Natural numbers are an obligatory part. Integers. natural series of numbers

Natural numbers are one of the oldest mathematical concepts.

In the distant past, people did not know numbers, and when they needed to count objects (animals, fish, etc.), they did it differently than we do now.

The number of objects was compared with parts of the body, for example, with the fingers on the hand, and they said: "I have as many nuts as there are fingers on the hand."

Over time, people realized that five nuts, five goats and five hares have a common property - their number is five.

Remember!

Integers are numbers, starting with 1, obtained when counting objects.

1, 2, 3, 4, 5…

smallest natural number — 1 .

largest natural number does not exist.

When counting, the number zero is not used. Therefore, zero is not considered a natural number.

People learned to write numbers much later than to count. First of all, they began to represent the unit with one stick, then with two sticks - the number 2, with three - the number 3.

| — 1, || — 2, ||| — 3, ||||| — 5 …

Then special signs appeared for designating numbers - the forerunners of modern numbers. The numbers we use to write numbers originated in India about 1,500 years ago. The Arabs brought them to Europe, so they are called Arabic numerals.

There are ten digits in total: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These digits can be used to write any natural number.

Remember!

natural series is the sequence of all natural numbers:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 …

In the natural series, each number is greater than the previous one by 1.

The natural series is infinite, there is no largest natural number in it.

The counting system we use is called decimal positional.

Decimal because 10 units of each digit form 1 unit of the most significant digit. Positional because the value of a digit depends on its place in the notation of a number, that is, on the digit in which it is written.

Important!

The classes following the billion are named according to the Latin names of numbers. Each next unit contains a thousand previous ones.

• 1,000 billion = 1,000,000,000,000 = 1 trillion (“three” is Latin for “three”)
• 1,000 trillion = 1,000,000,000,000,000 = 1 quadrillion (“quadra” is Latin for “four”)
• 1,000 quadrillion = 1,000,000,000,000,000,000 = 1 quintillion (“quinta” is Latin for “five”)

However, physicists have found a number that surpasses the number of all atoms (the smallest particles of matter) in the entire universe.

This number has a special name - googol. A googol is a number that has 100 zeros.

1.1 Definition

The numbers people use when counting are called natural(for example, one, two, three, ..., one hundred, one hundred and one, ..., three thousand two hundred twenty-one, ...) To write natural numbers, special signs (symbols) are used, called figures.

Nowadays accepted decimal notation. The decimal system (or way) of writing numbers uses Arabic numerals. These are ten different digit characters: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 .

Least a natural number is a number one, it written with a decimal digit - 1. The next natural number is obtained from the previous one (except one) by adding 1 (one). This addition can be done many times (an infinite number of times). It means that No greatest natural number. Therefore, it is said that the series of natural numbers is unlimited or infinite, since it has no end. Natural numbers are written using decimal digits.

1.2. The number "zero"

To indicate the absence of something, use the number " zero" or " zero". It is written with numbers. 0 (zero). For example, in a box all the balls are red. How many of them are green? - Answer: zero . So there are no green balls in the box! The number 0 can mean that something is over. For example, Masha had 3 apples. She shared two with friends, one she ate herself. So she has left 0 (zero) apples, i.e. none left. The number 0 could mean that something didn't happen. For example, a hockey match between the Russian team and the Canadian team ended with the score 3:0 (read "three - zero") in favor of the Russian team. This means that the Russian team scored 3 goals, and the Canadian team 0 goals, could not score a single goal. We must remember that zero is not a natural number.

1.3. Writing natural numbers

In the decimal way of writing a natural number, each digit can mean different numbers. It depends on the place of this digit in the notation of the number. A certain place in the notation of a natural number is called position. Therefore, the decimal notation is called positional. Consider the decimal notation 7777 of the number seven thousand seven hundred and seventy seven. There are seven thousand, seven hundred, seven tens and seven units in this entry.

Each of the places (positions) in the decimal notation of a number is called discharge. Every three digits are combined into Class. This union is performed from right to left (from the end of the number entry). Different ranks and classes have their own names. The number of natural numbers is unlimited. Therefore, the number of ranks and classes is also not limited ( endlessly). Consider the names of digits and classes using the example of a number with decimal notation

38 001 102 987 000 128 425:

So, classes, starting with the youngest, have names: units, thousands, millions, billions, trillions, quadrillions, quintillions.

1.4. Bit units

Each of the classes in the notation of natural numbers consists of three digits. Each rank has bit units. The following numbers are called bit units:

1 - digit unit of the digit of units,

10 - digit unit of the tens digit,

100 - bit unit of the hundreds digit,

1 000 - bit unit of the thousands place,

10,000 - digit unit of tens of thousands,

100,000 - bit unit of hundreds of thousands,

1,000,000 is the digit unit of the digit of millions, etc.

