How did numbers develop in the ancient world? The history of the development of numbers

The emergence of numbers in our lives is not an accident. It is impossible to imagine communication without the use of numbers. The history of numbers is fascinating and mysterious. Mankind managed to establish a number of laws and patterns of the world of numbers, unravel some mysteries and use their discoveries in Everyday life. Without the remarkable science of numbers - mathematics - neither the past nor the future is inconceivable today. And how many more unsolved!

Ancient people could not count. And they had nothing to count, because the items they used - tools - were very few: one ax, one spear. Gradually, the number of things increased, their exchange became more complicated and there was a need to count. For a long time, numbers seemed to people something mysterious. Every object could be seen and touched. A number cannot be touched, and at the same time, numbers really exist, since all objects can be counted. This oddity has led people to attribute supernatural properties to numbers.

In our high-speed fast-flying age - the age of great abundance of information, various printed publications and the virtual world, it is difficult to surprise people with anything. Write, create something, so that it is interesting to read! So

From early childhood, we get acquainted with numbers. And what are the numbers? I tried to answer this question in my work. My work is possible - this is a mini-guide for getting acquainted with such an interesting concept as "Numbers". Perhaps not everything is detailed, but in my work I tried to touch on all aspects related to the chosen topic. This work can be used by those who want to know more about mathematics than an ordinary student.

History of number development

At the first stages of the existence of human society, numbers served for the primitive counting of objects, days, steps. In primitive society, a person needed only the first few numbers. With the development of civilization, he needed to invent more and more numbers, this process continued for many centuries and required intense intellectual work. When exchanging products, it became necessary to compare numbers, the concepts of more, less, and equal arose. At the same stage, people began to add numbers, then they learned to subtract, divide, and multiply. When dividing two natural numbers, fractions appear, when subtracting, negative numbers appear.

The need to perform arithmetic operations led to the concept of rational numbers. In the IV century. BC e. Greek mathematicians discovered incommensurable segments, the lengths of which were not expressed either by an integer or a fractional number (for example, the length of the diagonal of a square with a side equal to 1). It took more than one hundred years for mathematicians to be able to work out a way to write such numbers as an infinite non-periodic decimal fraction. This is how irrational numbers appeared, which, together with rational numbers, were called real numbers.

But then it turned out that the simplest quadratic equations, for example, x2 + 1 = 0, have no solution in the set of real numbers. Mathematicians came to the need to expand the concepts of number so that the square root could always be extracted in the new set. The new set was called the set of complex numbers, introducing the concept of an imaginary unit: i2 = - 1.

An expression of the form a + bi was called a complex number. For a long time, many scientists did not recognize them as numbers. Only after finding the possibility of representing the imaginary number geometrically did the so-called imaginary numbers find their place in the set of numbers.

N- integers.

Q are rational numbers.

R are real numbers.

Complex numbers are called a + bi, where a and b are real numbers, i is the imaginary unit: i2 \u003d - 1. a is called the real part, bi is the imaginary part of the complex number.

Definition. Two complex numbers are called equal if their real parts and the coefficients of the imaginary parts are equal, i.e., a + bi = c + di a = c, b = d.

For complex numbers, there are no "greater than" or "less than" relationships.

mathematicians who contributed

Contribution to the development of number theory

We live in the world big numbers

Have you ever thought about how many kilometers a person travels in his life, how many goods are produced and become unusable hourly within a city, country? How many times does the speed of a passenger jet aircraft exceed the speed of a trained athlete-pedestrian? The answers to these and thousands of similar questions are expressed in numbers that often occupy a whole line or even more in the number of their decimal places.

To reduce the notation of large numbers, a system of quantities has long been used in which each of the following is a thousand times larger than the previous one:

1000 units is just a thousand (1000 or 1000)

1000 thousand - 1 million

1000 million - 1 billion (or 1 billion)

1000 billion - 1 trillion

1000 trillion - 1 quadrillion

1000 quadrillion - 1 quintillion

1000 quintillion - 1 sextillion

1000 sextillion - 1 septillion

1000 nonillion - 1 decillion, etc.

Thus, 1 decillion will be written in the decimal system as a unit with 3 * 11 = 33 zeros. 1. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000. 000.

“In vain they think that zero plays a small role”

Samuil Yakovlevich Marshak

The degree of a number is the product of itself by itself the required number of times, which is called the exponent (and the number itself is its base). For example, 3 * 3= 32 (here 3 is the base, 2 is the exponent), 2 * 2 * 2= 23, 10 * 10= 102=100, 105= 10 * 10 * 10 * 10 * 10= 100000.

Note that the number of zeros of the power of 10 is always equal to its exponent:

101=10, 102=100, 103=1000 etc.

And one more thing: mathematicians all over the world have long accepted that any number to the zero power is equal to one (a0 = 1). When writing large numbers, a power of 10 is often used.

Unit - 100=1

Thousand - 103= 1000

Million - 106= 1000 000

Billion - 109= 1000,000,000

Trillion - 1012=1000,000,000,000

Quadrillion - 1015 = 1000,000,000,000,000

Quintillion - 1018 = 1000,000,000,000,000,000

Sextillion - 1021 = 1000,000,000,000,000,000,000

Septillion - 1024 = 1000,000,000,000,000,000,000,000

Octillion - 1027 = 1000,000,000,000,000,000,000,000,000

Now for some interesting information:

The radius of the Earth is 6400 km.

The length of the Earth's equator is about 40 thousand km.

The area of ​​the globe is 510 million km.

The average distance from the Earth to the Sun is 150 million km.

The diameter of our Galaxy is 85 thousand light years.

A little over a billion seconds have passed since the beginning of our era.

Scheherazade number

There are numbers that bear the names of great mathematicians: the number of Archimedes -, Napier's number - the base of natural logarithms e \u003d 2, 718281 [Napier John (150-1617), Scottish mathematician, inventor of logarithms].

The number in question is no less popular. This is 1001. It is sometimes called the Scheherazade number, known to everyone who has read the tales of the Thousand and One Nights. The number 1001 has a number of interesting properties:

1. This is the smallest natural four-digit number that can be represented as the sum of cubes of two natural numbers: 1001=103+13.

2. Consists of 77 "ill-fated bloody dozens." (1001=77*13), out of 91 elevens or 143 sevens (remember that the number "7" was considered a magic number); further, if we assume that a year is 52 weeks, then 1001=143*7=(104+26+13)*7=2 years + ½ years + ¼ years

3. A method for determining the divisibility of a number by 7, by 11 and by 13 is based on the properties of the number 1001.

Consider this method with examples:

Is 348285 divisible by 7?

348285=348*1000+285=348*1000+348-348+285=348*1001-(348-285)

Since 1001 is divisible by 7, so that 348285 is divisible by 7, it is enough that the difference 348-285 is divisible by 7. Since 348-285=63, then 348285 is divisible by 7.

Thus, to find out if a number is divisible by 7 (by 11 or 13), it is necessary to subtract from this number without the last three digits the number from the last three digits; if this difference is divisible by 7 (11 or 13), then for given number also divisible by 7 (11 or 13).

Think about it, maybe you will find a fabulous number. Contribute to the queen of sciences - MATHEMATICS!!!

Reciprocal numbers

The reciprocal (reciprocal value, reciprocal value) is the number by which the given number must be multiplied to get one. Two such numbers are called reciprocals.

Examples: 5 and 1/5, −6/7 and −7/6, π and 1/π

For any number a not equal to zero, there is an inverse 1/a.

Birds live on the globe - the unmistakable compilers of the weather forecast for the summer. The name of these birds is encrypted with examples written on the board. Having consistently solved the examples and replacing the answers with letters, you will read the name of the birds - meteorologists.

1. 17/8 5/6 6/5;

2. 3,4 7/3 3/7;

3. 11/12 5,6 12/11;

4. 2,5 0,4 3;

5. 2/3 0,1 3/2;

6. 41/2 1/2 2;

8. 11/12 31/3 12/11.

