I wear a wristwatch all the time and feel very uncomfortable when I can’t see the time. I control when and where to come, what to do. How much time has already been spent on some process. Although there is such a saying that happy hours are not observed, even during rest, they are constantly with hours.
How did people in the old days get by without precise time control? But even then people could quite accurately determine the time, only with a small error.
Gnomon - the sun is shining, the time is known
Before mechanical watches became widespread, the time was determined using sundial... This device had three details: a gnomon, that is, an element that casts a shadow, a dial on which this shadow falls, and one more, conditional detail - the sun itself, which "winds" this very clock.
The dial has lines, and the gnomon has a size and shape, for the calculation of which geographic coordinates are used. That is, every sundial is created for a specific area. Making them is a painstaking and complex process that requires knowledge and skills. Therefore, such devices were not cheap.
In Russia, they acted easier: our ancestors simply dug a high pole into the ground, which cast a shadow. By observing the size of the shadow, it was possible to determine the time. Of course, this was not the ideal way. But comparing the length of the shadow, for example, at noon, in the evening or at dawn, as well as measuring it at different times of the year, our ancestors made up a fairly clear determinant of time.
Everything would be fine if the sun was constantly shining in Russia. Unfortunately, this is from the realm of fantasy - rains, clouds and other bad weather are very frequent here. If we add the peculiarity of the northern regions, where the sun does not rise too high above the horizon, which is why the shadow indicators of the gnomon are very long, it turns out that the sundial could not be considered as the only, accurate, year-round option.
We also have a huge sundial on the street in Belgorod. Once passing by, I compared them with my wristwatches - they definitely show the sun! That's it! literally up to a minute.
Clepsydras that steal water
The type of clock that does not depend on sunlight is water. They are called klepsydra. If we disassemble this word into parts, and it consists of klepto - to hide and hydor - water, then it is clear that in translation from Greek, but means nothing more than "water thief". The simplest clepsydra consists of two vessels installed at different levels. There is a hole in the upper one through which water drops drop by drop into the lower one. Time was determined by observing how the water level in the upper vessel decreases, or how it rises in the lower one. There is a version that the expression "the passage of time" originated from here.
Clepsydra
Since the pressure in the vessel affects the speed of water movement, the container began to be made in the form of a truncated cone. By its structure, the clepsydra has an advantage over the sundial, since the system of communicating vessels can be brought to perfection. They can be used at any time of the day, they determine the time more accurately.
Modern water clock.
But they can be used only when the water is in liquid state... Alas, in Russia there are often frosts, and she could simply freeze. Such constructions did not find wide distribution among the population, they were used mainly during church rituals and were called "hydrology".
Roosters, larks and flowers for the poor
Watches in Russia have long been a luxury item. Simple people used their own methods to avoid complex mechanisms. Our ancestors were observant, natural processes were not a secret for them.
For example, birds. It is clear that the alarm clock in ancient times (and in many villages to this day) was a rooster, which crowed three times a night: the first time after midnight, then at two in the morning, and the last time early in the morning, at about the beginning of the fifth. The oriole, the lark, the sparrow - these birds also woke up and began their chants at a certain time. It was only necessary to observe, listen to the advice of the elderly and remember the time.
As you know, larks start singing at 2 o'clock in the morning, willow at 3 o'clock, and sparrows fall asleep only at 6 o'clock in the morning. The main "clock" in the old days was the rooster. The first time the roosters crow at 1 am, the second time at 2 am, and the third time at 5 am.
The peasants watched the flowers, as they, turning to the sun, open and close at a strictly defined time. The life cycle of many plants and animals is related to the time of the day. Flowers of different plants open and close at different times, at a certain time. Most of the flowers bloom in the morning and close in the evening, but there are also those that close and open in the middle of the day or at night. Based on this principle, Karl Linnaeus invented and created a flower clock, which "worked" from three hours of the morning to midnight. Looking at them, it was possible to determine the time of the day with an accuracy of up to 30 minutes.
And, of course, the sun itself. The Slavs divided the day into day and night, focusing on the movement of the heavenly body. The middle of the day was noon when the sun was at its highest point. The later, the longer the shadows from objects.
