How ancient people counted bananas. Research work. How people learned to count. Presentation on the topic
How did they think in ancient times? How did they count in the old days?
For thousands of years, peoples have created legends and myths, reflecting their dreams and aspirations in them. Not being able to fly like birds or run faster than a deer, people came up with fairy tales about flying carpets or running boots. Suffering from hunger, they dreamed of a self-assembled tablecloth. But most of all they wanted to make their hard work easier. This is how tales arose about Emel and his miracle stove, Aladdin’s lamp, about wonderful mechanical and magical helpers and many others.
But while poets were writing poems, and writers were writing novels, scientists were taking the first steps to create automata. Even in ancient times, machines were invented that dispensed “holy” water in churches when a coin was dropped into them. Other machines opened doors when the priest approached and performed other “miracles” that made the people tremble before the omnipotence of the gods. Greek craftsmen built quite complex mechanical toys, including a mechanical theater in which entire performances were performed. These wonderful mechanisms were rare; they were not widely used, because the bulk of the population was uneducated. However, life forced people to learn to count and understand mechanisms.
At first, people counted “in their heads”, then they began to use improvised means - bone, clay and wooden beads, even their own fingers helped people.
The most ancient counting devices did not appear immediately. At first, the need for counting was small, and people had enough of their own fingers and the fingers of their neighbors in order to count military booty, the number of hunting trophies, knives, spears, warriors, etc. In ancient times, writing was poorly developed, and every person needed to count, so they had to use their own fingers, notches on bones, pebbles, beads and other small objects to count. But when people began to cultivate the land and domesticated some animals, they needed many more items for counting and the ability to perform operations with numbers.
To successfully engage in agriculture, arithmetic knowledge was necessary. Without counting days, it was difficult to determine when to sow fields, when to start watering, when to expect offspring from animals. It was necessary to know how many sheep were in the herd, how many bags of grain were put in the barns, etc.
Several decades ago, archaeological scientists discovered a camp of ancient people. In it they found a wolf bone, on which 30 thousand years ago some ancient hunter made fifty-five notches. It is clear that while making these notches, he was counting on his fingers. The pattern on the bone consisted of eleven groups, with five notches in each. At the same time, he separated the first five groups from the rest with a long line. The oldest artifact of this kind is the “Ishango bone”, found in the Congo (about twenty thousand years old). This is a baboon shin bone covered with serifs.
The word “tag” is still preserved in the Russian language. Now this is the name given to a tablet with a number or inscription, which is tied to sacks of goods, boxes, bales, etc. But two or three hundred years ago this word meant something completely different. This was the name given to pieces of wood on which the amount of debt or tax was marked with notches. The notched tag was split in half, after which one half remained with the debtor, and the other with the lender or tax collector. When calculating, the halves were added together, and this made it possible to determine the amount of debt or tax without disputes or complex calculations.
Ancient people invented the so-called “finger counting” - when not only numbers up to several hundred were depicted on the fingers, but even arithmetic operations were performed using the fingers (in Russian, the word “five” resembles “papal” - a part of the hand, a derivative of it - “wrist” - often used even now). The ancient Egyptians believed that in the afterlife, the soul of the deceased was tested by counting on their fingers. And in one of the ancient Greek comedies, the hero says that he prefers to calculate the taxes due on his fingers. Ancient people also learned to multiply single-digit numbers from 6 to 9 on their fingers.
In Rus', this method of counting on fingers was widespread: mentally number the fingers on both hands. Little finger - 6, ring finger - 7, middle finger - 8, index finger - 9, thumb - 10. Let's say you want to know how much 8 x 7 is. Connect the middle finger of your left hand (8) with the ring finger of your right hand (7). Now count. The two connected fingers plus those below them indicate the number of tens in the work. In this case - 5. Multiply the number of fingers above one of the closed fingers by the number of fingers above the other closed finger. In our case, 2 x 3 = 6. This is the number of units in the desired product. We add the tens with the ones, and the answer is ready - 56. Check the other options, and you will see that this old Russian method does not fail.