17/8 31/3 0,1 3,4 3 41/2 5,6 1

f o i l m n a g

prime numbers

"Prime numbers remain always ready to elude investigation"

If we write natural numbers in a row, and light lanterns in those places where prime numbers stand, then there was no place in this row where there would be complete darkness. Lanterns would be arranged very bizarrely. Between them there is only one number - even, this is 2, and the rest are odd. 2 and 3 consecutive natural numbers, the smallest prime - such a pair is unique, where one number is even and the other is odd.

1, 2, 3,4 ,5 ,6, 7,8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Two consecutive odd numbers, each of which is prime, are called numbers - twins.

The first twin primes are:

(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61),

(71, 73), (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193),

(197, 199), (227, 229), (239, 241), (269, 271), (281, 283), (311, 313), (347, 349),

(419, 421), (431, 433), (461, 463), (521, 523), (569, 571), (599, 601), (617, 619),

(641, 643), (659, 661), (809, 811), (821, 823), (827, 829), (857, 859), (881, 883)

The Greek scientist Euclid in his book "Beginnings" stated the following: "There is no greatest number." It is still unknown if there are the largest twin numbers. And there is still no answer to the question: are there infinitely many pairs prime numbers-twins.

The Russian mathematician Pafnuty Lvovich Chebyshev was the first to make deep research on how prime numbers are scattered among natural numbers. But until now, mathematicians do not know the formula by which you can get prime numbers one by one, there is not even a formula that gives only prime numbers.

The Alexandrian scholar Eratosthenes, who lived in the 3rd century BC, thought about how to make a list of prime numbers. His name entered science in connection with the method of finding prime numbers. In ancient times, they wrote on wax tablets with a sharp stick-style, so Eratosthenes "poked out" composite numbers pointy end style. After puncturing all the composite numbers, the table looked like a sieve. Hence the name "Sieve of Eratosthenes". Ancient Greek scientists were interested: how many prime numbers can there be in the natural series.

In 1750, Leonard Eimer established that the number 231 - 1 is prime. It remained the largest known prime number for over a hundred years. In 1876, the French mathematician Lucas established that a huge number

2127 - 1 = 170.141.183.460.469.231.731.678.303.715.884.105.727 is also prime. It contains 39 digits. To calculate it, mechanical desktop calculating machines were used. In 1957, the following prime number was found: 23217-1. And the prime number 244497-1 consists of 13000 digits.

Rational numbers

Rational number (lat. ratio - ratio, division, fraction) - a number represented by an ordinary fraction, where m is an integer and n is a natural number. In this case, the number m is called the numerator, and the number n is called the denominator of the fraction. Such a fraction should be intuitively understood as the result of dividing m by n, even if it cannot be completely divided. AT real life rational numbers can be used to count the parts of some whole but divisible objects, such as cakes or other foods that are cut into several pieces before consumption, or to roughly estimate the spatial relationships of extended objects.

Perfect Numbers

A perfect number (other Greek ἀριθμὸς τέλειος) is a natural number equal to the sum of all its own divisors (that is, all positive divisors other than the number itself).

The first perfect number is 6 (1 + 2 + 3 = 6), the next one is 28 (1 + 2 + 4 + 7 + 14 = 28). As the natural numbers increase, the perfect numbers become rarer. The third perfect number is 496, the fourth perfect is 8128, the fifth is 33,550,336, and the sixth is 8,589,869,056 (OEIS sequence A000396).

“Stop looking for interesting numbers!

Leave at least one uninteresting number for interest!”

From a letter from a reader to Martin Gardner

Among all the interesting natural numbers that have long been studied by mathematicians, a special place is occupied by perfect and closely related friendly numbers.

A perfect number is a number equal to the sum of all its divisors (including 1, but excluding the number itself). The smallest perfect number 6 is equal to the sum of its three divisors 1, 2 and 3. The next perfect number is 28=1+2+4+7+14. Early commentators on the Old Testament, writes Martin Gardner in his Mathematical Novels, saw a special meaning in the perfection of the numbers 6 and 28. Wasn't the world created in 6 days, they exclaimed, and isn't the moon renewed in 28 days?

The first major achievement in the theory of perfect numbers was Euclid's theorem that the number 2n-1(2n-1) is even and perfect if the number 2n-1 is prime 1. Only two thousand years later, Euler proved that Euclid's formula contains all even perfect numbers. Since no odd perfect number is known (readers have a chance to find it and glorify their name), when talking about perfect numbers, they usually mean an even perfect number.

Looking closely at Euclid's formula, we will see the connection of perfect numbers with members of a geometric progression 1, 2, 4, 8, 16. This connection is best seen in the example ancient legend, according to which the Raja promised the inventor of chess any reward. The inventor asked to put one grain of wheat on the first square of the chessboard, two grains on the second square, four on the third, eight on the fourth, and so on. On the last, 64th cell, 263 grains should be poured, and in total there will be a “heap” of 264-1 grains of wheat on the chessboard. This is more than has been collected in all the harvests in the history of mankind.

If we write on each cell of the chessboard how many grains of wheat would be due to the inventor of chess for it, and then remove one grain from each cell, then the number of grains remaining will exactly correspond to the expression in brackets in Euclid's formula. If this number is prime, then multiplying it by the number of grains on the previous cell (that is, by 2n-1), we get a perfect number! Prime numbers of the form 2n-1 are called Mersenne numbers after the 17th century French mathematician. On a chessboard with one grain removed from each cell, there are nine Mersenne numbers corresponding to nine prime numbers less than 64, namely: 2, 3, 5, 7, 13, 17, 19, 31 and 61. Multiplying them by the number of grains on the previous cells, we get the first nine perfect numbers. (Numbers n=29, 37, 41, 43, 47, 53, and 59 do not give Mersenne numbers, i.e. the corresponding numbers 2n-1 are composite.)

Euclid's formula makes it easy to prove numerous properties of perfect numbers. For example, all perfect numbers are triangular. This means that, taking the perfect number of balls, we can always add an equilateral triangle out of them. Another curious property of perfect numbers follows from the same formula of Euclid: all perfect numbers, except for 6, can be represented as partial sums of a series of cubes of successive odd numbers 13+33+53+ including itself, is always equal to 2. For example, taking the divisors of the perfect number 28, we get:

In addition, the representation of perfect numbers in binary form, the alternation of the last digits of perfect numbers, and other curious questions that can be found in the literature on entertaining mathematics. The main ones - the existence of an odd perfect number and the existence of the largest perfect number - have not yet been resolved.

From perfect numbers, the story inevitably flows to friendly numbers. These are two such numbers, each of which is equal to the sum of the divisors of the second friendly number. The smallest of the friendly numbers 220 and 284 were known to the Pythagoreans, who considered them a symbol of friendship. The next pair of friendly numbers 17296 and 18416 was discovered by the French lawyer and mathematician Pierre de Fermat only in 1636, and subsequent numbers were found by Descartes, Euler and Legendre. The sixteen-year-old Italian Niccolo Paganini (the namesake of the famous violinist) shocked the mathematical world in 1867 with the message that the numbers 1184 and 1210 are friendly! This pair, closest to 220 and 284, was overlooked by all the famous mathematicians who studied friendly numbers.

friendly numbers

Amicable numbers are two natural numbers for which the sum of all proper divisors of the first number is equal to the second number and the sum of all proper divisors of the second number is equal to the first number. Sometimes perfect numbers are considered a special case of friendly numbers: every perfect number is friendly to itself.

The following are pairs of friendly numbers less than 130,000.

6. 10744 and 10856

7. 12285 and 14595

8. 17296 and 18416

9. 63020 and 76084

10. 66928 and 66992

11. 67095 and 71145

12. 69615 and 87633

13. 79750 and 88730

14. 100485 and 124155

15. 122265 and 139815

16. 122368 and 123152

The tomb rests the ashes of Diophantus: marvel at her - and a stone

The age of the departed will tell him with wise art.

By the will of the gods, he lived a sixth of his life as a child

And he met half of the sixth with fluff on his cheeks.

Only the seventh passed, he got engaged to his girlfriend;

After spending five years with her, the sage waited for his son.

Only half the life of his father's beloved son lived,

He was taken from his father by his early grave.

Twice two years the parent mourned the heavy grief,

Here I saw the limit of my sad life.