The number in any of the digits shows the number of units of this digit. So, the number 9, in the hundreds of billions place, means that the number 38,001,102,987,000 128,425 includes nine billion (that is, 9 times 1,000,000,000 or 9 bit units of the billions). An empty hundreds of quintillions digit means that there are no hundreds of quintillions in this number or their number is equal to zero. In this case, the number 38 001 102 987 000 128 425 can be written as follows: 038 001 102 987 000 128 425.

You can write it differently: 000 038 001 102 987 000 128 425. Zeros at the beginning of the number indicate empty high-order digits. Usually they are not written, unlike zeros inside the decimal notation, which necessarily mark empty digits. So, three zeros in the class of millions means that the digits of hundreds of millions, tens of millions and units of millions are empty.

1.5. Abbreviations in writing numbers

When writing natural numbers, abbreviations are used. Here are some examples:

1,000 = 1 thousand (one thousand)

23,000,000 = 23 million (twenty-three million)

5,000,000,000 = 5 billion (five billion)

203,000,000,000,000 = 203 trillion (two hundred and three trillion)

107,000,000,000,000,000 = 107 sqd. (one hundred seven quadrillion)

1,000,000,000,000,000,000 = 1 kw. (one quintillion)

Block 1.1. Dictionary

Compile a glossary of new terms and definitions from §1. To do this, in the empty cells, enter the words from the list of terms below. In the table (at the end of the block), indicate for each definition the number of the term from the list.

Block 1.2. Self-training

In the world of big numbers

Economy .

1. The budget of Russia for the next year will be: 6328251684128 rubles.
2. Planned expenses for this year: 5124983252134 rubles.
3. The country's revenues exceeded expenses by 1203268431094 rubles.

Questions and tasks

1. Read all three given numbers
2. Write the digits in the million class of each of the three numbers

1. Which section in each of the numbers belongs to the digit in the seventh position from the end of the notation of numbers?
2. What number of bit units does the number 2 show in the first number?... in the second and third numbers?
3. Name the bit unit for the eighth position from the end in the notation of three numbers.

Geography (length)

1. Equatorial radius of the Earth: 6378245 m
2. Equator circumference: 40075696 m
3. The greatest depth of the world ocean (Marian Trench in the Pacific Ocean) 11500 m

Questions and tasks

1. Convert all three values ​​​​to centimeters and read the resulting numbers.
2. For the first number (in cm), write down the numbers in the sections:

hundreds of thousands _______

tens of millions _______

thousands of _______

billions of _______

hundreds of millions of _______

1. For the second number (in cm), write down the bit units corresponding to the numbers 4, 7, 5, 9 in the number entry

1. Convert the third value to millimeters, read the resulting number.
2. For all positions in the record of the third number (in mm), indicate the digits and digit units in the table:

Geography (square)

1. The area of ​​the entire surface of the Earth is 510,083 thousand square kilometers.
2. The surface area of ​​sums on Earth is 148,628 thousand square kilometers.
3. The area of ​​the Earth's water surface is 361,455 thousand square kilometers.

Questions and tasks

1. Convert all three values ​​​​to square meters and read the resulting numbers.
2. Name the classes and ranks corresponding to non-zero digits in the record of these numbers (in sq. M).
3. In the entry of the third number (in sq. M), name the bit units corresponding to the numbers 1, 3, 4, 6.
4. In two entries of the second value (in sq. km. and sq. m), indicate which digits the number 2 belongs to.
5. Write down the bit units for the number 2 in the records of the second value.

Block 1.3. Dialogue with a computer.

It is known that large numbers are often used in astronomy. Let's give examples. The average distance of the Moon from the Earth is 384 thousand km. The distance of the Earth from the Sun (average) is 149504 thousand km, the Earth from Mars is 55 million km. On a computer, using the Word text editor, create tables so that each digit in the record of the indicated numbers is in a separate cell (cell). To do this, execute the commands on the toolbar: table → add table → number of rows (put “1” with the cursor) → number of columns (calculate yourself). Create tables for other numbers (block "Self-preparation").

Block 1.4. Relay of big numbers

The first row of the table contains a large number. Read it. Then complete the tasks: by moving the numbers in the number entry to the right or left, get the next numbers and read them. (Do not move the zeros at the end of the number!). In the class, the baton can be carried out by passing it to each other.

Line 2 . Move all the digits of the number in the first line to the left through two cells. Replace the numbers 5 with the number following it. Fill in empty cells with zeros. Read the number.

Line 3 . Move all the digits of the number in the second line to the right through three cells. Replace the numbers 3 and 4 in the number entry with the following numbers. Fill in empty cells with zeros. Read the number.

Line 4. Move all digits of the number in line 3 one cell to the left. Change the number 6 in the trillion class to the previous one, and in the billion class to the next number. Fill in empty cells with zeros. Read the resulting number.

