How ancient peoples determined time. How different peoples counted time. What clock did man invent

The need to determine the time appeared when a person began to engage in agriculture. He needed to know when to sow and when it was time to harvest. At first, people were guided by time in general: winter was ending, which meant that it was necessary to sow. And as soon as there were signs of the first cold weather - it's time to collect.

It turns out that the record of time was very primitive: from sowing to harvesting. When a person was asked how old he was, he could answer: "I am fifteen winters." Until now, scientists are finding the remains of an account of this kind.

How was the timing point determined?

Different nationalities had their own point of reference for time. For example, in ancient Egypt it was the flood of the Nile River. When this process began again, it was clear that a year had passed. The Romans believed that time began to pass when their city of Rome was created. The inhabitants of ancient China calculated the time by the time of the ascension to the throne of the new emperor. As you can see, each nationality took a bright event and began to count the years from it.

Since each country had its own rules for how to count time, this was extremely inconvenient for their relationship. In addition, this creates difficulties for modern historians. To understand the year of the definition of some event, you need to delve into the culture of the people and find out how they counted time.

Due to the inconvenience of a different report of years, it was necessary to create single system, which would operate throughout the globe. It was decided to take as a basis the biblical message about the birth of Jesus Christ, the Son of God. This year was the start of the report.

Those countries that do not recognize the coming of Jesus did not agree with such a calculation. These were Muslim countries. The starting point of their calculation of years was the birth of their prophet Muhammad.

What were the first hours like?

It was noticed that many people determined in what position the sun was, and thus could tell what time it was. In this case, the errors were equal to a maximum of 10 minutes. Therefore, the first clocks for determining the time were solar devices made taking into account the movement of the sun. They consisted of a base and a mechanism such as a gnomon. The shadow from him performed the task of the arrow. Its end pointed to the north, and when the sun began to move, the shadow hand showed the time.

Despite the fact that the solar device was a very effective tool in ancient times, it had many drawbacks. They can only be used on sunny days. In addition, they could only show the time in a certain area.

People could also determine the time using sand, water and fire devices. Of course, any of these devices had relative accuracy, since many factors influenced them. For example, the accuracy of water clocks suffered due to atmospheric pressure or temperature. The measurement of time using the speed of the wick depended on the influx of air and the movement of the wind.

The most productive achievement of measuring time in antiquity was astronomical observation of the positions of the stars. The accuracy of time measurement is very high, so today such methods are very popular.

Not everyone could use the achievements of antiquity. Many lived in the countryside and had to tell the time without clocks or special facilities. They observed the surrounding nature, its phenomena and noticed that many actions are periodic. following life cycle animals and plants, you can find out what time it is with great accuracy.

How and how to measure time? The most ancient “clock” that never stopped or broke was the sun. Morning, evening, day - not very accurate measurements, but at first primitive man that was enough. Then people began to watch the sky more and found that after a certain time a bright star appears in the sky. These observations were made by the Egyptians, and they also named this starSirius . When Sirius appeared, the New Year was celebrated in Egypt. This is how the now well-known measure of time, the year, arose. It turned out that the interval between the appearances of Sirius consists of 365 days. As you can see, the calculations of the ancient Egyptians were quite accurate. After all, our year consists of 365 days. But a year is too long a measure of time. And in order to manage the economy: sowing, harvesting, preparing the harvest, smaller units of time were needed, and people again turned to the sky and the stars. This time the moon came to the rescue, or, in other words, the month. You have all watched the moon and you know that after a certain time it changes its shape: from a thin sickle to a bright round disk (full moon). The interval between two full moons is called a month. It turned out that the month consists of approximately 29 days. That's exactly how ancient world were able to tell the time.

And the seven-day week arose in Babylon thanks to those planets that appeared in the sky and were known to the Babylonians:

Saturday - the day of Saturn;

Sunday - the day of the sun;

Monday - the day of the moon;

Tuesday - the day of Mars;

Wednesday - the day of Mercury;

Thursday - the day of Jupiter;

Friday is the day of Venus.