How many years did Diophantus live?

curly numbers

A long time ago, helping themselves with counting with pebbles, people paid attention to the correct figures that can be laid out from pebbles. You can just put the pebbles in a row: one, two, three. If we put them in two rows to make rectangles, we will find that all even numbers are obtained. You can lay out the stones in three rows: you get numbers that are divisible by three. Any number that is divisible by something can be represented by such a rectangle, and only prime numbers cannot be "rectangular". What if you fold a triangle? The triangle is obtained from three pebbles: two in the bottom row, one in the top, in a hollow formed by the two bottom stones. If you add a stone to the bottom row, another hollow will appear; filling it, we get a hollow formed by two pebbles of the second row; putting a stone in it, we finally get a triangle. So we had to add three pebbles. The next triangle will be obtained by adding four pebbles. It turns out that at each step we add as many stones as there are in the bottom row. If we now assume that one stone is also a triangle, the smallest one, we get the following sequence of numbers: 1, 1+2=3, 1+2+3=6, 1+2+3+4=10, 1+ 2+3+4+5=15, etc. So, figurative numbers are the general name for numbers whose geometric representation is associated with one or another geometric figure. Numbers by the ancient Greeks, and together with them by Pythagoras and Pythagoreans, were conceived visibly, in the form of pebbles laid out on the sand or on a counting board - an abacus.

For this reason, the Greeks did not know zero, because it was impossible to "see" it. But the unit was not yet a full number, but was presented as a kind of “numerical atom”, from which all numbers were formed. The Pythagoreans called the unit the “border between the number and parts”, that is, between integers and fractions, but at the same time in it "the seed and the eternal root." The number was defined as a set composed of units. The special position of the unit as a "numerical atom" made it related to the point, which was considered a "geometric atom". That is why Aristotle wrote: "A point is a unit having position, a unit is a point without position." That. Pythagorean numbers in modern terminology, these are natural numbers. The numbers of pebbles were laid out in the form of regular geometric figures, these figures were classified. So there were numbers, today called curly. The ancient Greeks, when they had to multiply numbers, drew rectangles; The result of multiplying three by five was a rectangle with sides three and five. This is the development of counting on pebbles. Many patterns that arise when working with numbers were discovered by ancient Greek scientists when studying drawings. And for many centuries, the geometric method, with rectangles, squares, pyramids and cubes, was considered the best confirmation of the validity of such ratios. In the 5th - 4th centuries BC, scientists, combining natural numbers, made intricate series from them, giving the elements of these series one or another geometric interpretation. With their help, you can lay out the correct geometric shapes: triangles, squares, pyramids, etc. B. Pascal and P. Fermat were carried away, and independently of each other, by finding such numbers.

Even in the 17th century, when algebra was already well developed with the notation of quantities by letters, with signs of actions, many considered it a barbaric science, suitable for base purposes - everyday calculations, auxiliary calculations - but by no means for noble scientific papers. One of the greatest mathematicians of that time, Bonaventura Cavalieri, used algebra, because it is easier to calculate with its help, but to justify his scientific results all algebraic calculations were replaced by reasoning with geometric figures.

Among curly numbers, there are: Linear numbers (i.e., prime numbers) - numbers that are divisible only by one and by themselves and, therefore, can be represented as a sequence of points lined up: (linear number 5)

Flat numbers - numbers that can be represented as a product of two factors: (flat number 6)

Solid numbers expressed as a product of three factors: (solid number 8)

Triangular numbers: (triangular numbers 3,6,10)

Square numbers: (square numbers 4,9,16)

Pentagonal numbers: (pentagonal numbers 5,12)

It is from curly numbers that the expression "Square or cube a number" came from.

The representation of numbers in the form of regular geometric figures helped the Pythagoreans to find various numerical patterns. For example, to get a general expression for an n-coal number, which is nothing more than the sum of n natural numbers 1+2+3+. + n, it is enough to complement this number to a rectangular number n(n + 1) and see (with the eyes!) the equality

Having written a sequence of square numbers, it is again easy to see with your eyes the expression for the sum of n odd numbers:

Finally, breaking the n-th pentagonal number into three (n-1) triangular ones (after which n more "pebbles" remain), it is easy to find its general expression

By splitting into triangular numbers, the general formula for the nth k-gonal number is also obtained:

With k=3 we get triangular numbers, and k=4 we get square numbers, etc.

Similarly, you can represent a number in the form of a rectangle. For the number 12, this can be done in many ways (Fig.), And for the number 13 - only by placing all the objects in one line. Such ancients did not consider rectangular.

Thus, all composite numbers are rectangular numbers, and prime numbers are non-rectangular. The figurative representation of numbers helped the Pythagoreans discover the laws of arithmetic operations, as well as easily move on to the numerical characteristics of geometric objects - the measurement of areas and volumes.

So, representing the number 10 in two forms: 5*2=2*5, it is easy to "see" the commutative law of multiplication: a*b=b*a. In the same number 10: (2+3)*2=2*2+3*2=10, one can "see" the distributive law of addition with respect to multiplication: (a+b)c=ac+bc.

Finally, if the "pebbles" that form curly numbers are thought of as squares of equal area, then by fitting them into a rectangular number ab:. automatically we get the formula for calculating the area of ​​a rectangle: S=ab. Figured numbers also include pyramidal numbers, which are obtained if the balls are folded in a pyramid, as they used to put the balls near the cannon.

It is easy to see that the pyramidal number is equal to the sum of all triangular numbers - from the first to the nth. The formula for calculating the nth pyramidal number is:

"Number Fun"

This number is, first of all, remarkable in that it determines the number of days in a leap year. When divided by 7, it gives a remainder of 1, this feature of the number 365 has great importance for our seven-day calendar.

There is another feature of the number 365:

365=10×10×11×11×12×12, i.e. 365 equals the sum of the squares of three consecutive numbers starting from 10:

10²+11²+12²=100+121+144=365.

But that's not all. The number 365 is equal to the sum of the squares of the following two numbers, 13 and 14:

13²+14²=169+196=365.

If a person does not know the above properties of the number 365, then when solving the example:

10²+11²+12²+13²+14²

365 will begin to perform cumbersome calculations.

For example:

10²+11²+12²+13²+14² ‗ 100+121+144+169+196 ‗ 221+313+196 ‗ 730

A person who knows will solve this example in his mind instantly and get 2 in the answer.

10²+11²+12²+13²+14² ‗ 365+365 ‗ 730

The next number I will describe is 999.

It is much more amazing than its inverted image - 666 - "animal number"

Apocalypse that strikes fear into superstitious people, but it is in its own way arithmetic properties nothing stands out from other numbers.

The peculiarity of the number 999 is that it can be easily multiplied by three-digit numbers. Then you get a six-digit product: its first three digits are the multiplied number, reduced by one, and the remaining three digits are the additions of the first three to 9. For example,

One has only to look at the following line to understand the origin of this feature:

573×999=573×(1000-1)= 573

Knowing this feature, we can instantly multiply any three-digit number by 999.

For example:

947×999=946053, 509×999=508491, 981×999=980019,

543×999=542457, 167×999=166833, 952×999=951048 etc.

And since 999=9×111=3×3×3×37, then you can describe entire columns of six-digit numbers that are multiples of 37. Those who are not familiar with the properties of the number 999 will not be able to do this.

1. Number 1001

First, consider the number 1001. This is the number of fairy tales that Queen Sheherazade told King Shahriyar.

The number 1001 at first glance seems to be the most common. It can be decomposed into three successive prime factors 7, 11 and 13. Therefore, it is their product.

But the fact that 1001=7×11×13 is not interesting. The remarkable thing is that if you multiply it by any three-digit number, then the result will be the same number, written twice. We need to apply the distributive law of multiplication.

Let's decompose 1001 into the sum 1000+1.

For example:

247×1001=247×(1000+1)=247×1000+247×1=247000+247=247247

Number 111111

The next number I want to talk about is 111111.

Thanks to our acquaintance with the properties of the number 1001, we immediately see that

111 111=111×1001

But we know that

111=3×37, 1001=7×11×13.

It follows that our new numerical curiosity, consisting of only ones, is the product of five prime factors. Combining these 5 factors into two groups in all possible ways, we get 15 pairs of factors that give the same number in the product, 111 111.