In ancient times, there was no electricity, there was nothing to light the huts with. Yes, there were candles, but burning them constantly was economically unprofitable. Because as soon as the sun went down, darkness covered at home - you can go to bed with a clear conscience. In addition, our ancestors did not need a clear definition of time, in minutes, seconds. For what? Daybreak - you can go to the fields, work - until sunset, at dusk they return to the village. Shepherds, for example, measured the shadow of a tree with bast shoes worn on their feet. The shadow has reached seven bast shoes - you can collect the herd and lead home.
The process of replacing temporal concepts with spatial ones is interesting, for example: “Is this village far? - Yes, far away, two days of walking. The segment that could be traversed in one day was called the bottom.
Clear biological clock
Everyone knows about the biological clock today, this is the name of the internal circadian rhythm. It takes years for its formation, with its help all biological processes of the human body are formed. We are hungry, which means that it is time for lunch, to which we are accustomed. We want to sleep - you can look at the clock to make sure that it's already midnight (one in the morning, two, and so on), depending on the habit.
Artist K. Makovsky. Peasant lunch during the harvest.
Our ancestors lived according to a strict schedule. The peasant woman knew that she did not need to invite her husband to dinner. He came himself, or was already waiting for his wife in the field, because hard work and habit did their job, and the appetite was played very strongly.
The regularity, monotony characteristic of life in Russia, adherence to a certain routine, the absence of artificial lighting made orientation in time simple and partly arbitrary.
Flowers bloom and close at a specific time.
Watchmaking workshops began to develop in Russia in the 18th century. This event was marked by the opening of the Watch Dvor in Moscow. And today it is impossible to surprise even a kindergartner for hours - this subject has become so familiar, inexpensive and ubiquitous.
sources
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 as a whole: winter was over, which meant sowing had to be done. And as soon as there were signs of the first cold weather - it's time to collect.
It turns out that the timing 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 find account balances 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 overflow 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 reckoned the time by the time the new emperor ascended the throne. 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 the time, it was extremely inconvenient for their relationship. Moreover, it creates difficulties for modern historians. To understand the year of the definition of an event, you need to delve into the culture of the people and find out how their time was counted.
Due to the inconvenience of different reporting of years, it was necessary to create a single system that 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 this calculation. These were the countries professing Islam. The point of account of their reckoning 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 so they could tell what time it was. In this case, the errors were equal to a maximum of 10 minutes. Therefore, the first clocks to determine 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 served as an arrow. Its end pointed north, and when the sun began to move, the shadow hand indicated the time.
Despite the fact that the solar device has been very effective in ancient time, he had many shortcomings. They could only be used in sunny weather. In addition, they could only show the time in a certain area.
People could also tell time using sand, water and fire devices. Of course, any of these devices had relative accuracy, as they were influenced by many factors. For example, the accuracy of the water clock suffered from atmospheric pressure or temperature. The measurement of time with the speed of the wick depended on the flow of air and the movement of the wind.
The most productive achievement in measuring time in ancient times was astronomical observation of the position 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 rural areas, and they had to determine the time without clocks and special structures. They observed the surrounding nature, its phenomena and noticed that many actions are of a periodic nature. By following the life cycle of animals and plants, you can find out what time it is with great accuracy.
Federal Agency for Education
Branch of the state educational
higher professional institution
"Glazov State Pedagogical Institute
named after V.G. Korolenko "
izhevsk
ESSAY
From the history of development mathematical concepts
Completed by a student
4 courses of GSPIP and MDD
Checked
Izhevsk, 2010
The history of the development of mathematics is not only the history of the development of mathematical ideas, concepts and directions, but it is also the history of the relationship of mathematics with 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 agrarian, industrial and taxation. The first fields of application of mathematics were associated with gazing at the stars and agriculture. The study of the starry sky made it possible to lay trade sea routes, caravan roads to new regions 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 fenced off the field of mathematics. Mathematics gave a finished look to all sciences where it was applied.