A complete description of finger counting was compiled by the Irish monk Bede the Venerable, who lived in the 7th - 8th centuries AD. He described in detail how to represent various numbers on the fingers, up to a million. In some places, finger counting has survived even today. For example, at the world's largest Chicago grain exchange, brokers on their fingers, without uttering a single word, report offers, requests, and prices for goods. And Chinese merchants bargained by holding each other's hands and indicating the price by pressing certain knuckles. Is this where the words “shake hands”, which once meant concluding a trade deal, came from?
With the advent of the first states of Ancient Egypt, Mesopotamia, China, Ancient Rome, and the states of America, it was necessary to perform calculations with very large numbers - after all, it was necessary to calculate taxes, receipts of military booty in the treasury, tribute from conquered states, and calculate the construction of roads and temples. Merchants kept records of goods, profits received, etc. In those days, there was even a government position for those who carried out calculations - a scribe. The larger the numbers and more complex the calculations, the greater the chances of confusion and mistakes. And the most complex calculations were required to be carried out first by priests, and then by scientists for astronomical calculations - the movement of the moon, stars, sun on which agriculture, harvest and the well-being of the entire state depended!
How were ancient engineers, mathematicians and astronomers able to create machines and make calculations that are considered complex even today?
Counting devices.
In ancient states, scribes - the people who carried out calculations - were entrusted with a very difficult task - they had to keep records of government revenues and expenses, and these were always very large numbers that were difficult to calculate in the mind. And here the ancient people showed amazing ingenuity - they created hand-held devices for counting:
Skoda Lyubov Viktorovna
In the habitats of primitive man, archaeologists find objects with knocked out dots, scratched lines, and deep notches. These finds suggest that already in the Stone Age people were able not only to count, but also to record the results of their calculations.
As society developed, counting methods also improved. After all, such primitive techniques as notches on a stick, knots on a rope or pebbles folded into piles could not satisfy the needs of trade and production.
Approximately 3,000 years BC, a major discovery was made: people invented special signs to designate a certain number of objects. For example, the Egyptians denoted ten with the sign ∩ , a hundred − ⊂ . So, the number 123 was written as follows:
⊂∩∩||| .
In Ancient Rome, numbers were written using the following numbers:
I− one,
V− five,
X− ten,
L− fifty,
C− one hundred,
D− five hundred,
M− thousand.
Roman number system is based on the following principle: if, when reading from left to right, a smaller digit comes after a larger one, then it is added to the larger one: VI = 6, XXXII = 32; if a smaller number comes before a larger one, then it is subtracted from the larger one: IV = 4, VL = 45.
In the Roman number system, for example, the number 14 is written like this: XIV. Here, the number I stands between the larger numbers X and V. In such cases, the number I is subtracted from the number to the right of it (in our example, this is the number V).
The year in which the Great Patriotic War ended with the victory of our people can be written using Roman numerals as follows: MCMXLV. This system has survived to this day. You can often find entries using Roman numerals, for example: XXI century, chapter VI. They can also be seen on watch dials and on architectural monuments.
You've probably already noticed that even reading a number written in Roman numerals is not easy. It is even more difficult to perform arithmetic operations in such a notation. In addition, if you need to write down fairly large numbers (million, billion, etc.), then you need to come up with new numbers. Otherwise, the number will be too long. For example, if you use only the Roman numeral M to write the number 1,000,000, then the record will consist of a thousand such characters. All these shortcomings significantly reduce the possibility of using the Roman number system.
In Ancient Rus' they did not invent special icons to indicate numbers. They were obtained using letters of the alphabet. A wavy line was placed above the letter - the title.
For example, the number 241 was written like this:
The greatest achievement of mankind is the invention decimal positional number system. With this system, arbitrarily large numbers can be written using only ten different digits. This is possible because the same number has different meanings depending on its positions among.
Numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are called Arabic. However, the Arabs only spread the decimal positional system invented by the Hindus.
Some tribes and peoples used other positional number systems. For example, the Mayan Indians used the decimal system, and the ancient Sumerians used the sexagesimal system.