3×(7×11×13×37)=3×37037=111 111

7×(3×11×13×37)=7×15873=111 111

11×(3×7×13×37)=11×10101=111 111

13×(3×7×11×37)=13×8547=111 111

37×(3×7×11×13)=37×3003=111 111

(3×7)×(11×13×37)=21×5291=111 111

(3×11)×(7×13×37)=33×3367=111 111

(3×13)×(7×11×37)=39×2849=111 111

(3×37)×(7×13×11)=111×1001=111 111

(7×3)×(11×13×37)=21×5291=111 111

(7×11)×(3×13×37)=77×1443=111 111

(7×13)×(11×3×37)=91×1221=111 111

(7×37)×(11×3×13)=259×429=111 111

(11×13)×(7×37×3)=143×777=111 111

(37×11)×(13×7×3)=407×273=111 111

"Trick with a number"

Arithmetic tricks are honest, conscientious tricks. Here, no one seeks to deceive anyone, introduce a trance or lull the viewer's attention. To perform such a trick, neither miraculous manual dexterity, nor amazing dexterity of movements, nor any other artistic abilities, sometimes requiring many years of practice, are needed. A circle of comrades who are not initiated into mathematical secrets can be struck by the following tricks.

Focus number 1.

Write down the number 365 twice: 365 365.

Divide the resulting number by 5: 365 365÷5=73 0 73.

Divide the resulting quotient by 73: 73 0 73÷73=1001.

You will get the Scheherazade number, that is, 1001.

The clue to the trick is very simple: the number 365=5×73. That is, we divide the number 365365 by 365 and get 1001 in the answer.

Focus number 2.

Have someone write any three-digit number, and then add the same number again to it. You will get a six-digit number consisting of repeated digits.

Invite your friend to divide this number secretly by 7. The result must be passed on to the neighbor, who must divide it by 11. The result is passed on to the next student, whom you ask to divide this number by 13.

You give the result of the third division to the first comrade without looking. This is the intended number.

This trick is explained very simply. If you attribute it to a three-digit number, it means multiplying it by 1001, or by the product 7 × 11 × 13 = 1001. The six-digit number that your friend will receive after he assigns to given number itself, will have to be divided without a remainder by 7, and by 11, and by 13.

Focus number 3.

Write down any number three times in a row. Divide the resulting number by 37 and by 3. And you will get your number in the answer.

Solution: when we divide a three-digit number written in three the same numbers first by 37, and then by 3, then we, without noticing, divide by 111.

Focus number 4.

The number 111111 can also be used for tricks, just like the number 1001. In this case it is necessary to offer a friend a single-digit number, and ask him to write it down six times in a row. The divisors here can be five prime numbers: 3, 7, 11, 13, 37 and the resulting composites: 21, 33, 39, etc. This makes it possible to greatly diversify the performance of the trick.

For example: invite your comrades to think of any number other than zero. You need to multiply it by 37. Then multiply by 3. Attribute the result again on the right. The resulting number is divided by the originally conceived figure.

It turned out the number 111 111.

The key to the trick is based on the property of the number 111 111. When we multiply it by 1001 (we met the properties of the number 1001 in the previous chapter), we got the intended number written at the beginning. Further, when dividing by the intended number, six units are clearly obtained.

Focus number 5.

Have your friend write down any three-digit number. On the right, you need to attribute three zeros to it. From a six-digit number, suggest subtracting the original three-digit number. Then ask a friend to divide by the intended result. The quotient must be divided by 37.

The result is the number 27.

The secret of focus is easy to understand. It is based on the properties of the number 999.

The number 999 is the product of four prime factors:

3×3×3×37=999 and therefore 999÷37=27

When a three-digit number is multiplied by it, a result consisting of two halves is obtained: the first is the multiplied number, reduced by one, and the second is the result of subtracting the first half from the factor.

Focus number 6.

Number 111 111 111: can also be used for our number tricks:

Let's ask a classmate his favorite number (from 1 to 9).

Let's ask this figure to be multiplied by 9, and then the resulting product should be multiplied by the number 123456789. The result will be a number consisting of the classmate's favorite digits.

For example:

5 is the student's favorite number, then

45×123456789=555 555 555 i.e. 9×123456789=111 111 111

Conclusion

I think that my work is a mini-guide for the study of numerical diversity. Interesting ways of calculating numbers can be very helpful at school, at university, at work, and in general in life. So in the circle of comrades, you can think of interesting arithmetic tricks without deception and magic. Based on the foregoing, I conclude that it is desirable for everyone to know these and many other numerical curiosities. This knowledge will definitely be needed in life!

MOU Detchinsky secondary school

Introductory and indicative

project on:

The history of the emergence of numbers and figures.

Prepared by students of grade 6 "a":

Nikishina Veronica and

Romanova Ekaterina.

Teacher:

Kondratenko E.B.

Project structure:

1. Introduction. Relevance of the topic.

2. Goals and objectives of the project.

3. From the history of the emergence of numbers.

4.Conclusions.

1. Introduction. Relevance of the topic:

From a very early age, a person is faced with the need to count. However, having learned to count, people know little about where the numbers came from, who came up with the idea of ​​using this or that form of writing a number. Our survey showed that some students of our school, as well as our parents, could not answer the question: "How and where did the first numbers appear?" Meeting with numbers at every step, we are so used to their existence that we hardly think about where they came from. And, by the way, the history of their occurrence is extremely fascinating. Therefore, we decided to study the history of the emergence of numbers and present the material obtained to other students, which can also be used in mathematics lessons.

The topic is relevant and may be of interest both to the general public and to specialists in the field of algebra and geometry. In modern conditions, it is very important for every person to correctly understand the laws of numbers. Numbers are a necessary part of mathematics.

2. Goals and objectives of the project:

Purpose: To study the history of the emergence of numbers and numbers.

Tasks:

1. To study the available literature on the topic of the project.

2. Prepare a presentation on the topic of the project.

3. From the history of the emergence of numbers.

First there were…fingers. A very versatile, convenient and handy tool for. It is still used, however, only if it is necessary to show a small number limited to one ten (here we take into account only the capabilities of the hands, the toes do not count). Not surprisingly, the need for other, more advanced counting symbols quickly arose.

Primitive peoples had a developed number system. Back in the 11th century, many tribes in Australia and Polynesia had only two designations - for the number "one" and for the number "two". These designations were combined. They called the number "three" "two one", the number "four" - "two and two", the number "five" - ​​"two, two and one", the number "six" - "two, two and two". and numbers greater than six, they did not distinguish and called the word "many".

First similarity of numbers originated about five thousand years ago in Egypt and Mesopotamia and was a notch on a tree or stones. Egyptian priests papyrus was used for writing, and in Mesopotamia soft clay served for this purpose. The numbers of those times were indicated by dashes for units and various other marks for tens and higher orders.

It is interesting that the records were not only countable, but also mathematical: the ancient Egyptians, as you know, reached amazing heights in arithmetic and geometry. When hieroglyphs appeared, numbers began to be written through them.

The next step in history of numbers belongs to the ancient Romans. The number system they invented is based on the use of letters to represent numbers. So, they used the letters "I", "V", "L", "C", "D", and "M" in their system.

Not everyone needed so many characters to write numbers. For example, the Maya in the first millennium of our era wrote any number using only three characters: a dot, a line and an ellipse. A dot meant one, a line had a value of five, and an ellipse, being under any of these signs, increased its value by twenty times. Such minimization by no means led to a simplification of notation: to designate a particular number, one had to use long rows of symbols.

Modern familiar to usnumbers are of Arabic origin. Although the Arabs, in turn, borrowed them from the Indians, modifying them and adapting them to their writing. The character of the spelling of each of the nine can be clearly seen if they are written in an "angular" form. The number of corners of each digit corresponds to the number that this digit stands for. The forms of numbers familiar to us are more rounded. This is the influence of cursive writing: it is faster and more convenient to write numbers this way.

The decimal system, which is now widely used throughout the world, is more advanced. Instead of sticks taken from one to nine, the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 are used. New icons are not needed to indicate tens, hundreds, etc., since the same numbers are used and for writing tens, hundreds, etc. The same number has various meanings depending on the place (position) where it is recorded. Due to this property, the modern number system is called positional. The decimal positional number system allows you to write arbitrarily large natural numbers.