If other planets of ours were known in Babylon solar system, perhaps our week would not consist of 7, but of 9, 10 or 8 days. The change of these luminaries during the month occurred approximately 4 times. So it turned out that there are 4 weeks in a month. So, the most difficult thing - to find the measurements of time - was already done in the ancient world. These measures are still in use today. It's just that they are called differently. In Russia, the names of the days of the week came from the ordinal number of the day in the week:

Monday - per week; starter week;

Tuesday - second day;

Wednesday - middle of the week

Thursday - fourth day;

Friday - fifth day;

Saturday , Sunday - these names came from the church dictionary.

It turns out that people borrowed all the main measures of time (year, month, week) from nature many years ago. Although these measures could not be measured exact time, but the main step was nevertheless taken.

Residents who lived in sunny countries measured time using . The passage of time was measured with a stick that was stuck in the ground - people looked at the length of the shadow and its direction. Thus, the sundial was born.

Of course, they were not very convenient to use, since they could only be used during the day and if the sun did not hide behind the clouds.

Therefore, the first water clock, which people were called "night". Such watches included a reservoir of water, which had to drain in a certain time. With water, the float descended and led to the movement of the arrow.

Also, people began to use hourglasses, the trick of which was that the sand had to be poured from one vessel to another in a certain time.

But mechanical watches proved to be the most convenient to use. They were driven by an elastic spring and two weights. Christiano Huygens (a scientist who lived between 1629 and 1695) created a clock with a regulating mechanism and a pendulum.

The action of these clocks is based on the fact that a pendulum with a certain mass oscillates in different directions at the same time.

Nowadays, wind-up watches are practically not used. Now electronic or quartz watches are in use. The small one serves as a power source for watches of this type. Such watches are much more durable and much more accurate than mechanical ones.

At present, almost every inhabitant of our planet uses mobile phone and clock on his screen.

The most accurate are atomic clocks.

Task: come up with your own design of any clock (wrist, wall, electronic, alarm clock).

Question. What did people take as the basis for counting different periods of time? How did they learn to count days, months, years?

Answer. People took the Moon and the Sun as the basis for calculating time intervals, the main thing in this orientation was the Sun. To be more precise, rotation around its axis and rotation around the Sun. A day is the time it takes for the Earth to completely rotate on its axis. A month is the time it takes for the moon to revolve around the earth. A year is the time it takes for the Earth to revolve around the Sun.

Question. How long is the day?

Answer. The day lasts 24 hours.

Question. Why are there 7 days in a week?

Answer. The full moon is not seen every day. First, a narrow crescent appears in the sky, then the Moon becomes wider, grows fatter day by day and after a while becomes completely round. And then, after a few days, it starts getting smaller and smaller, becoming a narrow sickle again. Such changes of the moon occur every four weeks or 29 and a half days. This is called the lunar month. It served as the basis for the creation of the calendar. Therefore, the crescent of the moon began to be called the "month".

Historical sources date the first mention of the seven-day week to the period of Ancient Babylon (about 2 thousand years BC), from there this tradition passed to the Jews, Greeks, Romans and, of course, to the Arabs. It is believed that India also adopted the 7-day period from Babylon.

For Jews and Christians, the Old Testament gives answers to these questions, from where it becomes clear that the seven-day time structure is established by God. Let me remind you: on the first day of creation, light was created, on the second - water and firmament, on the third - land, seas and flora, on the fourth - luminaries and stars, on the fifth - animal world, on the sixth - man was created and commanded to multiply, the seventh day is consecrated for rest.

The seven-day week turned out to be very viable, even the transition from the Julian calendar to the Gregorian did not change the sequence of days, the rhythm was not disturbed. There is also an astronomical explanation for the 7-day period. 7 days is approximately a quarter of a lunar month, while observing the phases of the moon was for the ancients the most accessible and convenient way to measure time. A more subtle explanation can be found in the correspondence of the seven visible planets to the days of the week, and it is this logical move that sheds light on the origin of the modern calendar names of the days of the week.

Question. Why is there 365 days in a normal year and 366 in a leap year?