Line 5 . Move all the digits of the number in line 4 one cell to the right. Replace the number 7 in the “tens of thousands” place with the previous one, and in the “tens of millions” place with the next one. Read the resulting number.

Line 6 . Move all the digits of the number in line 5 to the left after 3 cells. Change the number 8 in the hundreds of billions place to the previous one, and the number 6 in the hundreds of millions place to the next number. Fill in empty cells with zeros. Calculate the resulting number.

Line 7 . Move all the digits of the number in line 6 to the right by one cell. Swap the digits in the tens of quadrillion and tens of billion places. Read the resulting number.

Line 8 . Move all the digits of the number in line 7 to the left through one cell. Swap the digits in the quintillion and quadrillion places. Fill in empty cells with zeros. Read the resulting number.

Line 9 . Move all the digits of the number in line 8 to the right through three cells. Swap two adjacent numbers in the number row from the millions and trillions classes. Read the resulting number.

Line 10 . Move all digits of the number in line 9 one cell to the right. Read the resulting number. Highlight the numbers indicating the year of the Moscow Olympiad.

Block 1.5. let's play

Light a fire

The playing field is a picture of a Christmas tree. It has 24 bulbs. But only 12 of them are connected to the power grid. To select the connected lamps, you must correctly answer the questions with the words "Yes" or "No". The same game can be played on a computer; the correct answer “lights up” the light bulb.

1. Is it true that numbers are special signs for writing natural numbers? (1 - yes, 2 - no)
2. Is it true that 0 is the smallest natural number? (3 - yes, 4 - no)
3. Is it true that in the positional number system the same digit can denote different numbers? (5 - yes, 6 - no)
4. Is it true that a certain place in the decimal notation of numbers is called a place? (7 - yes, 8 - no)
5. Given the number 543 384. Is it true that the number of the most significant digits in it is 543, and the lowest 384? (9 - yes, 10 - no)
6. Is it true that in the class of billions, the oldest of the bit units is one hundred billion, and the youngest one is one billion? (11 - yes, 12 - no)
7. The number 458 121 is given. Is it true that the sum of the number of the most significant digits and the number of the least significant is 5? (13 - yes, 14 - no)
8. Is it true that the oldest of the trillion-class units is one million times larger than the oldest of the million-class units? (15 - yes, 16 - no)
9. Given two numbers 637508 and 831. Is it true that the most significant 1 of the first number is 1000 times the most significant 1 of the second number? (17 - yes, 18 - no)
10. The number 432 is given. Is it true that the most significant bit unit of this number is 2 times greater than the youngest one? (19 - yes, 20 - no)
11. Given the number 100,000,000. Is it true that the number of bit units that make up 10,000 in it is 1000? (21 - yes, 22 - no)
12. Is it true that the trillion class is preceded by the quadrillion class, and that the quintillion class is preceded by that class? (23 - yes, 24 - no)

1.6. From the history of numbers

Since ancient times, man has been faced with the need to count the number of things, to compare the number of objects (for example, five apples, seven arrows ...; there are 20 men and thirty women in a tribe, ...). There was also a need to establish order within a certain number of objects. For example, when hunting, the leader of the tribe goes first, the strongest warrior of the tribe comes second, and so on. For these purposes, numbers were used. Special names were invented for them. In speech, they are called numerals: one, two, three, etc. are cardinal numbers, and the first, second, third are ordinal numbers. Numbers were written using special characters - numbers.

Over time there were number systems. These are systems that include ways to write numbers and various actions on them. The oldest known number systems are the Egyptian, Babylonian, and Roman number systems. In Russia in the old days, letters of the alphabet with a special sign ~ (titlo) were used to write numbers. The decimal number system is currently the most widely used. Widely used, especially in the computer world, are binary, octal and hexadecimal number systems.

So, to write the same number, you can use different signs - numbers. So, the number four hundred and twenty-five can be written in Egyptian numerals - hieroglyphs:

This is the Egyptian way of writing numbers. The same number in Roman numerals: CDXXV(Roman way of writing numbers) or decimal digits 425 (decimal notation of numbers). In binary notation, it looks like this: 110101001 (binary or binary notation of numbers), and in octal - 651 (octal notation of numbers). In hexadecimal notation, it will be written: 1A9(hexadecimal notation). You can do it quite simply: make, like Robinson Crusoe, four hundred and twenty-five notches (or strokes) on a wooden pole - IIIIIIIII…... III. These are the very first images of natural numbers.

So, in the decimal system of writing numbers (in the decimal way of writing numbers), Arabic numerals are used. These are ten different characters - numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 . In binary, two binary digits: 0, 1; in octal - eight octal digits: 0, 1, 2, 3, 4, 5, 6, 7; in hexadecimal - sixteen different hexadecimal digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F; in sexagesimal (Babylonian) - sixty different characters - numbers, etc.)