The peoples came to this system gradually. It originated in India in the 5th century. In the 10th century it was already owned by the Arabs, in the 10th century it reached Spain, and in the 12th century it appeared in other European countries, but it became widespread in the 16th century. For a long time, the development of the positional number system was hampered by the absence of a number and the digit zero in it. It was only after the introduction of zero that the system became perfect.

In Russia, the decimal number system began to spread in the 17th century. In 1703, the first printed textbook of mathematics was published - "Arithmetic" by L. F. Magnitsky, in which all calculations were carried out in decimal notation.

Until this date, they were written in the letters of the Slavic alphabet. The numbers from 1 to 9 were written like this:

A special sign (title) was placed over one or more letters to emphasize that the resulting entry was not a letter, not a word, but a number:

Interestingly, the numbers from 11 (one-by-ten) to 19 (nine-by-ten) were written in the same way as they were said. That is, the "number" of units was put before the "number" of tens.

In some countries, number systems with other bases -5, 12, 20, 60 were used. For example, the ancient Babylonian number system was sexagesimal. Traces of this system are now preserved in units of time:

1 h=60 min, 1 min=60 s.

An example of a non-positional number system without zero is the Roman system. The numbers are written using next digits:

I=1, V=5, X=10, L=50, C=100, D=500, M=1000

If the smaller number comes after the larger one, then it is added to the larger one: ХV = 15, ХVI = 16. If the smaller number comes before the larger one, then it is subtracted from the larger one: IV = 4, I X = 40, XC = 90, CD = 400, CM = 900. In other cases, the subtraction rule does not apply. Numbers from 1 to 21 are denoted as follows:

I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII, XIX, XX, XXI.

Using the Roman numeral system, we write down the year of publication of "Arithmetic" by L.F. Magnitsky-MDCCIII. This is 1000+500+200+3=1703.

The Roman numbering system is still used today to designate centuries, chapters in books, etc.

Electronic computers use a binary number system, in which there are only two digits 0 and 1. For example, let's write numbers from 0 to 9 in two systems:

Addition and multiplication tables for single-digit binary numbers are very simple:

Here are some examples of addition, subtraction and multiplication in binary:

Task.

On one of the old streets of Moscow there are two houses, on the facades of which the dates of their construction are indicated:

MDCCCCV and MDCCCLXXXXIX

What year was each house built?

Answer.

The first house was built in 1905.

The second house was built in 1899.

Here is what tells us about the origin of numbers :

"The oldest numbers known to us are the numbers of the Babylonians and Egyptians. The Babylonian Ts. (2nd millennium BC - the beginning of our era) are cuneiform characters for the numbers 1, 10, 100 (or only for 1 and 10 ), all other natural numbers are written by combining them.In the Egyptian hieroglyphic numbering (its appearance dates back to 2500 - 3000 BC), there were separate signs for denoting units of decimal places (up to 107). Egyptian hieroglyphic numberings are Phoenician, Syriac, Palmyrene, Greek, Attic, or Herodian."

"The emergence of Attic numbering dates back to the 6th century BC: numbering was used in Attica until the 1st century AD, although in other Greek lands it had long before been supplanted by the more convenient alphabetic Ionian numbering, in which units, tens and hundreds were designated by letters of the alphabet.All other numbers up to 999 - by their combination (the first records of numbers in this numbering date back to the 5th century BC.) Alphabetical designation of numbers also existed among other peoples, for example, among the Arabs, Syrians, Jews, Georgians, Armenians.Old Russian numbering (which originated around the 10th century and was found until the 16th century) was also alphabetic using the Slavic Cyrillic alphabet.The most durable of the ancient digital systems was the Roman numbering, which arose among the Etruscans around 500 BC. .: it is used sometimes and now.
"The prototypes of modern C. (including zero) appeared in India, probably not later than the 5th century AD [before that, C. karoshti and numbering along with them were used in India. C. which are similar to the letters of the Brahmi alphabet, numbers from inscriptions in the Nazik (or Nasik) cave]. their other name, which has survived to this day, is “Arabic” C.) and became widespread from the 2nd half of the 15th century.

4.Conclusions:

We studied the history of the emergence of numbers and figures. Prepared a presentation on the topic of the project and showed it to the students of our school.

Used Books:

Great Soviet Encyclopedia

Internet resources:
http://en.wikipedia.or g

We all know what we use when counting Arabic numerals. However, how did they appear and reach us? The process of the emergence of Arabic numbers is very interesting and entertaining.

How did they originate?

The decimal system of the Arabic counting includes 10 basic numbers from 0 to 9. With their help, you can write a number of any size.

Before the origin of numbers, people used their fingers for counting, but one day they needed to count such a large number of objects that there were no longer enough fingers. This is how numbers were written.

The history of numbers began 5 thousand years ago in Egypt and Mesopotamia. And although these two cultural layers overlapped little with each other, their systems of calculation are very similar. Initially, stone was used for records or notches were made on wood. Subsequently, in Mesopotamia they began to use clay tablets, and in Egypt they wrote on papyrus. Appearance numbers in these cultures is different, but one thing is for sure: the artifacts found by archaeologists confirm that these were not just numbers, but mathematical operations.

The history of the origin of Arabic numerals as we know them today is rather confusing. The exact time of their occurrence is unknown, but scientists know for sure that for the first time astronomers began to use numbers. Between the 2nd and 6th centuries AD Indian astronomers learned about the Greek sexagesimal number system and adopted zero from the Greeks. Then the foundations of the Greek calculus were combined in India with the decimal system borrowed from China.

It was in India that they began to designate numbers with one character. The Indian notation was popularized by a scholar named Al-Khwarizmi, who wrote a work called "On the Indian Account". Subsequently, the book on calculus was translated into Latin, which led to the spread of the decimal system in Europe.

It is India that we today owe the emergence of Arabic numbers, which happened around the 5th century AD. e. Already in the 10th-12th centuries, Arabic numerals became known to Europe. This happened due to the capture of Spain by the Moors, who brought with them Muslim culture and Arabic books. A scientist named Sylvester, arriving in Muslim Cordoba, could gain access to such literature that Europe did not yet know. Since part of Spain was still Christian, the translation of an Indian book into Latin allowed it to be popularized in Christian Europe.

In Russia, almost until the time of Peter the Great, Old Slavonic letters were used to designate numbers. With the advent of European culture, the Arabic recording system began to take root. Since the Old Slavonic alphabet has changed significantly since ancient times, Arabic numerals have deeply entered our lives.

Arabic numerals were much more convenient than Roman numerals and quickly gained popularity. Today we use them in all areas of our activity. Take a closer look: we use numbers to watch TV, talk on the phone, get money from a bank account, measure time, buy groceries, and more. Ours without numbers modern life is simply impossible.

So why did the numbers invented in India come to be called Arabic?

In the 7th century AD, a new state was formed - the Arab Caliphate, which captured the north-west of India in its dominance. The Arabs planted their culture on these lands, but as a result, it was the achievements of Indian astronomers that gave the world decimal calculus, and the Arab scientist Al-Khwarizmi only popularized it. So it turned out that the Europeans already knew about the numbers from the Arabs.

History of numbers (presentation slides)

How do they look?

Children often have a question: why do the numbers look exactly the way we know them? What is the history of the appearance of numbers in this form, as we know them now?

Writing on paper significantly changed the original appearance of Arabic numerals. Since ancient people were forced to write numbers on clay, wood or papyrus, hand movements were difficult. It was easier to draw not rounded shapes, but lines and angles. That is why the original figures were made up of traits. Their combinations are not random: each number contained as many angles in writing as the number itself indicated. For example, in a unit we see one corner, in a deuce - two corners, etc. Partially restore the ancient style of Arabic numerals will help Digital Watch, where the designations differ significantly from uppercase and also consist of lines and angles.

We all know the numbers from 0 to 9. But how did they appear? Where did these familiar 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 come from, which we constantly use in everyday life? What are they called and why do they have such a name? Let's dive into history and find out the answers to these and many other questions.