Answer. The true year is 365 days 5 hours 46 minutes 48 seconds. Thus, in 4 years, one more day accumulates. It is in this year that February has 29 days and is called a leap year.

What is a day

Question. What was the first measure of time? How did the ancient peoples celebrate it?

Answer. The most ancient “clock”, which, moreover, never stopped or broke, turned out to be the Sun. Morning afternoon Evening Night. Not very accurate measurements, but at first this was enough for primitive man. People made notches on posts, notches on mammoth tusks. Others extruded circles on clay pots, or tied knots on leather straps. So the first records of the lived days appeared. The ancient Egyptians divided the night, and then the day into 12 parts - according to the number of constellations they read, which could be observed during the night.

Then people learned to determine the time more accurately: during the day - according to the Sun, and at night - according to the stars. People have noticed that the stars in the sky are moving slowly. All of them seem to be tied with invisible threads to a bright star, which is always in the same place. Perhaps that is why some nations call it the Nail of Heaven. We call this star the Polaris; it shows the direction to the north, to the North Pole. Not far from the North Star in the sky, you can always find seven stars arranged in the form of a ladle or saucepan with a long handle. This is the constellation Ursa Major. During the day, the Big Dipper goes around the Polar Star in a full circle, in the night half a circle. So it turns out that there is a real night clock with a star arrow in the sky.

Question. Try to explain why we do not notice the rotation of the Earth.

No wonder for a long time people believed that the Earth is flat, like a table or like a pancake, rests on three whales (or three elephants). With the development of science, people's ideas about the Earth have changed. Now we know that the Earth is involved in several movements at the same time.

Without noticing the rotation of the Earth, we observe and feel its consequences - the change of day and night. If the Earth did not rotate, then on the side that faces the light, there would always be day, and the opposite side would always be in darkness. We also do not notice the movement of the Earth around the Sun, but, nevertheless, we see and feel the change of seasons. The Earth revolves around the Sun in 365.25 days. This period of time is called a year.

Our planet is involved in several other types of motion: relative to the Milky Way. Milky Way moving relative to other galaxies. There is nothing immovable, unchanging, once and for all given in the Universe.

Question. Think about whether it is possible to organize the life of a family, a city, a state without knowing time. What happens if all the clocks suddenly disappear?

Answer. It is impossible to organize the life of a family, a city, a state without knowing the time. Time organizes people's lives; modes of work, study, and the armed forces are subordinate to it. The work of computers is tied to time. Time determines the work of transport and much, much more.

The task. Think about whether you can increase or decrease the length of the day. How is it defined?

Answer. It is not possible to increase or decrease the length of the day. It is equal to 24 hours and this is the time of a complete rotation of the Earth around its axis. Now a person is not able to slow down and speed up this rotation.

The task. Discuss why in different places on the Earth the length of the day is the same, but the length of daylight hours is different? What does it depend on?

Answer. The rotation of the Earth around its axis is a day and they are equal in all points of the globe. But the length of daylight hours depends on the height of the Sun above the horizon. And it is different in different parts of the world. That is why somewhere the daylight hours are longer, and somewhere shorter.

The task. Consider the drawing on p. 12. Think about where on Earth it is noon, midnight, morning, evening.

Answer. On Earth, noon in Africa, midnight in America, evening in Australia, morning in Western Europe.

How the years count

Question. What movement of the Earth is taken as the basis for counting years?

Answer. The calculation of years is based on the movement of the Sun around the Earth. One full rotation is equal to one year.

Question. Explain why we do not notice the movement of the Earth around the Sun.

Answer. Because it is impossible to notice the rotation of the Earth, being on its surface. Man is too small compared to the globe. In addition, we rotate with the Earth. Rotation can only be seen from the side.

The task. Think about whether the duration of winter is the same everywhere on Earth.

Answer. The duration of winter on Earth is not the same in different parts. This is due to the tilt of the Earth's axis and the distance from the equator. Due to this, the height of the Sun above the horizon is not the same. The farther from the equator, the lower the Sun above the horizon, so the winter in these places will be longer.

