Slide 3

Primitive peoples believe

Until recently, there were tribes in whose language there were names of only two numbers: one and two. The natives believed as follows: 1 - "urapun" 2 - "okoza" 3 - "okoza - urapun"

4 - "okoza - okoza" 5 - "okoza - okoza - urapun". ... ... ... ...

All other numbers are "LOT"! It can be seen that people have mastered only a small number of integers.

The first concepts of mathematics were "less", "more" and "the same". If one tribe exchanged the caught fish for stone knives made by people of another tribe, there was no need to count how many fish were brought and how many knives. It was enough to put a knife next to each fish for the exchange between the tribes to take place.

Slide 4

Many Russian proverbs say that the same was the case with our ancestors:

• "Seven nannies have a child without an eye"
• "Seven troubles - one answer"
• "Seven do not wait for one"
• "Seven times measure cut once"

The natives of New Guinea bend their fingers one by one, saying "be - be - be ...". Having counted to FIVE, he says "ibon - be" (RUKA). Then they bend the fingers of the other hand "be - be .." until it comes to "ibon - ali" (TWO HANDS). For further counting, the toes are used, and then…. someone else's arms and legs!

The number is used in the sense

• "many"
• "seven"

Slide 5

However, among most peoples, the numbers that were considered "money" (and mainly livestock served as money), gradually replaced all the rest. It was they who became those universal numbers that made it possible to count any objects.

People gradually became accustomed to counting to arrange objects in stable groups of two, ten or twelve.

But the numbers did not yet have separate names. In Florida natives, 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 start to get names

Slide 6

So, individual names received numbers less than 10, as well as ten, one hundred, one thousand.

Operations on numbers

People dealt with operations of addition and subtraction long before numbers got names. When several groups of root gatherers or fishermen put their prey in one place, they performed the addition operation.

People got acquainted with the multiplication operation when they began to sow grain and saw that the harvested crop was several times greater than the number of seeds sown.

They said: they harvested "twenty themselves", that is, they harvested twenty times more than they sowed.

Finally, when the mined animal meat or the harvested nuts were divided equally among all the "mouths", a division operation was performed.

Slide 7

In the middle of the 5th century. BC In Asia Minor, where there were ancient Greek colonies, a new type of number system appeared - Ancient Greece

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 denote numbers from 1 to 9, the next nine are 10, 20 ... 90 and the next nine are numbers 100, 200..900. So it was possible to designate any number up to 999. alphabetical numbering

Slide 8

For thousands, the first nine letters were used again, but with a slash at the bottom left. For the number 10000, the sign M was used,

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. this number was called MYRIAD

The great mathematician, mechanic and engineer of antiquity devoted a whole work to giving a general method for naming arbitrarily large numbers.

ARCHIMEDES (III century BC)

Slide 9

Often in fairy tales there is an "unsolvable" problem: to count 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 tasks can be solved. He called his work that way

("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 digits. Archimedes' method is close to the positional one, "Calculus of sand" before mankind managed to create a decimal positional number system. BUT it took another 1000 years, OKTAD

Slide 10

NUMBERS IN ANCIENT ROME

In the Roman system, there are special signs for:

• I - 1 VI - 6
• II - 2 VII - 7
• III - 3 VIII - 8
• IV - 4 IX - 9
• V - 5 X - 10
• L - 50 D - 500
• C - 100 M -1000

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 notation is less convenient than the one we use. Writing numbers is much longer. The Roman system has another existing drawback: it does not provide a way to write arbitrarily large numbers.

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Slide 11

Sumerian cuneiform

Here the farmer brought the onion grown by him to the tax collector in the village of the countries of Sumer. “Sum!” - said the collector, because “sum” in Sumerian means “onion” - and drew a bunch of onions on a damp clay tablet that he held in his hand.

Sumerian bookkeepers have painted fish and birds, livestock and plants for years. 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 the clay so that they turned out to be turned to one side.

Why did it happen? The fact is that at first they wrote on the clay in columns from top to bottom, and each next column began to the left of the previous one. But at the same time, they smeared with their hand what had been written before. Therefore, they began to turn the tile by a quarter of a turn and began to write the same signs in lines, from left to right (and each next line began below the previous one).

Slide 12

The inverted birds and animals were unlike anything else. This led the bookkeepers to an interesting discovery. They realized that there was no need to make similar drawings at all.

The changes did not end there. We also got rid of the sinuous lines, but simply pressed the style into the clay and immediately took it away. Clear wedge-shaped marks remained on the clay. This is what is called - CLEANING.

Any icon is good, as long as everyone has agreed on what it will mean.

Slide 13

"And for a low life there were numbers, Like livestock for livestock, Because a clever number conveys all the shades of meaning."

Russian poet Nikolai Gumilev expressed the meaning of this discovery in the words:

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Slide 14

This is one of the oldest numberings. The inscriptions of the Egyptians consist of pictures - hieroglyphs.

Two mathematical papyri have survived, allowing us to judge how the ancient Egyptians believed. It is believed that the hieroglyph for a hundred represents a measuring rope, for a thousand, lotus flowers,

Slide 15

It turns out that they performed multiplication and division by sequentially doubling numbers - in fact, representing a number in the binary system for ten thousand - a raised finger, one hundred thousand - a frog, a million - a man with raised hands, ten million - the entire Universe. How did the ancient Egyptians think?

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Slide 16

BABYLONIA

The first known positional number system was

The Babylonians did this: wrote down all the numbers

1 to 59 in decimal using the principle of addition. At the same time, they always used two signs: a straight wedge for designating 1 and a recumbent wedge for 10. These signs served as numbers in their system. The number 60 was again indicated by the same sign as 1, i.e. ...

Babylonians, which arose about 2500 - 2000 BC. The number 60 served as its basis.