The history of the emergence of numbers

Even in ancient times, a person needed an account. Even when there were no letters and numbers yet, when ancient man did not know what two or five were, he had to perform simple actions on the division of prey, determining the number of people for hunting, and many others.

Initially, he used his hands, and sometimes even his legs, showed on his fingers. Do you remember the saying “I know like my own 5 fingers”? It is possible that it was invented in those distant times. It was the fingers that were the first tools for counting.

Life went on as usual, everything changed, people needed some other signs besides fingers. The numbers were getting bigger, it was difficult to keep them in my head, I had to somehow label and write them down. That's how the numbers came about. Moreover, different countries came up with their own. The first were the Egyptians, then the Greeks and the Romans. Now we sometimes use Roman numerals. However, the most popular and used by us to this day are the numbers invented in India before the beginning of the 5th century.

Why are they called that

Why are the usual numbers called Arabic, because they were invented in India? And all because they received distribution precisely thanks to the Arab countries, which began to actively use them. The Arabs took the Indian numbers, changed them a little and began to actively use them. Among those who helped the world to discover the Arabic numerals we know so well were the Frenchman Alexander de Villiers, the British teacher John Halifax and the famous mathematician Fibonacci, who often traveled to the East and studied the works of Arab scientists.

The very word "number" Arabic origin. The consonant Arabic word "sifr" denotes those icons that we are used to using 0.1, 2 ... 9.

Let's take a closer look at the numbers

Number 1

Guess the riddle:

Sister with a sly nose
The account will open ... ( unit)

That's right, this is the number 1. The very first number. It's easy to write. It is with her that acquaintance with numbers always begins. Any number can be made from units, for example, 1+1=2, etc. In China, one is the beginning of everything. However, so do we. The school year starts on September 1st and the new year starts on January 1st.

The number 1 symbolizes the beginning, unity, integrity, like God, the sun, the universe, the cosmos. It is an indivisible and unique number.

Number 2

Next riddle:

Neck, tail and head
Like a swan figure ... ( two)

Number 2. Look at it carefully. She really looks like a swan. In some countries, the deuce is considered a symbol of opposites, and in some, on the contrary, a symbol of pairing. And also integrity. Millions of creations without a pair are not a whole... For example, two wings, two eyes, two ears and other parts of the body. Every family starts with two...

Often the number two is found in the literature. Remember Krylov's fables "Two Doves", "Two Dogs" or the fairy tale of the Brothers Grimm "Two Brothers", Nosov's fairy tale "Two Frosts". Two is the smallest prime number. Also the worst grade in school. In order not to get deuces, you need to study well.

Number 3

Let's solve another riddle:

What a miracle
What a number!
Every tomboy knows.
Even in our alphabet
She has a twin sister... three)

Number 3. You probably noticed that the number three is very common in many fairy tales: “Father had three sons”, “traveled for three days and three nights”, “spit three times”, “knock on wood three times”, “ clap your hands three times”, “turn around your axis three times”, “say something three times”, “three heroes”, “three wishes”, etc. It is believed that the number "three" is sacred. The number really looks like the letters of the Russian alphabet "З".

Number 4

I stand after the number 3,
And I'm a little behind the number five.
What kind of number am I?

Number 4. They say that the four is the most magical of the numbers. In most states, it is a symbol of integrity. But in Asian countries they treat it with apprehension. In life, we meet the number 4 very often: 4 seasons, 4 cardinal points, 4 natural elements, 4 times of day, etc.

Number 5

How many fingers are on the hand
And a penny in a patch,
At the starfish rays,
The beaks of five rooks,
Blades near maple leaves
And the corners of the bastion
Tell about it all
The numbers will help us... (five)

Number 5. In most schools, this is the best score! Although, for example, in Germany, the top five is put on the contrary to those who do not try hard. Where can we meet the five? For example, there are 5 continents on Earth, and the symbol Olympic Games 5 rings, and 5 fingers on the hands and feet.

Number 6

How many letters does the dragon have
And zeros for a million
Various chess pieces
Wings of three white hens,
May beetle legs
And the sides of the chest.
If we can't count ourselves
Will tell us number ... (six)

Number 6. The trickiest number. If you stand on your head, the number 6 will become a nine. A cube has 6 sides, all insects have 6 legs, many musical instruments have 6 holes - these are examples of where the number 6 occurs in life.

Number 7

How many colors are in a bright rainbow?
How many wonders of the world are there on earth?
How many hills does Moscow have?
This figure is so suitable for us to answer!

Number 7. Easy to write, reminiscent of an ax or a question mark. Perhaps everyone knows that this figure is considered the most successful. Each week has 7 days, music has 7 notes, and the rainbow has 7 colors, world civilization has 7 wonders of the world. As you can see, the number 7 is also very common in life.

I also love the number 7. folk beliefs and loves to live in fairy tales. Well, who doesn’t know such favorite fairy tales as “The Wolf and the Seven Kids”, “The Seven-Flower Flower”, “Snow White and the Seven Dwarfs”, “The Tale of the Princess and the Seven Bogatyrs”.

The most desired word in the world also contains the number 7 - Family.

Number 8

It's necessary! We carry the number
On the nose, take a look, please.
This figure plus hooks -
Points are earned...

Number 8. The number 8 is an inverted infinity sign. For many peoples, this figure is special. For example, in China, it means prosperity and wealth. The famous mathematician Pythagoras also believed that the number 8 is harmony, balance and prosperity. Do you remember what holiday we celebrate on March 8? How many hooves do two cows have? How many legs does a spider have?

Number 9

A kitten walked across the bridge
He sat on the bridge and hung his tail.
"Meow! It makes me feel better…”
The kitten has become a number ...!

Number 9. Remember, we recently studied the number 6? Is it true that the number 9 looks like it? This is the last number in the row.

Number 0

The numbers stood up like a squad,
In a friendly numerical series.
First in order role
The number will play for us ...

The number 0. This is the only number that cannot be divided by. The number zero is neither positive nor negative. Al-Khwarizmi, a medieval Persian scientist, was the first to use the figure.

We have already found out that the history of numbers and numbers is as old as the world. For the entire time of existence, figures and numbers have grown into the most various myths and legends. They are associated with many interesting facts. The most interesting of them are presented below.

1. Translated from Arabic, the word "figure" means "emptiness, zero." Agree, this is very symbolic.
2. Is it possible to write zero in Roman numerals? And here it is not. You cannot write “zero” in Roman numerals, it does not exist in nature. The Roman count starts from one.
3. Most big number at the moment - centillon. It is a 1 followed by 600 zeros. It was first written down on paper back in 1852.
4. What do you associate the number 666 with? Did you know that this is the sum of all the numbers on the casino roulette?
5. Around the world it is believed that 13 - unlucky number. In many countries, floor number "13" is skipped and the twelfth is followed by the fourteenth or, for example, 12A. But in Asian countries (China, Japan, Korea), the unlucky number is 4, so the floor is also skipped. In Italy, for some reason, another unloved number is 17.
6. On the contrary, 7 is considered to be the luckiest and most successful number.
7. The Arabs themselves write numbers from right to left, and not as we are used to doing from left to right.
8. An interesting theory of one mathematician is that the numerical value is directly related to the number of angles in writing the number. Indeed, earlier the figures were written in an angular way, they acquired their rounded habitual outlines over time.

Practical work

Mathematics and Calculus

AT modern world a person constantly uses numbers, without even thinking about their origin. Without knowledge of the past it is impossible to understand the present. Therefore, the purpose of this work is to study the history of the emergence of numbers, associated with the need to express all numbers with signs.

Page 11

MOU "Volchikhinskaya high school№2"

Altai Territory

Research

THE APPEARANCE OF NUMBERS

Performed:

Potekhina Anastasia

with. She-wolf

MOU "VSH №2", 9 "A" class

Supervisor:

Potapenko Svetlana Vladimirovna

mathematics teacher, MOU "High School No. 2"

second qualification category

She-wolf

2011

1. Introduction…………………………………………………………………………. 3

2. Research part…………………………………….……...…… 5

1. The emergence of the word "mathematics"………………………………………. 5
2. Account at primitive people……………………………………...………… 5
3. Numbers y different peoples…………………………………………….…….. 6

3.1. The appearance of numbers………………………………………………..…….. 6

3.2. Roman numeration………………………………………..……...… 11

3.3. Figures of the Russian people……………………………………….…. ...eleven

4) The world of large numbers…………………………………………………………… 12

3. Conclusion…………………………………………………………………...… .14

4. List of used literature……..…….………………....………...…. 17

INTRODUCTION

Who wants to be limited to the present,

without knowledge of the past,

he will never understand...

