In what cases is a negative number greater than a positive one? Application of positive and negative numbers in human life

Is it possible to subtract a larger number from a smaller number? We have begun to consider this issue.

In order to clarify the situation, let's draw a vertical line and mark the position of the city with a dot on it. We will consider this point starting point or zero. Now let’s plot several equal divisions on the straight line above and below the zero point. Let each division correspond to one kilometer.

We will call numbers above the reference point (that is, north of the city) normal (or positive), and numbers below the reference point (that is, south of the city) will be called numbers less than zero, or negative.

Now we need a special symbol that will help distinguish between positive and negative numbers. This is usually done using a notation system based on the way in which the number can be obtained. Any positive number is obtained by adding other positive numbers. The addition symbol is "+" sign, so positive numbers are denoted by +1, +2, +3 and so on. The very name “positive number” suggests that this number really exists.

Negative numbers are obtained as a result of subtraction, say, when subtracting (2-3) we get a number by one less than zero. It is designated -1. Thus, negative numbers represent - -1, -2, -3, and so on.

It is no coincidence that numbers less than zero are called negative. Even when mathematicians mastered operations with numbers, less than zero, it was necessary to emphasize that these numbers do not exist in reality.

Please note zero is neither a positive nor a negative number.

Now we have a vertical marked line, that is, a scale, and we can use it for addition and subtraction operations. Since positive numbers increase up the scale, and adding positive numbers causes numbers to increase, we will assume that addition is a movement up the scale. Subtraction is the opposite operation of addition, so subtraction is a movement down the scale.

Suppose we need to add +2 and +5. This expression can be written as follows: (+2) + (+5). We needed parentheses for the reason that it is necessary to separate the plus as a sign from the pluses denoting positive numbers. But since we are accustomed to what we usually deal with positive numbers mi, then often the “+” signs in front of positive numbers are simply omitted. Then we get: 2+5. It is necessary to put “+” signs in front of positive numbers only in cases where it is necessary to attract special attention to the sign of the number.

Now let's put two divisions up on our scale. This number is 2. Let's add 5 more divisions and stop at division 7, that is, 2+5=7. We can start at 5 and add two divisions. We will again get 7. Here I once again want to draw your attention to the fact that changing the places of the terms does not change the sum.

Now let's do the subtraction. Suppose we need to subtract 2 from 5. From point 5 on the scale, we put two divisions down and end up at point 3. Thus, we get 5-2 = 3.

Now we need to figure out how to handle negative numbers. Is it possible to perform the same operations with them as with positive numbers? If so, they will be very useful, even though they are not “real” numbers. And indeed, negative numbers have found wide application not only in science and engineering practice, but also in everyday activities. They are used, for example, in accounting, where inventories and income are denoted by positive numbers, and expenses by negative numbers.

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

Application of positive and negative numbers in human life

Research work

Completed:

6B 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 positive and negative numbers 4

2.Use of 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 provides general methods for solving arithmetic problems, regardless of the specific content and initial numerical data.

Having studied 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 a wide application.

This research work aims to study 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.

Purpose of the work: Explore 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 research:Positive and negative numbers.

Research method:reading and analyzing 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 applications of positive and negative numbers in various fields.

4. Draw conclusions.

1. 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. Chinese scientist Zhang Can wrote the book Arithmetic in Nine Chapters. From the contents of the book it is clear that this is not a completely independent work, but a reworking of other books written long before Zhang Can. In this book, negative quantities are encountered for the first time in science. They are understood differently from the way we understand and apply them. He does not have a complete and clear understanding of the nature of negative and positive quantities and the rules for operating with them. He understood every negative number as a debt, and every positive number as property. He performed operations with negative numbers not the same way as we do, but using reasoning about debt. For example, if you add another debt to one debt, then the result is debt, not property (i.e., according to ours (- a) + (- a) = - 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 the numbers expressing property (positive). In Chinese mathematics, positive quantities were called “chen” and were depicted in red, and negative ones were “fu” and were depicted in black. This method of representation was used in China. until the middle of the 12th century, until Li Ye proposed a more convenient designation for negative numbers - the numbers that depicted negative numbers were crossed out diagonally from right to left. Although Chinese scientists explained negative quantities as debt, and positive quantities as property, they still avoided the broad one. using them, since these numbers seemed incomprehensible, the actions with them were unclear. If the problem led to a negative solution, then they tried to replace the condition (like the Greeks) so that in the end a positive solution would be obtained. In the V-VI centuries, negative numbers appear and spread very widely in Indian mathematics. Unlike China, the rules of multiplication and division were already known in India. In India, negative numbers were used systematically, much as we do now. Already in the work of the outstanding Indian mathematician and astronomer Brahmagupta (598 - about 660) we read: “property and property is property, the sum of two debts is a debt; the sum of property and zero is property; the sum of two zeros is zero... Debt, which is subtracted from zero, becomes property, and property becomes debt. If it is necessary to take away property from debt, and debt from property, then they take their sum.”

The "+" and "-" signs were widely used in trade. Winemakers put a “-” sign on empty barrels, indicating decline. If the barrel was filled, 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 since 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 rights to positive ones, and the fact that they cannot 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 the history of the development of this mathematical concept begins.

1. Using positive and negative numbers

1. Medicine

Myopia and farsightedness

Negative numbers express eye pathology. Myopia (myopia) is manifested by decreased visual acuity. In order for the eye to see distant objects clearly in case of myopia, diverging (negative) lenses are used.Myopia (-), farsightedness (+).

Farsightedness (hyperopia) is a type of eye refraction in which the image of an object is focused not on a certain area of ​​the retina, but in the plane behind it. This state of the visual system leads to blurred images perceived by the retina.