Traces of the base-20 system can be found in some European languages. So, the French say “four times twenty” instead of “eighty” ( quatre−uingts ). The division of one hour into 60 minutes, and one minute into 60 seconds is an example of a clear legacy of the sexagesimal system.
Counting with ten fingers led to the emergence of the decimal system. The total number of fingers and toes was the basis for the creation of the base-20 system. The duodecimal system also has a “finger” origin: try using your thumb to count the phalanges on other fingers of the same hand, the result will be the number 12 (Fig. 2). This is how the account arose by the dozens.
And today in Europe they sell dozens of handkerchiefs, buttons, and chicken eggs. The number of items in cutlery and sets (forks, knives, spoons, plates, cups, glasses, etc.) is usually 6 (half a dozen), 12, 24, etc.
There are other positional number systems. Thus, the structure and operation of a computer is based on a binary number system that uses only two digits - 0 and 1.
1 of 20
Presentation on the topic:
Slide no. 1
Slide description:
Slide no. 2
Slide description:
Slide no. 3
Slide description:
Primitive peoples believe that the first concepts of mathematics were “less,” “more,” and “the same.” If one tribe exchanged caught fish for stone knives made by people of another tribe, there was no need to count how many fish and how many knives they brought. It was enough to place a knife next to each fish for the exchange between the tribes to take place. Until recently, there were tribes whose language had the names of only two numbers: one and two. The natives thought like this: 1 - “urapun” 2 - “okosa” 3 - “okosa - urapun” 4 - “okosa - okosa” 5 - “okosa - okosa - urapun”. . . . .All other numbers are “MUCH”! It can be seen that people have mastered only a small number of integers.
Slide no. 4
Slide description:
Many Russian proverbs say that this was also the case with our ancestors: “Seven nannies have a child without an eye” “Seven troubles - one answer” “Seven do not expect one thing” “Measure seven times, cut once” The number is used in the sense of Natives New Guinea people bend their fingers one after another, saying “be-be-be...”. Having counted to FIVE, he says “ibon-be” (HAND). Then they bend the fingers of the other hand “be - be..” until they reach “ibon - ali” (TWO HANDS). For further counting, use your toes, and then….the hands and feet of someone else!
Slide no. 5
Slide description:
People gradually got used to placing objects in stable groups of two, ten or twelve when counting. But the numbers did not yet have separate names. Among the natives of Florida, the word “na-kua” meant 10 eggs, “na-banara” - 10 baskets, but the word “na,” which seemed to correspond to the number 10, was not used separately. Numbers begin receive names However, among most peoples, numbers, which were considered “money” (and cattle were mainly used as money), gradually replaced all others. They became those universal numbers that made it possible to count any objects.
Slide no. 6
Slide description:
Thus, numbers less than 10, as well as ten, one hundred, and a thousand received individual names. Operations on numbers People dealt with the operations of addition and subtraction long before numbers received names. When several groups of root gatherers or fishermen put their catch in one place, they performed an addition operation. People became familiar with the operation of multiplication when they began to sow grain and saw that the harvest was several times greater than the number of seeds sown. They said: they reaped the harvest “twenty times”, that is, they reaped twenty times more than they sowed. Finally, when the harvested animal meat or collected nuts were divided equally among all the "mouths", the operation of division was performed.
Slide no. 7
Slide description:
Ancient GreeceIn the middle of the 5th century. BC. In Asia Minor, where there were ancient Greek colonies, a new type of number system appeared - alphabetical numbering. It is usually called Ionian. In this system, numbers were designated using the alphabet, over which dashes were placed. The first nine letters denoted the numbers 1 to 9, the next nine 10, 20...90 and the next nine the numbers 100, 200...900. This could be used to represent any number up to 999.
Slide no. 8
Slide description:
For thousands, the first nine letters were used again, but with a slash at the bottom left. For the number 10,000, the sign M was used; this number was called MYRIAD. Above the sign was a number indicating the number of myriads. So it was possible to designate all numbers up to a myriad of myriads, i.e. 108. The great mathematician, mechanic and engineer of antiquity ARCHIMEDES (III century BC) devoted an entire work to giving a general method for naming arbitrarily large numbers.