How months are counted

Question. Observing which cosmic bodies can one count days, weeks, months, years?

Answer. Watching the Moon, the Sun, you can count days, weeks, months, years.

Question. Why does the appearance of the moon in the sky change and repeat?

Answer. The Moon is a natural satellite of the Earth. During its movement, it occupies a different position relative to the Sun and the Earth. During its movement, it occupies a different position relative to the Sun and the Earth. Therefore, its appearance in the sky changes. The time of one revolution of the Moon around the Earth is another measure of time - a month.

Question. Why are there 12 months in a year?

Answer. 12 months in a year is equal to the number of revolutions of the Moon around the Earth during the year.

The task. Review the drawings. At the beginning or end of the month, does the student observe the moon?

Answer. the schoolboy has to observe the moon at the beginning of the month or at the new moon.

The task. Discuss what the images of the moon might have been on ancient objects that marked the weeks of the month.

Answer. In places of ancient settlements, objects with images of views of the moon with notches depicting months are often found. Different nations gave them their names. The ancients noted four types of the moon, which change during the month every seven days. Images could be as follows: a light circle - a full moon. Half of the circle - the direction depending on the moon's waxing or waning, dark circle - the moon is not in the sky.

What clock did man invent

Question. What is the hand in a sundial?

Answer. On a sundial, the arrow is the shadow of the sun. Ancient people measured time during the day with the help of a gnomon - a high vertical pole. During the day, the shadow from him slowly turns and its length changes. Over time, a dial was placed under the gnomon, on which the shadow from it indicated the time. Thus, the sundial was born.

Question. What time does the clock show at noon?

Answer. To determine the onset of noon, you need to take a twig 1 meter high and notice when it casts the shortest shadow. This will take place between 11 am and 1 pm. Perhaps the time of noon will not coincide with 12 o'clock on the dial.

Question. How to check the accuracy of your watch?

Answer. Radio time signals are given by special quartz clocks. They can get ahead or behind by only 7 seconds in 274 years. Even more accurate clocks, by which you can correct the course of all other clocks, are atomic clocks. They are kept at a constant temperature, and sometimes even placed underground, in special deep mines. Despite all possible precautions, even atomic clocks can be slightly faster or slower. Therefore, they are adjusted according to the most important natural clock - according to the star.

The task. Look at the drawings of the clock. Explain how they are set up. Which of them are convenient to use? What clock is shown in the center?

Answer. On the image:

Fire clock, time is determined as the candle burns out

Hourglass - as the sand pours out

Clock with a weight - the weight moves the hands on the dial

Water clock - the mechanism of the clock is powered by falling water

Mechanical watch - clock mechanism consists of gears

Electronic clock - based on semiconductors

Star clock - determines the time by the position of the stars

It is most convenient to use an electronic clock - they are the most accurate and reliable. The Kremlin chimes are depicted in the center.

The text of the work is placed without images and formulas.
Full version work is available in the "Files of work" tab in PDF format

Introduction.

How old are you? How many friends do you have? How many paws does a cat have?

To calculate all this, you need to know the numbers. Teachers and textbooks, parents and older friends help us with this. Meanwhile, earlier people couldn't count! It's hard to imagine, but it's a fact. And it became interesting to me, what did the ancient people think, because they did not know the numbers. How did people learn to write them down?

Research topic: "How did people learn to count?"

Target: understand how people learned to count.

Tasks:

Collect material about numbers and numbers, consider the history of the emergence of numbers.

What symbols are used to write a number.

Find out what numbers we use today.

See what role they play in our lives.

Ancient people obtained their food mainly by hunting. The whole tribe had to hunt for a large animal - a bison or an elk: you cannot cope with it alone. The leader of the raid was usually the oldest and most experienced hunter. In order for the prey not to leave, it had to be surrounded, well, at least like this: five people on the right, seven behind, four on the left. Here you can't do without an account! And the leader of the primitive tribe coped with this task. Even in those days when a person did not know such words as "five" or "seven", he could show the numbers on his fingers.