G.W. Leibniz

In the modern world, a person constantly uses numbers, without even thinking about their origin. Without knowledge of the past it is impossible to understand the present. Therefore, the purpose of this work is to study the history of the emergence of numbers, associated with the need to express all numbers with signs. It was decided to investigate the history of the emergence of numbers on the example of natural numbers.

The first stage of the research work was to determine the origin of the word "mathematics". After studying the literature, it became known that this word arose in Ancient Greece in V century BC.

The second stage of this work was the study of counting techniques among primitive people. It is noted that knots, pebbles, and sticks were used in counting. All these methods were not convenient, which led to the appearance of conventional signs.

At the third stage of the study, conventional signs are considered - the numbers of different peoples. It is noted that different peoples had their own images, but the initial figures were gradually turning into ours. modern numbers. A separate place is occupied by Roman numeration, based on the principles of addition and subtraction.

The appearance of numbers among the Russian people is also considered. It is noted that our ancestors first used Slavic numbering (numbers were denoted by letters) and only with XVIII centuries began to use Arabic numbers.

To solve the tasks, the following methods were used:

1. Research;
2. Interviewing;
3. Computer data processing;
4. Mathematical.

In the study of the history of the emergence of numbers, a relationship was established between the emergence of numbers and the need to express all numbers with signs. This dependence influenced the appearance of digits, which replaced other not very convenient ways of designating numbers.

Numbers are an expression of a certain amount of something. For thousands of years, people have used their fingers and toes, but this was not very convenient when indicating a large number. There was a need for a more convenient way of expressing quantity. In this way, numbers are written using special characters - numbers.

The topic "History of the emergence of numbers" is relevant in the modern world, and is very important for our development, since at present our society constantly uses numbers.

The material of this work can be recommended for use in mathematics lessons or in the classroom of a school mathematical circle as additional material in order to generate interest in subject and awakening the desire to study mathematics in students, as well as to expand their horizons.

RESEARCH PART

1. The origin of the word "mathematics"

The word "mathematics" originated in ancient Greece around V century BC. It comes from the word "mathema" - "teaching", "knowledge gained through reflection" (3, p. 10).

The ancient Greeks knew four "maths":

1. the doctrine of numbers (arithmetic);
2. music theory (harmony);
3. the doctrine of figures and measurements (geometry);
4. astronomy and astrology.

There were two directions in ancient Greek science. Representatives of the first of them, headed by Pythagoras, considered knowledge intended only for initiates. No one had the right to share their discoveries with outsiders. Representatives of the second direction, on the contrary, believed that mathematics is accessible to everyone who is capable of productive thinking. They called themselves mathematicians. The second direction won.

1. An account with primitive people

People have learned to count since time immemorial. At first they distinguished just one or many objects. Hundreds of years passed before the number 2 appeared. Counting in pairs turned out to be very convenient, and it is no coincidence that some tribes in Australia and Polynesia until recently had only two numerals: one and two, and all numbers greater than two were named in the form of a combination of these two numerals . For example, three is "one, two"; four - "two, two"; five - "two, two, one." Later, special names for numbers appeared. First for small numbers, and then for bigger and bigger ones. Number is one of the basic concepts of mathematics that allows you to express the results of counting or measuring. Fingers are always with us, then they began to count on the fingers. Thus, the most ancient and simple "calculating machine" has long been the fingers and toes (3, p. 13).

It was difficult to memorize large numbers, and therefore, in addition to the fingers and toes, other “devices” were “involved”. For example, the Peruvians used multi-colored laces with knots tied to them. Rope abacus with knots were in use in Russia, as well as in many European countries. Until now, they sometimes tie knots on handkerchiefs as a keepsake.

Serifs on sticks were used in commercial transactions. After the end of the calculations, the sticks were broken in half, one half was taken by the creditor, and the other by the debtor. The half played the role of a "receipt". In the villages they used scores in the form of notches on sticks.

At a higher stage of development, people began to use different objects when counting: they used pebbles, grains, a rope with tags. These were the first counting devices, which, in the end, led to the formation of different number systems and to the creation of modern high-speed electronic computers.

1. Numbers from different nations

The idea of ​​expressing all numbers with signs

so simple that it is because

this simplicity is hard to comprehend,

how amazing she is.

Pierre Simon Laplace (1749-1827), French astronomer, mathematician, physicist.

Numbers are conventional signs for designating numbers. The first records of numbers can be considered notches on wooden tags or bones, and later - dashes. But it is inconvenient to depict large numbers in this way, so special signs (numbers) began to be used.

1. The appearance of numbers

Until recently, there were tribes whose language contained the names of only two numbers: “one” and “two”. The natives of the islands located in the Torres Strait knew two numbers: "urapun" - one, "okoz" - two and knew how to count up to six. The islanders thought so: “okoz-okoz” - three, “okoz-okoz” - four, “okoz-okoz-urapun” - five, “okoz-okoz-okoz” - six. About numbers, starting from 7, the natives spoke "many", "many". Our ancestors, for sure, also started with this. In old proverbs and sayings such as, for example, “Seven do not expect one”, “Seven troubles - one answer”, “Seven nannies have a child without an eye”, “One with a bipod, seven with a spoon” 7 also meant “a lot”.

In ancient times, when a man wanted to show how many animals he owned, he put as many pebbles in a big bag as he had animals. The more animals, the more stones. This is where the word "calculator" came from, "calculus" in Latin means "stone"(3, p. 17).

At first they counted on their fingers. When the fingers on one hand ended, they switched to the other, and if there were not enough on both hands, they switched to the legs. Therefore, if in those days someone boasted that he had “two arms and one leg of chickens,” this meant that he had fifteen chickens, and if it was called “the whole man,” that is, two arms and two legs, then it meant twenty.

The Peruvian Incas kept track of animals and crops by tying knots on straps or cords of various lengths and colors (Fig. 1). These knots were called quipu. Some rich people accumulated several meters of this rope "account book", try it, remember in a year what 4 knots on a cord mean! Therefore, the one who tied the knots was called the rememberer.

Rice. one.

The ancient Sumerians were the first to come up with the notation of numbers. They used only two numbers. A vertical line denoted one unit, and an angle of two recumbent lines denoted ten. These lines they obtained in the form of wedges, because they wrote with a sharp stick on damp clay tablets, which were then dried and fired. This is what these boards looked like (Fig. 2).

Fig.2.

After counting by notches, people invented special symbols called numbers. They have been used to refer to various quantities any items. Different civilizations created their own numbers(4, p. 12).

So, for example, in the ancient Egyptian numbering, which originated more than 5000 years ago, there were special signs (hieroglyphs) for writing the numbers 1, 10, 100, 1000, ...: (Fig. 3).

Rice. 3.

In order to depict, for example, the integer 23145, it is enough to write two hieroglyphs in a row representing ten thousand, then three hieroglyphs for a thousand, one for a hundred, four for ten and five hieroglyphs for a unit: (Fig. 4).

Rice. 4.

This one example is enough to learn how to write numbers as they were depicted by the ancient Egyptians. This system is very simple and primitive.

In a similar way, numbers were designated on the island of Crete, located in the Mediterranean Sea. In Cretan writing, units were denoted by a vertical bar |, tens by a horizontal bar - , hundreds by a circle ◦, thousands by a ¤.

The peoples (Babylonians, Assyrians, Sumerians) who lived in the Mesopotamia Tigris and Euphrates in the period from II millennium BC BC, at first they denoted numbers using circles and semicircles of various sizes, but then they began to use only two cuneiform signs - a straight wedge (1) and lying wedge (ten). These peoples used the sexagesimal number system, for example, the number 23 was depicted as follows:   The number 60 was again denoted by the sign , for example, the number 92 was written like this: (4, p. 17).