§ 2. Development of counting activities
The oldest mathematical activity was counting. An account was needed to keep track of livestock and trade. Some primitive tribes counted the number of objects by comparing them with different parts of the body, mainly the fingers and toes. The rock drawing, which has survived to our times from the Stone Age, depicts the number 35 in the form of a series of 35 sticks-fingers lined up in a row. The first significant advances in arithmetic were the conceptualization of number and the invention of four basic operations: addition, subtraction, multiplication and division. The first advances in geometry were associated with such simple concepts as line and 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 letters of the alphabet. The Attic system, which was in use from the 6th – 3rd centuries. BC, used the vertical bar to denote the unit, and the initial letters of their Greek names to denote the numbers 5, 10, 100, 1000 and 10,000. In the later Ionic number system, 24 letters of the Greek alphabet and three archaic letters were used to denote numbers. Multiples of 1000 to 9000 were denoted in the same way as the first nine integers from 1 to 9, with a vertical bar in front of each letter. Tens of thousands were designated by the letter M (from Greek myrioi - 10,000), after which the number by which ten thousand had to be multiplied was put
The deductive nature of Greek mathematics was fully formed 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, with whose name the development of mathematics is associated, was Pythagoras (c. 585-500 BC). It is believed that he may have become acquainted with Babylonian and Egyptian mathematics during his long travels. Pythagoras founded the movement, which 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 "stone". Numbers 3, 6, 10, etc. The Pythagoreans called it 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 a square, etc.
Some of the properties of integers arose from simple geometric configurations. For example, the Pythagoreans found 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 \u003d (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
Of those that have come down to us mathematical papers East, we can conclude that in ancient Egypt, the branches of mathematics related to the solution of economic problems were highly developed. The Rynda Papyrus (c. 2000 BC) began with a promise to teach "a perfect and thorough investigation of all things, an understanding of their essence, the knowledge of all secrets."
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 most often found on papyri.
§four. How did you learn to measure different quantities
For one or two centuries the Greeks managed to master the mathematical heritage of their predecessors, but they were not content with assimilating knowledge; the Greeks created abstract and deductive mathematics. They were, above all, geometers, whose names and even compositions have come down to us. These are Thales of Miletus, the school of Pythagoras, Hippocrates of Chios, Democritus, Eudoxus, Aristotle, Euclid, Archimedes, Apolonius.
The main merit of the Pythagoreans in the field of science is the significant development of mathematics, both in content and in form. Content - the discovery of new mathematical facts. In form - the construction of geometry and arithmetic as theoretical, demonstrative sciences that study the properties of abstract concepts about numbers and geometric forms.
The Pythagoreans developed and substantiated the planimetry of rectilinear figures: the doctrine of parallel lines, triangles, quadrangles, regular polygons. The elementary theory of the circle and circle was developed.
The fact that the Pythagoreans have the doctrine of parallel lines indicates that they owned the method of proof by contradiction and were the first to prove the theorem on the sum of the angles of a triangle. The pinnacle of the Pythagorean achievements 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, a leading mathematical figure of the 15th century. He translated Ptolemy, Heron, Archimedes. He put a lot of work on calculating trigonometric tables, compiled a table of sines with an interval of one minute. The sine values \u200b\u200bwere considered as line 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. She included in algebra the entire area of \u200b\u200bclassical 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 took place primarily under the influence of the insurance business.
In the XVII century. a new period in the history of mathematics begins - the period of mathematics of variables. Its origin is associated primarily with the successes of astronomy and mechanics.
The first decisive step in the creation of mathematics of variables was the appearance of Descartes' book "Geometry". The main merits of Descartes before mathematics are the introduction to him variable and the creation of analytical geometry. First of all, he was interested in the geometry of motion, and, applying algebraic methods to the study of objects, he became the creator of analytic geometry.
Analytical 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, measuring scales entered on them and the origin - 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 to calculate the areas enclosed by various curved lines. Only one impetus was needed to create a single integral calculus out of disparate methods.
Differential methods solved the main problem: knowing a curved line, find its tangents. Many practical tasks led to the formulation of an inverse problem. In the process of solving the problem, it turned out that integration methods were applicable to it. This established a deep connection 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 fluxia, built by Newton.
In the 18th century. a number of important mathematical disciplines emerged from mathematical analysis: the theory of differential equations, calculus of variations.
§five. Number systems, types of number systems
Notation - a symbolic method of writing numbers, representing numbers using written signs.