By the way, fingers played a significant role in the history of counting. Especially when people began to exchange objects of their labor with each other. So, for example, wanting to exchange a spear made by him with a stone tip for five skins for clothes, a person put his hand on the ground and showed that a skin should be placed against each finger of his hand. One five meant 5, two - 10. When the hands were not enough, the legs were also used. Two arms and one leg - 15, two arms and two legs - 20.

They often say: "I know like the back of my hand." Is it not from this distant time that this expression went, when to know that there were five fingers meant the same thing as to be able to count?

Fingers were the first images of numbers. It was very difficult to add and subtract. Bend your fingers - add, unbend - subtract. When people did not yet know what numbers were, both pebbles and sticks were used when counting. In the old days, if a poor peasant borrowed several sacks of grain from a rich neighbor, he would give out a stick with notches instead of a receipt - a tag. They made as many notches on a stick as there were bags taken. This wand was split: the debtor gave one half to a rich neighbor, and kept the other for himself, so that he would not later demand five bags instead of three. If they lent money to each other, they also marked it on a stick. In a word, in the old days the tag served as something like a notebook.

How people learned to write numbers. IN different countries and in different times it was done in different ways. These “numbers” are very different and sometimes even funny for different peoples. IN Ancient Egypt the numbers of the first ten were written down with the corresponding number of sticks. Instead of the number "3" - three sticks. But for dozens there is already a different sign - like a horseshoe.

The ancient Greeks, for example, had letters instead of numbers. Letters denoted numbers in ancient Russian books: “A” is one, “B” is two, “C” is three, etc.

The ancient Romans had other numbers. We still sometimes use Roman numerals. They can be seen both on the clock face and in the book, where the chapter number is indicated. If you look closely, Roman numerals look like fingers. One is one finger; two - two fingers; five is five with the thumb set aside; six is ​​five and one more finger.

The Maya Indians managed to write any number using only a dot, a line and a circle.

How did modern numbers come to us. The writing of Arabic numerals, which we use every day, consisted of segments of straight lines, where the number of angles corresponded to the size of the sign. Probably, one of the Arab mathematicians once proposed the idea - to connect numerical value figures with the number of angles in its writing.

Let's look at the Arabic numerals and see that

0 - a number without a single corner in the outline.

1 - contains one acute angle.

2 - contains two sharp corners.

3 - contains three sharp corners (correct, Arabic, the outline of the number is obtained by writing the number 3 when filling in the postal code on the envelope)

4 - contains 4 right angles (this is what explains the presence of a "tail" at the bottom of the number, which does not affect its recognition and identification in any way)

5 - contains 5 right angles (the purpose of the lower tail is the same as for number 4 - completion of the last corner)

6 - contains 6 right angles.

7 - contains 7 straight lines and sharp corners(the correct, Arabic, spelling of the number 7 differs from that shown in the figure by the presence of a hyphen that intersects at right angles the vertical line in the middle (remember how we write the number 7), which gives 4 right angles and 3 angles gives the upper broken line)

8 - contains 8 right angles.

9 - contains 9 right angles (this is what explains such an intricate lower tail at the nine, which had to complete 3 corners so that their total number became equal to 9.

The modern word "zero" appeared much later than "digit". It comes from the Latin word "nulla" - "none". The invention of zero is considered one of the most important mathematical discoveries. With the new way of writing numbers, the value of each written digit began to directly depend on the position, place in the number. With the help of ten digits, you can write down any, even the most big number, and it is immediately clear which figure means what.

The modern word "zero" appeared much later than "digit". It comes from the Latin word "nulla" - "none". The invention of zero is considered one of the most important mathematical discoveries. With the new way of writing numbers, the value of each written digit began to directly depend on the position, place in the number. With the help of ten digits, you can write down any, even the largest number, and it is immediately clear which number means what. Numbers and numbers in our life. The number of life is able to tell a person about what his life mission is. The number of the birthday is a constant companion of life. Fate each time presents new obstacles and difficulties. At such moments, the number of life helps to resist the blow and overcome obstacles without difficulty.