Slide no. 9
Slide description:
Often in fairy tales there is an “unsolvable” problem: counting how many stars are in the sky, drops in the sea or how many grains of sand are on the ground. Archimedes showed that such problems can be solved. He called his work “Calculus of Sand” (“Psammit”). To solve the problem, Archimedes combines all numbers less than a myriad of myriads into the first and calls them the first numbers. The second numbers are from 108 to 1016...And then you can increase the ranks. Archimedes' method is close to the positional one, BUT it took about 1000 more years before humanity managed to create a decimal positional number system.
Slide no. 10
Slide description:
NUMBERS IN ANCIENT ROME The rest of the numbers are written using these symbols using addition and subtraction. The number 444 will be written in the Roman system as follows. This form of notation is less convenient than the one we use. Writing numbers turns out to be much longer. There is another existing flaw in the Roman system: it does not provide a way to write arbitrarily large numbers. In the Roman system there are special signs for:I - 1 VI - 6II - 2VII - 7III - 3VIII - 8IV - 4IX - 9V - 5X - 10 L - 50D - 500C - 100M -1000
Slide no. 11
Slide description:
Sumerian cuneiform So a farmer brought an onion he had grown to a tax collector in a village in the Sumerian countries. “Sum!” said the collector, because “sum” meant “onion” in Sumerian, and he drew a bunch of onions on a damp clay tablet that he held in his hand. Sumerian accountants spent years drawing pictures of fish and birds, livestock and plants. Clear, smooth lines required a lot of work, and still they did not retain their shape well. Then they began to draw all the signs on clay so that they turned out to be on their side. Why did this happen? The fact is that they first wrote on clay in columns from top to bottom and each subsequent column began to the left of the previous one. But at the same time, they smeared with their hands what was written before. Therefore, they began to turn the tile a quarter turn and began to write the same characters in lines, from left to right (and each subsequent line began lower than the previous one).
Slide no. 12
Slide description:
The upside-down birds and animals turned out to be unlike anything else. This is what led the accountants to an interesting discovery. They realized that there was no point in making similar drawings. The changes didn't end there. They also got rid of the curvy lines, and simply pressed the style into the clay and immediately took it away. Clear wedge-shaped marks remained on the clay. This is called Cuneiform. Any icon will do, as long as everyone agrees on what it will mean.
Numbers rule the world. Pythagoras Abstract on the topic: As they believed in ancient times Completed by: student of the 5th “a” class Veronika Melnikova
Until recently, there were tribes whose language had the names of only two numbers: one and two. The natives thought like this: 1 - “urapun” 2 - “okosa” 3 - “okosa - urapun” 4 - “okosa - okosa” 5 - “okosa - okosa - urapun”. . . . . All other numbers are “A LOT”! It can be seen that people have mastered only a small number of integers. The first concepts of mathematics were "less", "more" and "same". If one tribe exchanged caught fish for stone knives made by people of another tribe, there was no need to count how many fish and how many knives they brought. It was enough to place a knife next to each fish for the exchange between the tribes to take place.
People's lives were constantly changing, their knowledge about the world was growing. In order to remember the number of animals, people put aside as many stones as there were sheep. Then they could find out what the offspring were like or how many sheep died in the flood.
Some primitive tribes counted the number of objects by matching them with various parts of the body, mainly fingers and toes. To remember the counting results, notches, knots, etc. were used. Incas counting device
In Ancient China and Japan, calculations were carried out on a special counting board, using a principle similar to Russian abacus. The abacus is a counting board used for arithmetic calculations from approximately the 5th century BC. in Ancient Greece, Ancient Rome. Chinese (top) and Japanese (bottom) abacus
In Ancient Babylon, numbers were written using cuneiform characters. They had a wedge-shaped appearance, since the Babylonians wrote on clay tablets with triangular sticks. These signs were repeated the required number of times, for example
In Ancient Rome they counted as fives, i.e. their main number was the number 5. Then they also switched to counting in tens, but in the system of recording numbers, five still remained. Perhaps the basis of such a recording was counting with fingers. Look closely at the Roman numeral 5 - V: four fingers are pressed against each other, and one is pointed to the side. And the Roman numeral 10 is X, two fives put together by angles.