Notation:
gives representations of a set of numbers (integers or real)
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 positional may not be considered; counting on fingers, notches, nodules "for memory", etc.);
2 - binary (in discrete mathematics, computer science, programming);
3 - ternary;
4 - quaternary;
5 - fivefold;
8 - octal;
10 - decimal (used everywhere);
12 - duodecimal (counting in dozens);
16 - hexadecimal (used in programming, computer science, as well as in fonts);
60 - sexagesimal (units of measurement of time, measurement of angles and, in particular, coordinates, longitude and latitude).
Binary number system is a positional number system with base 2. In this number system, numbers are written using two characters (1 and 0).
The hieroglyphic number system has a base 10 and is not positional: to denote the numbers 1, 10, 100, etc. it uses different symbols, each symbol is repeated a certain number of times, and in order to read a number, you need to sum the values \u200b\u200bof all the symbols included in its record. Thus, their order is irrelevant and they are written either horizontally or vertically.
The hieratic number system is also decimal, but special additional characters help avoid the repetition adopted in the hieroglyphic system.
The mathematics of Babylon, like the Egyptian, was brought to life by the needs of industrial activity, since problems related to the needs of irrigation, construction, economic accounting, property relations, and the calculation of time were solved. The surviving documents show that, based on the 60-ary number system, the Babylonians could perform four arithmetic operations, there were tables of square roots, cubes of cube roots, sums of squares and cubes, degrees this number, the rules for summing progressions were known. Remarkable results have been obtained in the field of numerical algebra. The solution of the problems was carried out according to the plan, the problems were reduced to a single "normal" type and then solved according to general rules. There were problems that reduce 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 the sexagesimal and decimal systems using the positional principle; it uses only two different symbols: one denotes one, the second - the number 10; all numbers are written using these two symbols, taking into account the positional principle. In the earliest texts (about 1700 BC) there is no symbol for zero; thus, the numerical value assigned to the symbol depended on the conditions of the problem, and the same symbol could denote 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 BL .. NI Lobachevsky and his geometry. -M .: Education, 1976.
Rybnikov K.A .. History of mathematics.- Moscow: Nauka, 1994.
Samarskiy A.A. Mathematical modeling. -M .: Nauka, 1986.
Stoll RR .. Set, Logic, Axiomatic theory. -M .: Education, 1968.
D.Ya. Stroyk Brief outline History of Mathematics), Moscow: Nauka, Fizmatlit, 1990.
Tikhonov AN, Kostomarov DP Stories about applied mathematics. -M .: Vita-Press, 1996.
Yushkevich A.P. Mathematics in its history. -M .: Nauka, 1996.
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Throughout life, each person always learns something, and the knowledge gained after a while seems so natural that it is perceived as a familiar fact. The thought does not even creep into my head: how did it all begin? How did people learn to count and How long ago did society come to the understanding that practically everything in the world obeys numbers?
How a man learned to count time
It's in modern world 365 days a year, 30 days a month and 24 hours a day are a natural fact. Previously, when there was no knowledge about the amount of time, a person was content with methods invented on his own, and the sun was the means for this. A dial with markings and a pole were installed on some surface, the shadow of which moved in a circle. Dependence on weather conditions was a significant drawback of such a device: even the rains did not make it possible to determine the time. The analogue of this design in the modern world is a watch that has firmly conquered its niche and has become an irreplaceable item in human life.
Determination of time by stars, water and fire
Stars - a symbol of romance and dreams of something distant and beautiful, also served as a kind of time marker in the night period. For this, maps of the starry sky were invented, the measurement of which was carried out using a transit instrument.
In addition to the sidereal clock and sundial, popular among almost all peoples and differing only in design, water exhibits were used quite massively, representing a cylindrical container, from which water dripped dropwise. It was by the amount of drained water that people measured time. Such watches were popular in Egypt, Rome, Babylon. How did a person learn to count time in Asian countries? Here in devices water type the opposite principle was used: the floating vessel was filled with water coming through a small hole.
Trying to bring into his life not only water, but also the fire element, man also invented a fire clock, which originated in China and gained popularity throughout Europe over time. The basis of these time-determining devices was a combustible material (in the form of a stick or a spiral) and metal balls attached to it, falling when a certain fraction of the material was burned. In Europe, candle clocks were mainly used, preferring them to lamp and wick clocks. The time for them was determined by the amount of burned wax. Such clocks were especially common in churches and monasteries.