The number of life is a kind of key to the code of fate, which occupies an important place in the construction of important plans. The code of fate is able to prepare a person for the fact that more than once he will have to face "abrupt" turns. But the number of life exists so that this does not happen.

I was interested to know how my classmates feel about numbers. To do this, I conducted a survey among students in grade 5, and this is what I got.

The favorite number of the majority turned out to be 5.

Today, many people ascribe magical properties to numbers, associate them with various events that occur in life, and I decided to find out how my classmates relate to such numbers.

As you can see from the diagrams, for the most part, my classmates are not superstitious.

Well, at the end of my survey, I asked, perhaps, the most important question, for which I chose this topic.

To the question "Why do people need an account?" The guys responded like this:

This means that my classmates also often meet with numbers and understand that we cannot do without counting.

Conclusion.

Modern life cannot be imagined without numbers, they are around us, we live among them, we need them like the sun, air and water.

We use numbers day after day, year after year. They are with us at home and at school, in the classroom and after school.

For a conscious understanding of the world around you, mathematical knowledge about numbers is necessary, further development of mathematical thinking is necessary.

Theoretical knowledge can be deep and solid only if it is directly connected with the living activity of people.

Federal Agency for Education

Branch of the state educational

higher professional institution

"Glazovsky State Pedagogical Institute

named after V.G. Korolenko"

Izhevsk

ESSAY

From the history of the development of mathematical concepts

Completed by a student

4 courses GGPIP and DMD

checked

Izhevsk, 2010

The history of the development of mathematics is not only the history of the development of mathematical ideas, concepts and trends, but it is also the history of the relationship between mathematics and human activity, the socio-economic conditions of different eras.

The formation and development of mathematics as a science, the emergence of its new sections is closely related to the development of society's needs for measurements, control, especially in the fields of agriculture, industry and taxation. The first areas of application of mathematics were associated with contemplation of the stars and agriculture. The study of the starry sky made it possible to build trade sea routes, caravan roads to new areas and dramatically increase the effect of trade between states. The exchange of goods led to the exchange of cultural values, to the development of tolerance as a phenomenon underlying peaceful coexistence. different races and peoples. The concept of number has always been accompanied by non-numeric concepts. For example, one, two, many... These non-numerical concepts have always protected the realm of mathematics. Mathematics gave a finished look to all the sciences where it was applied.

§ 2. Development of counting activities

The most ancient mathematical activity was counting. The account was necessary to keep track of livestock and trade. Some primitive tribes counted the number of objects by comparing them with various parts of the body, mainly fingers and toes. The rock drawing, preserved to our times from the Stone Age, depicts the number 35 in the form of a series of 35 finger sticks lined up in a row. The first significant advances in arithmetic were the conceptualization of number and the invention of the four basic operations: addition, subtraction, multiplication, and division. The first achievements of geometry are associated with such simple concepts as a straight line and a circle. Further development of mathematics began around 3000 BC. thanks to the Babylonians and Egyptians.

The Greek number system was based on the use of the letters of the alphabet. The Attic system, which was in use from the 6th-3rd centuries. BC, used a vertical line to designate a unit, and to designate the numbers 5, 10, 100, 1000 and 10,000 the initial letters of their Greek names. The later Ionic number system used 24 letters of the Greek alphabet and three archaic letters to represent numbers. Multiples of 1000 to 9000 were denoted in the same way as the first nine integers from 1 to 9, but each letter was preceded by a vertical bar. Tens of thousands were denoted by the letter M (from the Greek myrioi - 10,000), after which the number by which ten thousand had to be multiplied was put.

The deductive character of Greek mathematics was fully developed by the time of Plato and Aristotle. The invention of deductive mathematics is usually attributed to Thales of Miletus (c. 640–546 BC), who, like many ancient Greek mathematicians of the classical period, was also a philosopher. It has been suggested that Thales used deduction to prove some results in geometry, although this is doubtful.

