particular, to indicate 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 the 17th century, with the advent of analytic geometry, negative numbers received a visual geometric representation on the number line. From this moment comes their complete equality. Nevertheless, the theory of negative numbers was in its infancy for a long time. For example, the strange proportion 1: (-1) = (-1): 1 was actively discussed - 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 larger is equal to the smaller ("Arnaud's paradox"). It was also not clear what meaning the multiplication of negative numbers has, and why the product of negative numbers is positive; there were heated discussions on this topic. A complete and quite rigorous theory of negative numbers was created only in the 19th century by William Hamilton and Hermann Grassmann.
Properties of negative numbers Negative numbers are subject to almost the same algebraic rules, which are natural, but have some features. If any set of positive numbers is bounded below, then any set of negative numbers is bounded above. When multiplying integers, the sign rule applies: the product of numbers with different signs negative, with the same - positive. When both sides of the 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 defined). For every natural number (n) there is one and only one negative number, denoted by (-n), which completes n to zero: Both numbers are called opposites of each other. Subtracting an integer (a) from another integer (b) is equivalent to adding b with the opposite sign of a: (b)+ (-a)
Basic Rules Rule 1. The sum of two negative numbers is a negative number equal to the sum of the modules of these numbers. Example - The sum of the numbers (-3) and (-8) is equal to minus 11. Rule 2. The product of two numbers with different signs is a negative number, the modulus of which is equal to the product of the moduli of the factors. Example - The product of minus three and five is equal to minus fifteen, because when multiplying two numbers with different signs, a negative number is obtained, and its modulus is equal to the product of the moduli of factors, that is, three and five. Rule 3. To mark negative numbers, coordinate beam complement it with a ray opposite to it and put on it the corresponding coordinates. Example. The numbers located on the coordinate line to the right of zero are called positive, and to the left - negative.
Module of negative number Distance from point A(a) to the origin, i.e. to the point O(o), is called the modulus of the number a and is denoted /a/ The modulus of the negative number is equal to the number, its opposite. The module, doing nothing with positive numbers and zero, takes away the minus sign from negative numbers. The module is indicated by vertical lines, which are written on both sides of the number. For example / -3 / = 3; / -2.3 / = 2.3; / -526/7 / = 526/7. Of two negative numbers, the greater is the one whose modulus is less, and the less is the one whose modulus is greater. (On this occasion, they usually joke that negative numbers are not like people, on the contrary)
Conclusion Negative numbers are common these days: they are used, for example, to represent temperatures below zero. Therefore, it seems surprising that a few centuries ago there was no specific interpretation of negative numbers, and negative numbers that appeared in the course of calculations were called "imaginary". Negative numbers are needed not only when measuring temperature. For example, if an enterprise received an income of 1 million rubles, or, conversely, suffered a loss of 1 million rubles, how should this be reflected in financial documents? In the first case, 1,000,000 rubles are recorded. or + 1,000,000 rubles. And in the second, respectively, (- 1,000,000 rubles).
Thank you for your attention! -
Let's say Denis has a lot of sweets - a whole big box. First Denis ate 3 sweets. Then dad gave Denis 5 sweets. Then Denis gave Matvey 9 sweets. Finally, mom gave Denis 6 sweets. Question: Did Denis eventually have more or less sweets than he had at the beginning? If more, how much more? If less, how much less?
In order not to get confused with this task, it is convenient to apply one trick. Let's write out all the numbers from the condition in a row. At the same time, we will put a “+” sign in front of the numbers that indicate how much candy Denis has increased, and a “-” sign in front of the numbers that indicate how much Denis has decreased candies. Then the whole condition will be written out very briefly:
− 3 + 5 − 9 + 6.
This entry can be read, for example, like this: “First, Denis received minus three candies. Then plus five sweets. Then minus nine candies. And finally, plus six sweets. The word "minus" changes the meaning of the phrase to the exact opposite. When I say: “Denis got minus three candies,” it actually means that Denis lost three candies. The word "plus", on the contrary, confirms the meaning of the phrase. "Denis got plus five candies" means the same as simply "Denis got five candies".
So, first Denis got minus three candies. This means that Denis has minus three more candies than he had at the beginning. For brevity, we can say: Denis has minus three candies.
Then Denis got plus five sweets. It is easy to figure out that Denis had plus two sweets. Means,
− 3 + 5 = + 2.
Then Denis got minus nine candies. And here's how many candies he got:
− 3 + 5 − 9 = + 2 − 9 = − 7.
Finally, Denis got +6 more candies. And all the candies became:
− 3 + 5 − 9 + 6 = + 2 − 9 + 6 = − 7 + 6 = − 1.