In ancient times, systems in which numbers were designated by letters of the alphabet were widespread. These included the Greek alphabetic system, also called Ionic. It came to the Slavic tribes along with Christianity and writing. The Slavic numbering was created by the Greek monks Cyril and Methodius in the 9th century, following the Greek model.
Together with the alphabet, such a system of writing numbers came to Ancient Rus'. But instead of a dash in Rus' they put a wavy line - a title. -1 -2 -1000 -3000 -10000 - darkness -100000 - legion -1000000 - leodr
Evolution of Indian numerals
To record numbers, the ancient Egyptians used the following hieroglyphs, meaning (sequentially): one, ten, one hundred, thousand, ten thousand, one hundred thousand (frog), million (man with raised hands), ten million. It is believed that the hieroglyph for a hundred depicts a measuring rope, for a thousand - a lotus flower, for 10,000 - a raised finger, and for 10,000,000 - the entire Universe. All other numbers were composed from the basic ones using only one operation - addition.
In this case, the recording was made not from left to right, as with us, but from right to left. The number 15, for example, was written like this: And the number 444 was written like this: 
We see that ancient Egyptian numbering is similar to Roman, only subtraction is not used when writing numbers. Getting acquainted with Roman numbering, we saw how inconvenient it is to multiply numbers written in a non-positional system. How did the ancient Egyptians think? It turns out that they performed multiplication and division by sequentially doubling numbers. Let, for example, you need to multiply 19 by 37. The Egyptians successively doubled the number 37, and in the right column they wrote down the results of doubling, and in the left column - the corresponding powers of two.
1 37 2 74 4 148 8 296 16 592 Doubling continued until it turned out that a multiplier could be made from the numbers in the left column (in our example 19=1+2+16). The Egyptians marked the corresponding lines with vertical lines and added up the numbers that appear on the same lines to the right. In this case, you need to add 37+74+592=703. This is how the work was received; If now the number 703 needed to be divided by 19, then the Egyptians began to successively double the divisor and continued this until the numbers in the right column remained less than 703. Then from the numbers in the right column they tried to form the dividend, and then the sum of the corresponding numbers in the left column gave divisor: In this case, 703=608+76+19, i.e. the quotient will be 1+4+32=37. If the dividend were not divisible by the divisor without a remainder, then it would not be possible to compose it from the numbers in the right column. We would get both the quotient and the remainder.
The Egyptian method of multiplication is not difficult, but it requires a very large number of operations, even when multiplying two-digit numbers. 
If we had to multiply very large numbers in the same way, we could not do without the help of a machine. Note also that for multiplication and division, the Egyptians actually used the representation of numbers in the binary system.
Alphabetical numbering. Psammit We have seen that non-positional numbering is inconvenient: writing numbers in them is very long, and arithmetic operations are difficult to perform. As trade and crafts developed, these inconveniences became more and more sensitive, and in Asia Minor, where there were ancient Greek colonies that conducted brisk trade, in the middle of the 5th century. BC e. A new type of number system appeared, the so-called alphabetical numbering. It is usually called Ionian.
In this system, numbers were designated using. letters of the alphabet over which dashes were placed: the first nine letters denoted the numbers from 1 to 9, the next nine - the numbers 10, 20, 30 to 90 and the next nine - the numbers 100, 200 to 900. In this way it was possible to denote any number to 999. To designate the numbers 1000, 2000, ..., 9000, the Greeks used the same letters as for the numbers 1, 2, ..., 9, but only when writing them they put a slash at the bottom left.
How this was done can be seen from the attached figure. Further, for the number 10,000 a sign was used - this number was called a myriad; two myriads, i.e. 20,000, were designated as follows: . In this way, it was possible to designate all numbers up to a myriad of myriads, that is, up to 108. Higher decimal places could no longer be written in Ionian numbering and had no name in the ancient Greek language. The great mathematician, mechanic and engineer of antiquity Archimedes (3rd century BC) devoted an entire essay to giving a general method for naming arbitrarily large numbers.