Another great Greek, whose name is associated with the development of mathematics, was Pythagoras (c. 585-500 BC). It is believed that he could become acquainted with Babylonian and Egyptian mathematics during his long wanderings. Pythagoras founded a movement that flourished in the period ca. 550–300 BC The Pythagoreans created pure mathematics in the form of number theory and geometry. They represented integers in the form of configurations of dots or pebbles, classifying these numbers in accordance with the shape of the emerging figures (“curly numbers”). The word "calculation" (calculation, calculation) originates from the Greek word meaning "pebble". Numbers 3, 6, 10, etc. the Pythagoreans called them triangular, since the corresponding number of pebbles can be arranged in the form of a triangle, the numbers 4, 9, 16, etc. - square, since the corresponding number of pebbles can be arranged in the form of a square, etc.

Some properties of integers arose from simple geometric configurations. For example, the Pythagoreans discovered that the sum of two consecutive triangular numbers is always equal to some square number. They discovered that if (in modern notation) n2 is a square number, then n2 + 2n +1 = (n + 1)2. A number equal to the sum of all its own divisors, except for this number itself, was called perfect by the Pythagoreans.

§3. Development of written numbering

From the mathematical documents of the East that have come down to us, we can conclude that in ancient Egypt the branches of mathematics associated with the solution of economic problems were strongly developed. The Rhinda Papyrus (c. 2000 BC) began with a promise to teach "the perfect and thorough investigation of all things, the understanding of their essence, the knowledge of all mysteries."

The Egyptians used two writing systems. One - hieroglyphic - is found on monuments and gravestones, each symbol depicts an object. In another system - hieratic - conventional signs were used, which originated from hieroglyphs as a result of simplifications and stylizations. It is this system that is more commonly found on papyri.

§4. How did you learn to measure different quantities

The Greeks, within one or two centuries, managed to master the mathematical heritage of their predecessors, but they were not satisfied with the assimilation of knowledge; the Greeks created abstract and deductive mathematics. They were, first of all, geometers, whose names and even writings have come down to us. These are Thales of Miletus, the school of Pythagoras, Hippocrates of Chio, Democritus, Eudoxus, Aristotle, Euclid, Archimedes, Apollonius.

The main merit of the Pythagoreans in the field of science is the significant development of mathematics, both in content and in form. In terms of content - the discovery of new mathematical facts. In form - the construction of geometry and arithmetic as theoretical, evidence-based sciences that study the properties of abstract concepts about numbers and geometric shapes.

The Pythagoreans developed and substantiated the planimetry of rectilinear figures: the doctrine of parallel lines, triangles, quadrangles, and regular polygons. The elementary theory of the circumference and the circle was developed.

The presence among the Pythagoreans of the doctrine of parallel lines suggests that they possessed the method of proving by contradiction and for the first time proved the theorem on the sum of the angles of a triangle. The pinnacle of the achievements of the Pythagoreans in planimetry is the proof of the Pythagorean theorem.

Mathematics developed mainly in the growing trading cities. The townspeople were interested in counting, arithmetic, calculations. Typical of this period is Johann Müller, the host mathematical figure 15th century. He translated Ptolemy, Heron, Archimedes. He put a lot of work into calculating trigonometric tables, compiled a table of sines with an interval of one minute. The values ​​of the sines were considered as segments representing the half-chords of the corresponding angles in the circle, so they depended on the length of the radius.

The development of analysis received a powerful impetus when Descartes' Geometry was written. It included in algebra the entire field of classical geometry. Descartes created analytic geometry. Fermat and Pascal became the founders of the mathematical theory of probability. The gradual formation of interest in problems related to probabilities occurred primarily under the influence of the insurance business.

In the 17th century a new period in the history of mathematics begins - the period of mathematics of variables. Its origin is connected, first of all, with the successes of astronomy and mechanics.

The first decisive step in the creation of the mathematics of variables was the appearance of Descartes' book "Geometry". The main merits of Descartes before mathematics are the introduction of a variable and the creation of analytic geometry. First of all, he was interested in the geometry of motion, and, having applied algebraic methods to the study of objects, he became the creator of analytic geometry.