In the usual language, this means that in the end Denis had one less candy than he had at the beginning. Problem solved.
The trick with the "+" or "-" signs is very widely used. Numbers with a "+" sign are called positive. Numbers with the "−" sign are called negative. The number 0 (zero) is neither positive nor negative because +0 is no different from −0. Thus, we are dealing with numbers from the series
..., −5, −4, −3, −2, −1, 0, +1, +2, +3, +4, +5, ...
Such numbers are called whole numbers. And those numbers that have no sign at all and with which we have dealt so far are called natural numbers(only zero does not apply to natural numbers).
Integers can be thought of as rungs on a ladder. The number zero is landing located flush with the street. From here you can climb step by step to the higher floors, or you can go down to the basement. As long as we do not need to go into the basement, we are quite satisfied with natural numbers alone and zero. Natural numbers are essentially the same as positive integers.
Strictly speaking, an integer is not a step number, but a command to move up the stairs. For example, +3 means to go up three steps, and -5 means to go down five steps. It's just that the step number is taken to be such a command that moves us to the given step if we start moving from the zero level.
Integer calculations are easy to do by simply mentally jumping up or down the steps - unless, of course, you need to make too big jumps. But what about when you have to jump a hundred or more steps? After all, we will not draw such a long staircase!
And yet, why not? We can draw a long staircase from such a distance that the individual steps are no longer visible. Then our staircase will turn into just one straight line. And to make it more convenient to place it on the page, let's draw it without tilt and separately mark the position of the step 0.
Let's first learn how to jump along such a straight line using the example of expressions, the values of which we have been able to calculate for a long time. Let it be required to find
Strictly speaking, since we are dealing with integers, we should write
But a positive number at the beginning of a line usually does not have a “+” sign. Ladder jumping looks something like this:
Instead of two large jumps drawn above the line (+42 and +53), you can make one jump drawn below the line, and the length of this jump, of course, is
Such drawings in mathematical language are usually called diagrams. Here's what the chart looks like for our usual subtraction example.
First we made a big jump to the right, then a smaller jump to the left. As a result, we remained to the right of zero. But another situation is also possible, as, for example, in the case of the expression
This time the jump to the right turned out to be shorter than the jump to the left: we flew over zero and ended up in the "basement" - where the steps with negative numbers are located. Let's take a closer look at our jump to the left. In total, we climbed 95 steps. After we climbed 53 steps, we caught up with mark 0. How many steps did we climb after that? Well, of course
Thus, once on step 0, we went down another 42 steps, which means that in the end we came to step number −42. So,
53 − 95 = −(95 − 53) = −42.
Similarly, by drawing diagrams, it is easy to establish that
−42 − 53 = −(42 + 53) = −95;
−95 + 53 = −(95 − 53) = −42;
and finally
−53 + 95 = 95 − 53 = 42.
In this way, we have learned to freely travel the whole ladder of integers.
Consider now such a problem. Denis and Matvey exchange candy wrappers. At first, Denis gave Matvey 3 candy wrappers, and then he took 5 candy wrappers from him. How many candy wrappers did Matvey get in the end?
But since Denis got 2 candy wrappers, then Matvey got -2 candy wrappers. We attributed a minus to Denis's profit and got Matvey's profit. Our solution can be written as a single expression
−(−3 + 5) = −2.
Everything is simple here. But let's slightly modify the condition of the problem. Let Denis first give Matvey 5 candy wrappers, and then take 3 candy wrappers from him. The question is, again, how many candy wrappers did Matvey end up with?
Again, first we calculate the "profit" of Denis:
−5 + 3 = −2.
So Matvey got 2 candy wrappers. But now how can we write our solution as a single expression? What would you add to a negative number −2 to get a positive number 2? It turns out that this time it is also necessary to assign a minus sign. Mathematicians are very fond of uniformity. They strive to ensure that the solution of similar problems is written in the form of similar expressions. In this case, the solution looks like this:
−(−5 + 3) = −(−2) = +2.
So mathematicians agreed: if positive number If you add a minus, then it becomes negative, and if you add a minus to a negative number, then it becomes positive. This is very logical. After all, going down minus two steps is the same as going up plus two steps. So,
−(+2) = −2;
−(−2) = +2.
To complete the picture, we also note that
+(+2) = +2;
+(−2) = −2.
This gives us the opportunity to take a fresh look at long-familiar things. Let the expression
The meaning of this entry can be imagined in different ways. You can, in the old fashioned way, consider that a positive number +3 is subtracted from a positive number +5:
In this case +5 is called reduced, +3 - deductible, and the whole expression difference. That's how they teach in school. However, the words "reduced" and "subtracted" are not used anywhere except at school, and they can be forgotten after the final control work. About the same entry, we can say that a negative number -3 is added to the positive number +5:
The numbers +5 and −3 are called terms, and the whole expression sum. This sum has only two terms, but, in general, the sum can consist of any number of terms. Likewise, the expression
can be considered with equal right as the sum of two positive numbers:
and as the difference between positive and negative numbers:
(+5) − (−3).