Hourglass is a rare pride of our time
Of course, the most popular was the hourglass, which is still actively used to fulfill its main function, as well as as a decorative item. The accuracy of the calculated time in devices of this type depends on the quality of the sand, which determines the uniformity of its flowability.
The history of the emergence of counting science
Understanding time in its quantitative indicator was the determining factor for the knowledge of numbers and the ability to count. Moreover, the history of the emergence of the account is so old that it looks more like a fairy tale. How did people learn to count? Many centuries ago, humanity lived in tribes, led a herd lifestyle, dressed in the skins of killed animals and ate what its representatives could get themselves.
Accordingly, the simplest tools: sticks and stones were also the improvised tools for survival and food extraction. Perhaps the constant dangers and the need for food became the main impetus for the need for counting, which in our time is not only perceived as a natural fact, but is also facilitated by modern computing technology.
One, two and many
The first concepts to denote quantity and to explain how people learned to count were "one" and "many." "One" - an object or individual separately distinguished according to certain criteria: the leader of the pack, grain in an ear, etc. “Many” is the total mass in which this object is located.
The emergence of the number "two", meaning "pair": eyes, ears, paws, wings, hands, explains how man learned to count in the days of non-existent numbers. Talking about the two ducks caught, the hunter pointed to his eyes, thus explaining the amount of the trophy.
In the counting science of the ancient world, gradual progress was observed: the numbers "one", "two" and "many" were already known. Soon, a person came to the conclusion that he began to single out three, four, five or more objects from the total mass, and this number did not have a name, but was explained as the sum of the numbers known at that time: "2" and "1". For example: "3" is "1" and "2" in total; “4” is the sum of “2” and “2”; and "5" is "2", "2" and "1" combined. In Tibet, the number "2" is the wings, in India - the eyes, for some peoples "1" is the moon, "5" is the hand. That is, each number had at first a visual-associative perception, before it was given a name.
Account as a vital necessity
How did people learn to count, if the ability to this "art" at every stage of human development became simply a necessity? In the process of hunting, when the animal was surrounded, the senior hunter needed to place people correctly in order to take the animal in the ring. To do this, he showed on his fingers where and how many people needed to take the right positions.
In trading, the mathematics of the fingers (and feet, if the cost was high) were also used to indicate the price. For example, when exchanging a spear made for animal skins, the seller put his hand on the ground and showed that a skin should be placed in front of each finger. By the way, bending the fingers meant addition, and their extension meant subtraction. These were the first mathematical examples to explain how people learned to count in the distant past.
Counting science in different countries
Many countries that have preserved in their history models of how people learned to count are still using the legacy of the past: in Japan and China, household items are considered fives and tens; in England and France - twenty.
The ancient Egyptians, who depicted any action in the form of a picture on papyrus, did not write down numbers as such. Residents Ancient Rome numbers were indicated by dashes. So "I" is one, "V" is an image of a hand with a finger sticking out to the side, or rather five fingers in a simplified version, "X" is two fingers folded together.
With the advent of letters, the alphabet began to be used to denote numbers. For example: B-
With the advent of letters, the alphabet began to be used to denote numbers. For example: V is "2", G is "3", D is "4", E is "5". To distinguish letters and numbers, an icon called "titlo" was placed above the latter. The method was not very convenient, since it did not allow writing large numbers. Over time, people began to separate numbers from letters and perceive them separately, regardless of objects.
Modern ones, which are widely used today everywhere, were invented in India, and found their application in our country in the 18th century. Roman numbers have not lost their popularity, they are still found on the dials of watches, and are used to indicate centuries and chapters in books.
Distinguished by the way of counting Ancient Babylon, in which, 6 thousand years before our era, mathematical accounting of business transactions was already conducted. Records of this kind were depicted as pictures (hieroglyphs) in the form of narrow horizontal and vertical wedges, hence the name "cuneiform".
One was designated by one wedge, two by two, and so on. The number "10" stood out with a wide wedge and had a special name. The mathematician of Babylon experienced its heyday during the reign. Written sources from that time period reveal evidence of how people learned to write and count long before our times. These are records of complex computational operations, as well as the solution of quadratic and cubic equations.