At the beginning of our era, the Maya Indians, who lived on the Yucatan Peninsula in Central America, used a different number system - vigesimal. They denoted 1 dot, and 5 - a horizontal line. The Mayan number system also had a sign for zero. In its shape, it resembled a half-closed eye.

In ancient Greece, at first the numbers 5, 10, 100, 1000, 10000 were denoted by the letters G, H, X, M, and the number 1 - by a dash /. These symbols were used to designate   G (35), etc. Later numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, 10000, 20000 began to be denoted by the letters of the Greek alphabet, to which three more obsolete letters had to be added. To distinguish numbers from letters, a dash was placed above the letters.

The ancient Indians invented their own sign for each number. Here is how they looked (Fig.5) (4, p. 18).

Rice. 5.

However, India was cut off from other countries - thousands of kilometers of distance and high mountains lay on the way. The Arabs were the first "strangers" who borrowed numbers from the Indians and brought them to Europe. A little later, the Arabs simplified these icons, they began to look like this (Fig. 6).

Rice. 6.

They are similar to many of our numbers. The word "number" also came to us from the Arabs by inheritance. The Arabs called zero, or "empty", "sifra". Since then, the word "digit" has appeared. True, now all ten icons for writing numbers that we use are called numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

The gradual transformation of the original figures into our modern figures.

1. Roman numeration

Roman numbering is based on the principles of addition (for example, VI=V+I ) and subtraction (for example, IX=X -one). The Roman numbering system is decimal, but non-positional. Roman numerals did not come from letters. Initially, they were designated, as in many nations, with "sticks" ( I - one, X - 10 - crossed out stick, V - 5 - half of ten, one hundred - a circle with a dash inside, fifty - half of this sign, etc.).

Over time, some signs have changed: C - one hundred, L - fifty, M - one thousand, D - five hundred. For example: XL - 40, LXXX - 80, XC - 90, CDLIX - 459, CCCLXXXII - 382, ​​CMXCI - 991, MCMXCVIII - 1998, MMI - 2001 (4, p. 13).

3.3. Figures of the Russian people

Arabic numbers in Russia began to be used mainly from the 18th century. Before that, our ancestors used Slavic numbering. Titles (dashes) were placed above the letters, and then the letters denoted numbers (4, p. 15).

In one of the Russian manuscripts of the 18th century it is written: “... Know that there is a hundred and that there is a thousand, and that there is a darkness, and that there is a legion, and that there is a leodr ...; ... a hundred is ten ten, and a thousand is ten hundred, and darkness is ten thousand, and a legion is ten, and a leodrus is ten legions ... ”(4, p. 15).

The first nine numbers were written like this:

Hundreds of millions were called "decks".

The “deck” had a special designation: square brackets were placed above the letter and below the letter. For example, the number 108 was written as

Numbers from 11 to 19 were designated as follows:

The remaining numbers were written in letters from left to right, for example, the numbers 5044 or 1135 had the designation

In the given system, the notation of numbers did not go beyond thousands of millions. Such an account was called a "small account". In some manuscripts, the authors also considered the “great count”, which reached the number 10 50 . Further it was said: “And more than this the human mind should understand” (4, p. 15).

1. World of Big Numbers

How many kilometers does a person walk in his life, how many goods are produced and become unusable hourly within the city, country? How long would it take for the fastest calculator to perform a million computational operations that a modern computer performs in ... a second? How many times does the speed of a passenger jet aircraft exceed the speed of a trained athlete-pedestrian? The answers to these and thousands of similar questions are expressed in numbers that often occupy a whole line or even more in the number of their decimal places.

To reduce the notation of large numbers, a system of quantities has long been used in which each of the following is a thousand times larger than the previous one:

1000 units is just a thousand (1000 or 1000)

1000 thousand - 1 million (1 million)

1000 million - 1 billion (or billion, 1 billion)

1000 billion - 1 trillion

1000 trillion - 1 quadrillion

1000 quadrillion - 1 quintillion

1000 quintillion - 1 sextillion

1000 sextillion - 1 septillion

1000 nonillion - 1 decillion

etc. (4, p. 127).

Thus, 1 decillion will be written in the decimal system as a unit with 3 x 11 = 33 zeros:

1 000 000 000 000 000 000 000 000 000 000 000.

As Samuil Yakovlevich Marshak wrote: "In vain do they think that zero plays a small role."

When writing large numbers, a power of 10 is often used.

Note that the number of zeros of the power of 10 is always equal to its exponent:

10 1 = 10, 10 2 = 100, 10 3 = 1000, etc.

And one more thing: mathematicians around the world have long accepted that any number to the zero power is equal to one.(a 0 = 1) (4, p. 127).

Thus,

unit - 10° =1

thousand -10 3 =1 000

million -10 6 =1 000 000

billion - 10 9 = 1,000,000,000

trillion - 10 12 = 1,000,000,000,000

quadrillion - 10 15 = 1,000,000,000,000,000

quintillion - 10 18 = 1,000,000,000,000,000,000

sextillion - 10 21 = 1 000 000 000 000 000 000 000

septillion - 10 24 =1 000 000 000 000 000 000 000 000

octillion - 10 27 = 1 000 000 000 000 000 000 000 000 000

Decillion - 10 33 = 1 000 000 000 000 000 000 000 000 000 000 000

Conclusion

It is interesting to note that the word NUMBER in reverse side read as a combination of two individual words- [Ol] and [Sich], which are consonant with the two English words "All" [all] and "Search" [sought]. Therefore, this combination of Russified words of the English language "Ol Sich" in the framework of my research can be perceived as a new semantic concept, for example - "everything you are looking for", and it should be understood as "literally everything".

When doing research, I was interested in finding out how many separate words - quantitative names of numerals, which are "simple" names of numbers, in order to spell out all the numbers from 1 to 999. It turns out that only 36 separate words will be required. This category of words, which form the basic basis of the system for writing numbers in words, is traditionally divided into three types: simple non-derivative words, simple derivatives and complex derivatives. But within the framework of the method, they are all reduced to one category of quantitative names of numerals - “simple” (single-word) names of numbers.

If, by analogy with the alphabetic Alphabet, the concept of "Digital Alphabet" is introduced, then its basic basis will be ten initial (single) signs-symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. They can be called "simple" digital representations of numbers. In the writing system, they stand for a total of 9 numbers - from 1 to 9. The numeric character "0" is used in the writing system to indicate the absence of a number. To designate all other numbers exceeding the number 9, it is necessary to use a combination of initial symbols, which, in relation to the "simple" images of numbers, are "composite".

I was interviewed. The question was asked "What is the largest number you know?". With this question, I turned to classmates, students of other classes, teachers and acquaintances. The interview results were processed and presented in the form of a diagram. From which it can be seen that 40% of the respondents know the largest number trillion, 25% - a billion, 20% - a million, 10% are familiar with a quadrillion and 5% with a sextillion. These data are presented in the form of a diagram (see Annex 1). And many have never even heard of such numbers as septillion, octillion and decillion.

At the end of the work, the following conclusions can be drawn:

1. The word mathematics originated in ancient Greece in V century BC.
2. People have learned to count since time immemorial.
3. At first, fingers and toes were used for counting.
4. At a higher stage of development, people began to use different objects when counting: pebbles, grains, a rope with tags.
5. The need to designate numbers led to the formation of special characters-numbers.
6. Large numbers are also written using numbers.
7. There are various theories about the origin of numbers.

Appendix 1

LIST OF USED LITERATURE

1. Big Mathematical Encyclopedia / Yakusheva G.M. and others - M .: Philol. O-vo "SLOVO": OLMA-PRESS, 2005. - 639 p.: ill.
2. The emergence and development of mathematical science: Book. For the teacher. – M.: Enlightenment, 1987. – 159 p.: ill.
3. Sheinina O. S., Solovieva G. M. Mathematics/O. S. Sheinina, G. M. Solovieva - M .: Publishing House of NC ENAS, 2007. - 208p.
4. Encyclopedia for children. T.11. Mathematics / Chapter. ed., M.D. Aksyonova. – M.: Avanta+, 1998. - 688 p.: ill.
5. Encyclopedia. The wisdom of the millennia. - M.: OLMA-PRESS, 2004. -