Analytic geometry began with the introduction of a coordinate system. In honor of the creator, a rectangular coordinate system consisting of two axes intersecting at right angles, measurement scales entered on them and a reference point - the point of intersection of these axes - is called a coordinate system on a plane. Together with the third axis, it is a rectangular Cartesian coordinate system in space.

By the 60s of the XVII century. Numerous methods have been developed for calculating areas bounded by various curved lines. Only one push was needed to create a unified integral calculus from disparate methods.

Differential methods solved the main problem: knowing a curved line, find its tangents. Many practical problems led to the formulation of an inverse problem. In the process of solving the problem, it turned out that integration methods are applicable to it. Thus, a deep connection was established between differential and integral methods, which created the basis for a unified calculus. The earliest form of differential and integral calculus is the theory of fluxes constructed by Newton.

In the XVIII century. from mathematical analysis a number of important mathematical disciplines emerged: the theory of differential equations, the calculus of variations.

§five. Number systems, types of number systems

Notation- a symbolic method of writing numbers, representing numbers using written characters.

Notation:

gives representations of a set of numbers (integers or reals)

gives each number a unique representation (or at least a standard representation)

reflects the algebraic and arithmetic structure of numbers.

The most commonly used positional systems are:

1 - single (as a positional one it may not be considered; counting on the fingers, notches, nodules “for memory”, etc.);

2 - binary (in discrete mathematics, computer science, programming);

3 - ternary;

4 - quaternary;

5 - quinary;

8 - octal;

10 - decimal (used everywhere);

12 - duodecimal (counting in dozens);

16 - hexadecimal (used in programming, computer science, and also in fonts);

60 - sexagesimal (time units, measurement of angles and, in particular, coordinates, longitude and latitude).

The binary number system is a positional number system with base 2. In this number system, numbers are written using two symbols (1 and 0).

The hieroglyphic number system has base 10 and is not positional: to denote numbers 1, 10, 100, etc. it uses different characters, each character is repeated a certain number of times, and in order to read a number, you need to sum up the values ​​​​of all the characters included in its entry. Thus, their order does not matter, and they are written either horizontally or vertically.

The hieratic number system is also decimal, but special additional characters help to avoid the repetition adopted in the hieroglyphic system.

Mathematics of Babylon, like Egyptian, was brought to life by the needs of production activities, since problems related to the needs of irrigation, construction, economic accounting, property relations, and time calculation were solved. The surviving documents show that, based on the 60-decimal number system, the Babylonians could perform four arithmetic operations, there were tables square roots, cubes of cube roots, sums of squares and cubes, powers of a given number, the rules for summing progressions were known. Remarkable results have been obtained in the field of numerical algebra. Problem solving was carried out according to the plan, the problems were reduced to a single "normal" form and then solved according to general rules. There were problems that reduced to solving equations of the third degree and special types of equations of the fourth, fifth and sixth degrees.

The Babylonian number system is a combination of sexagesimal and decimal systems using the positional principle; it uses only two different characters: one indicates a unit, the second - the number 10; all numbers are written using these two symbols, taking into account the positional principle. In the most ancient texts (about 1700 BC) there is no symbol for zero; thus, the numerical value that was given to the symbol depended on the conditions of the problem, and the same symbol could mean 1, 60, 3600 or even 1/60, 1/3600

List of used literature

Binary number system. - Electronic access mode: http://ru.wikipedia.org/wiki/

Laptev B.L.. N.I. Lobachevsky and his geometry. -M.: Enlightenment, 1976.

Rybnikov K.A. History of mathematics. - M.: Nauka, 1994.

Samarsky A.A. Mathematical modeling. -M.: Nauka, 1986.

Stoll R.R. Set, Logic, Axiomatic Theory. -M.: Enlightenment, 1968.

Stroyk D.Ya. Brief essay on the history of mathematics.- M.: Nauka, Fizmatlit, 1990.

Tikhonov A.N., Kostomarov D.P. Stories about applied mathematics. -M.: Vita-Press, 1996.

Yushkevich A.P. Mathematics in its history. -M.: Nauka, 1996.

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