How did the Babylonians write down their numbers?

Slide 17

All other degrees of 60 were denoted in the same way. Thus, "numbers", i.e. all numbers from 1 to 59, the Babylonians wrote down according to the decimal non-positional system, and the number as a whole - according to the positional system with base 60. That is why we call them the sexagesimal system. But the numbering of the Babylonians had another important feature:

And if a straight wedge was depicted, then without additional explanations it was impossible to determine which number was written: 1, 60, 3600 or some other power of 60. Subsequently, there was no sign for ZERO in it, the Babylonians introduced a special symbol to denote the missing sixty-decimal place.

Slide 18

In India and China.

Positional number systems arose independently of one another in the ancient Mesopotamia, the Maya and India.

Ancient India and China had recording systems based on principle. In such systems, the same symbols are used to record the same number of units, tens, hundreds or thousands, but after each symbol the name of the corresponding digit is written.

What led people to this discovery?

MULTIPLICATIVE

Slide 19

Indians have long had a deep interest in large numbers and the way they were recorded. royal brides competed not only in wrestling or archery, but also in writing and arithmetic.

Between the 2nd and 6th centuries A.D. The Indians became acquainted with Greek astronomy. At the same time, they became familiar with 60-ary numbering and the Greek round zero.

If tens are denoted by the symbol D, and hundreds - by C, then the number 325 will look like this: 3S2D5.

The Indians also combined the Greek numbering principles with their decimal multiplicative system.

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In the habitats of primitive man, archaeologists find objects with embossed dots, scribbled lines, and deep notches. These findings suggest that already in the Stone Age, people were able not only to count, but also to record the results of their calculations.

With the development of society, the methods of counting have also improved. After all, such primitive techniques as notches on a stick, knots on a rope or pebbles piled in piles could not satisfy the needs of trade and production.

Around 3,000 BC, an important discovery was made: people invented special signs to indicate a number of objects. For example, the Egyptians denoted ten by 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- a thousand.

Roman numeral system is based on the following principle: if, when reading from left to right, the smaller digit is after the larger one, then it is added to the larger one: VI = 6, XXXII = 32; if the smaller digit is in front of the larger one, then it is subtracted from the larger one: IV = 4, VL = 45.

In the Roman numeral system, for example, the number 14 is written as follows: XIV. Here, the number I stands between the larger numbers X and V. In such cases, the number I is subtracted from the number on 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 in Roman numerals as follows: MCMXLV. This system has survived to this day. You can often find records using Roman numerals, for example: XXI century, chapter VI. They can also be seen on watch dials, on architectural monuments.

You've probably already noticed that even reading a number written in Roman numerals is not easy. It is all the more difficult to perform arithmetic operations in such a record. In addition, if you need to write down large enough numbers (million, billion, etc.), then you need to come up with new numbers. Otherwise, the number will be recorded too long. For example, if only the Roman numeral M is used to record the number 1,000,000, then the record will consist of a thousand such characters. All these disadvantages significantly reduce the possibility of using the Roman numeral system.

In Ancient Russia, they did not invent special icons to denote numbers. They were obtained using the letters of the alphabet. A wavy line was placed above the letter - titlo.

For example, the number 241 was written like this:

The greatest achievement of humanity is the invention decimal positional number system... With this system, numbers as large as desired are written using only ten different digits. This is possible because the same number has different meanings depending on its positions in list.

Numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are called Arabic. However, the Arabs only extended the decimal positional system invented by the Indians.

Some tribes and peoples used other positional number systems. For example, the Maya Indians used the decimal system, and the ancient people of the Sumerians used the hexadecimal system.

Traces of the decimal system can be found in some European languages. So, instead of "eighty" the French say "four times twenty" ( quatre − uingts ). Breaking one hour into 60 minutes and one minute into 60 seconds is an example of the 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 vague system. The twelve-digit system also has a "finger" origin: try using your thumb to count the phalanges on the other fingers of the same hand, the result will be the number 12 (Fig. 2). This is how the account came into being by the dozen.

And today in Europe dozens of handkerchiefs, buttons, chicken eggs are sold. 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. So, the structure and operation of a computer is based on a binary number system that uses only two digits - 0 and 1.

Ancient Greek numbering In the 5th century BC. alphabetical numbering appeared.

Non-positional The value of a digit does not depend on its position in the number The value of a digit depends on its position in the number. Roman XXX Decimal Binary Duodecimal 333 = 3 * * Octal Hexadecimal Unary

I (1), V (5), X (10), L (50), C (100), D (500), M (1000). IX (9) XI (11) 1998 = MCMXCVIII = 1000 + () + (100-10)

Decimal number system Numbers developed in India around 400 AD. NS. The Arabs began to use this numbering around 800 AD. NS. Around 1200 A.D. NS. this numbering began to be applied in Europe. The famous Persian mathematician Al-Khwarizmi published a textbook in which he outlined the basics of the Hindu decimal system.

The Aztecs and Mayans, the peoples who inhabited vast areas of the American continent for many centuries and created a high culture there, adopted the decimal number system. The same system was adopted by the Celts who inhabited Western Europe from the second millennium BC. The number 20 is found in the French monetary system: the basic monetary unit, the franc, is divided by 20 sous.

Binary number system Two digits are used - 0 and 1 Used in technical devices

Decimal Binary Eight Hex A B C D E F

She was 1,100 years old. She went to the 101st grade. She carried 100 books in a portfolio. All this is true, not nonsense. When a dozen feet were dusty, She walked along the road, A puppy always ran after her With one tail, but moaning. She caught every sound With her ten ears, And 10 tanned hands She held a briefcase and a leash. And 10 dark blue eyes They looked around the world as usual. But everything will become quite ordinary when you understand our story.