How to learn to count in your head
If such complex actions were within the power of our ancestors, then for the modern generation, mathematical calculation, improved by time and many great minds, should not be particularly difficult. True, the presence of computing technology capable of performing digital actions instead of a person greatly facilitates the mental work of the latter. Therefore, verbal counting, which helps to develop memory and train skills, should be owned by everyone. Training in this type of mental activity will be successful if:
- abilities that, together with mental concentration, help to focus attention on the task at hand and keep complex numbers in memory;
- knowledge of the formulas that determine the ease of computational operations;
- practice, which, along with constant training, allows you to develop and improve skills.
Examples of uncomplicated mental counting
Multiplication by 4
An easy way, in which the number needs to be multiplied by 2, and the result obtained is doubled again. For instance:
35 * 4 = 35* 2 = 70 * 2 = 140
Multiplication by 11
The digits of a two-digit number, multiplied by 11, need to be moved apart.
For instance:
48 * 11 \u003d 4 and 8 * 11
Then you need to add the digits of the number, in this case 4 and 8, and the result will be the answer. It is important to remember that if, when summing, the result is a two-digit number, then you need to leave only units, and add 1 to tens.
4 (12) 8 \u003d 5 2 8 \u003d 528. That is, from the result obtained, 12 were left with units - this is 2, and 1 was added to the ten.
Division by 5
To make this action easy, you need to double the number and move the comma one digit back.
For instance:
125/5 \u003d 125 * 2 \u003d 250 (comma offset) \u003d 25
Division by 50
In this case, the pattern is similar: the number is multiplied by 2 and divided by 100.
600/50 = 600 * 2 / 100 = 12
Division by 25
The number is multiplied by 4 and divided by 100.
700/ 25 = 700*4 / 100 = 28
Addition and subtraction of natural numbers
When adding, you should know such a trick that if one of the terms is increased by a certain number (to facilitate the calculation), then the same number must be subtracted from the result.
For instance:
787 + 193 \u003d (787 + 193+ 7 (to round 193 to 200)) - 7 \u003d (787 + 200) - 7 \u003d 980
How and with what to measure time? The most ancient "clock" that never stopped or broke was the sun. Morning, evening, day are not very accurate measurements, but at first this was enough for primitive man. Then people began to observe 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 is 365 days. As you can see, the calculations of the ancient Egyptians were quite accurate. After all, our year also consists of 365 days. But a year is too long a measure of time. And in order to manage the economy: sowing, collecting, 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. All of you have 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 was called the month. It turned out that the month consists of approximately 29 days. This is how exactly the ancient world knew how to determine 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 - day of the sun;
monday - the day of the moon;
tuesday - Mars day;
wednesday - the day of Mercury;
thursday - day of Jupiter;
friday - the day of Venus.
If other planets of ours were known in Babylon Solar systemperhaps our week would not be 7, but 9, 10 or 8 days. The change of these luminaries occurred about 4 times during the month. So it turned out that there are 4 weeks in a month. So, the most difficult thing - to find the measures of time - was already done in the Ancient World. These measures are used to this day. Only now they are called differently. In Russia, the names of the days of the week come from the ordinal number of the day in the week:
monday - by week; beginner week;
tuesday - second day;
wednesday - the 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 measurements could not measure the exact time, the main step was nevertheless taken.
Residents who lived in sunny countries measured their 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 appeared.
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 a watch included a reservoir of water, which had to drain out for a certain time. The float descended with water and caused the arrow to move.
Also, people began to use an hourglass, the focus of which was that the sand had to be poured from one vessel to another within a certain time.
But the most convenient to use were mechanical watches. They were set in motion by means of 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 this clock is based on the fact that a pendulum with a certain mass oscillates in different directions at the same time
In our time, watches with a winding drive are practically not used. Electronic or quartz watches are now in use. The small one serves as a power source for this type of watch. Such watches are much more durable and much more accurate than mechanical ones.
At present, almost every inhabitant of our planet uses a mobile phone and a clock on its screen.
The most accurate are atomic clocks.
Assignment: come up with your own design of any clock (wrist, wall, electronic, alarm clock).