The cause of farsightedness can be a shortened eyeball, or a weak refractive power of the optical media of the eye. By increasing it, you can ensure that the rays will focus where they focus during normal vision.

With age, vision, especially near vision, increasingly deteriorates 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 low degrees of farsightedness, high vision is usually maintained both at distance and near, but there may be complaints of fatigue, headache, and dizziness. With moderate hypermetropia, distance vision remains good, but near vision is difficult. With high farsightedness, there is poor vision both far and near, since all the eye’s ability to focus images of even distant objects on the retina has been exhausted.

Farsightedness, including age-related, can only be detected through carefuldiagnostic examination (with medicinal dilation of the pupil, the lens relaxes and the true refraction of the eye appears).

Myopia is an eye disease in which a person has difficulty seeing objects located far away, but sees objects that are close well. Nearsightedness is also called myopia.

It is believed that about eight hundred million people are myopic. Everyone can suffer from myopia: both adults and children.

Our eyes contain a cornea and a lens. These components of the eye are capable of transmitting rays by 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 appears blurry and unclear. This may happen for the following reasons:

– if the eye elongates greatly, the retina moves away from the stable focus location. In people with myopia, the eye reaches thirty millimeters. And the normal one healthy person the size of the eye is twenty-three to twenty-four millimeters; - if the lens and cornea refract the light rays 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 located close to the eyes of a person who is myopic, then he will see these objects well.

2) Thermometers

Let's look at the scale of a regular outdoor thermometer.

It has the form shown on scale 1. Only positive numbers are printed on it, and therefore, when indicating the numerical value of the temperature, it is necessary to additionally explain 20 degrees Celsius (above zero). This is inconvenient for physicists - after all, you can’t put words into a formula! Therefore, in physics a scale with negative numbers is used (scale 2).

3) Balance on the phone

When checking the balance on your phone or tablet, you can see a number with a sign (-), this means that this subscriber has a debt and cannot make a call until he tops up his account, a number without a sign (-) means that he can call or make any -or other function.

1. Sea level

Let's look at physical card peace. The land areas on it are painted in various shades of green and brown colors, 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) a particular color means, for example, this:

Scale of depths and heights in meters

Deeper 5000 2000 200 0 200 1000 2000 4000 higher

On this scale we see only positive numbers and zero. The height (and depth too) at which the surface of the water in the World Ocean is located is taken as zero. Using only non-negative numbers in this scale is inconvenient for a mathematician or physicist. The physicist comes up with such a scale.

Height scale in meters

Less -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 answer various places on land above the surface of the sea; negative numbers correspond to points below the sea surface.

In the height scale we considered, the height of the water surface in the World Ocean is taken as zero. This scale is used in geodesy and cartography.

In contrast, in everyday life we ​​usually take the height of the earth’s surface (in the place where we are) as zero height.

5) Human qualities

Each person is individual and unique! However, we do not always think about what character traits define us as a person, what attracts people to us and what repels us. Highlight positive and negative qualities person. For example, positive qualities activity, nobility, dynamism, courage, enterprise, determination, independence, courage, honesty, energy, negative, aggressiveness, hot temper, competitive, criticality, stubbornness, selfishness.

6) Physics and comb

Place several small pieces of tissue 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 crackling sound. Then slowly move the comb towards 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. This attraction is easy to observe if you bring a comb to a thin stream of water flowing calmly from a tap. You will see that the stream is noticeably bent.

Now roll up two tubes 2-3 cm long from thin paper (preferably tissue paper). and a diameter of 0.5 cm. Hang them side by side (so that they lightly touch each other) on silk threads. After combing your hair, touch the paper tubes with the comb - they will immediately move apart and remain in this position (that is, the threads will be deflected). We see that the tubes repel each other.

If you have a glass rod (or tube, or test tube) and a piece of silk fabric, 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” onto the stick in the same way as on the comb, and then slide off it. The stream of water is also deflected by the glass rod, and the paper tubes that you touch with the rod 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, attractive and repulsive forces are manifested. In experiments, we saw that charged objects (physicists say charged bodies) can be attracted to each other, and can also repel each other. This is explained by the fact that there are two types, two varieties electric charges, and charges of the same type repel each other, and charges different types are attracted.

7) Counting time

IN different countries differently. For example, in Ancient Egypt Every time a new king began to rule, the counting of years began anew. The first year of the king's reign was considered the first year, the second - the second, and so on. When this king died and a new one came to power, the first year began again, then the second, the third. The counting of years used by the residents of one of the ancient cities peace-Rome. The Romans considered the year the city was founded to be the first, the next year to be the second, and so on.

The counting of years that we use arose a long time ago and is associated with the veneration of Jesus Christ, the founder Christian religion. Counting years from the birth of Jesus Christ was gradually adopted in different countries. In our country, it was introduced by Tsar Peter the Great three hundred years ago. We call the time calculated from the Nativity of Christ OUR ERA (and we write it in abbreviated form NE). Our era continues for two thousand years. Consider the “time line” in the figure.

Foundation Beginning First mention of Moscow Birth of A. S. Pushkin

Rome revolt

Spartak

Conclusion

Working with various sources and exploring various phenomena and processes, we found that negative and positive are used in medicine, physics, geography, history, 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 people.

This activity can be used in math lessons to motivate students to learn about positive and negative numbers.

List of used literature

1. Vigasin A.A., Goder G.I., “History ancient world", textbook 5th grade, 2001.
2. Vygovskaya V.V. “Lesson-based developments in Mathematics: 6th grade” - M.: VAKO, 2008.
3. Newspaper "Mathematics" No. 4, 2010.
4. Gelfman E.G. "Positive and Negative Numbers" training manual in mathematics for the 6th grade, 2001.