After we got acquainted with integers, we definitely need to clarify the rules for opening brackets. If there is a “+” sign in front of the brackets, then such brackets can simply be erased, and all numbers in them retain their signs, for example:
+(+2) = +2;
+(−2) = −2;
+(−3 + 5) = −3 + 5;
+(−3 − 5) = −3 − 5;
+(5 − 3) = 5 − 3
etc.
If there is a “-” sign before the brackets, then by erasing the bracket, we must also change the signs of all the numbers in it:
−(+2) = −2;
−(−2) = +2;
−(−3 + 5) = +3 − 5 = 3 − 5;
−(−3 − 5) = +3 + 5 = 3 + 5;
−(5 − 3) = −(+5 − 3) = −5 + 3;
etc.
At the same time, it is useful to keep in mind the problem of the exchange of candy wrappers between Denis and Matvey. For example, the last line can be obtained like this. We believe that Denis first took 5 candy wrappers from Matvey, and then another -3. In total, Denis received 5 - 3 candy wrappers, and Matvey - the same number, but with opposite sign, that is, −(5 − 3) candy wrappers. But after all, the same problem can be solved in another way, bearing in mind that whenever Denis receives, Matvey gives. This means that at first Matvey received -5 candy wrappers, and then another +3, which ultimately gives -5 + 3.
Like natural numbers, integers can be compared with each other. Let's ask, for example, the question: which number is greater: -3 or -1? Let's look at the staircase with integers, and it will immediately become clear that -1 is greater than -3, and therefore -3 is less than -1:
−1 > −3;
−3 < −1.
Now let's clarify: how much more is -1 than -3? In other words, how many steps do you have to climb to get from step -3 to step -1? The answer to this question can be written as the difference between the numbers −1 and −3:
− 1 − (−3) = −1 + 3 = 3 − 1 = 2.
Jumping up the steps, it is easy to check that this is so. And here is another curious question: how much the number 3 more number 5? Or, which is the same: how many steps do you have to climb to get from step 5 to step 3? Until recently, this question would have puzzled us. But now we can easily write out the answer:
3 − 5 = − 2.
Indeed, if we are on step 5 and go up another −2 steps, then we will find ourselves on step 3.
Tasks
2.3.1. What is the meaning of the following phrases?
Denis gave dad minus three sweets.
Matvey is older than Denis by minus two years.
To get to our apartment, you have to go down minus two floors down.
2.3.2. Do such phrases make sense?
Denis has minus three candies.
Minus two cows graze in the meadow.
Comment. This problem does not have a unique solution. Of course, it would not be wrong to say that these statements are meaningless. And at the same time, they can be given a very clear meaning. Suppose Denis has a large box filled to the top with sweets, but the contents of this box do not count. Or suppose that two cows from the herd did not go out to graze in the meadow, but for some reason remained in the barn. It should be borne in mind that even the most familiar phrases can be ambiguous:
Denis has three sweets.
This statement does not exclude that Denis has a huge box of sweets hidden somewhere else, but they simply keep silent about those sweets. In the same way, when I say: “I have five rubles,” I do not mean that this is my entire fortune.
2.3.3. The grasshopper jumps up the stairs, starting from the floor where Denis' apartment is located. First he jumped 2 steps down, then 5 steps up, and finally 7 steps down. How many steps and in what direction did the grasshopper move?
2.3.4. Find expression values:
− 6 + 10;
− 28 + 76;
etc.
− 6 + 10 = 10 − 6 = 4.
2.3.5. Find expression values:
8 − 20;
34 − 98;
etc.
8 − 20 = − (20 − 8) = − 12.
2.3.6. Find expression values:
− 4 − 13;
− 48 − 53;
etc.
− 4 − 13 = − (4 + 13) = − 17.
2.3.7. For the following expressions, find the values by doing the calculations in the order given by the brackets. Then open the brackets and make sure that the meanings of the expressions remain the same. Make up problems about sweets that are solved in this way.
25 − (−10 + 4);
25 + (− 4 + 10);
etc.
25 − (− 10 + 4) = 25 − (−(10 − 4)) = 25 − (−6) = 25 + 6 = 31.
25 − (− 10 + 4) = 25 + 10 − 4 = 35 − 4 = 31.
“Denis had 25 sweets. He gave dad minus ten candies, and four candies to Matvey. How many candies did he have?