Decimal digits came to European countries from the Middle East, Arab countries. Hence the name - Arabic numerals. But they came to the Arabs from India, where they were invented around the middle of the first millennium.

1.7. Roman numeral system

One of the ancient number systems in use today is the Roman system. We give in the table the main numbers of the Roman numeral system and the corresponding numbers of the decimal system.

The Roman numeral system is addition system. In it, unlike positional systems (for example, decimal), each digit denotes the same number. Yes, record II- denotes the number two (1 + 1 = 2), notation III- number three (1 + 1 + 1 = 3), notation XXX- the number thirty (10 + 10 + 10 = 30), etc. The following rules apply to writing numbers.

1. If the smaller number is after larger, then it is added to the larger one: VII- number seven (5 + 2 = 5 + 1 + 1 = 7), XVII- number seventeen (10 + 7 = 10 + 5 + 1 + 1 = 17), MCL- the number one thousand one hundred and fifty (1000 + 100 + 50 = 1150).
2. If the smaller number is before greater, then it is subtracted from the greater: IX- number nine (9 = 10 - 1), LM- the number nine hundred and fifty (1000 - 50 = 950).

To write large numbers, you have to use (invent) new characters - numbers. At the same time, the entries of numbers turn out to be cumbersome, it is very difficult to perform calculations with Roman numerals. So the year of the launch of the first artificial Earth satellite (1957) in Roman notation has the form MCMLVII .

Block 1. 8. Punch card

Reading natural numbers

These tasks are checked using a map with circles. Let's explain its application. After completing all the tasks and finding the correct answers (they are marked with the letters A, B, C, etc.), put a sheet of transparent paper on the card. Mark the correct answers with “X” marks on it, as well as the combination mark “+”. Then lay the transparent sheet on the page so that the alignment marks match. If all the "X" marks are in the gray circles on this page, then the tasks are completed correctly.

1.9. Reading order of natural numbers

When reading a natural number, proceed as follows.

1. Mentally break the number into triplets (classes) from right to left, from the end of the number entry.

1. Starting from the junior class, from right to left (from the end of the number entry), they write down the names of the classes: units, thousands, millions, billions, trillions, quadrillions, quintillions.
2. Read the number, starting with high school. In this case, the number of bit units and the name of the class are called.
3. If the digit is zero (the digit is empty), then it is not called. If all three digits of the called class are zeros (the digits are empty), then this class is not called.

Let's read (name) the number written in the table (see § 1), according to steps 1 - 4. Mentally divide the number 38001102987000128425 into classes from right to left: 038 001 102 987 000 128 425. Let's indicate the names of the classes in this number, starting from the end its entries are: units, thousands, millions, billions, trillions, quadrillions, quintillions. Now you can read the number, starting with the senior class. We name three-digit, two-digit and one-digit numbers, adding the name of the corresponding class. Empty classes are not named. We get the following number:

• 038 - thirty-eight quintillion
• 001 - one quadrillion
• 102 - one hundred and two trillion
• 987 - nine hundred and eighty seven billion
• 000 - do not name (do not read)
• 128 - one hundred twenty eight thousand
• 425 - four hundred and twenty five

As a result, the natural number 38 001 102 987 000 128 425 is read as follows: "thirty-eight quintillion one quadrillion one hundred and two trillion nine hundred and eighty-seven billion one hundred and twenty-eight thousand four hundred and twenty-five."

1.9. The order of writing natural numbers

Natural numbers are written in the following order.

1. Write down three digits for each class, starting with the highest class to the units digit. In this case, for the senior class of numbers, there can be two or one.
2. If the class or rank is not named, then zeros are written in the corresponding digits.

For example, number twenty five million three hundred two written in the form: 25 000 302 (thousand class is not named, therefore, zeros are written in all digits of the thousand class).

1.10. Representation of natural numbers as a sum of bit terms

Let's give an example: 7 563 429 is the decimal representation of the number seven million five hundred sixty-three thousand four hundred twenty-nine. This number contains seven million, five hundred thousand, six tens of thousands, three thousand, four hundred, two tens and nine units. It can be represented as a sum: 7,563,429 \u003d 7,000,000 + 500,000 + 60,000 + + 3,000 + 400 + 20 + 9. Such an entry is called the representation of a natural number as a sum of bit terms.

Block 1.11. let's play

Dungeon Treasures

On the playing field is a drawing for Kipling's fairy tale "Mowgli". Five chests have padlocks. To open them, you need to solve problems. At the same time, when you open a wooden chest, you get one point. When you open a tin chest, you get two points, a copper one - three points, a silver one - four, and a gold one - five. The winner is the one who opens all the chests faster. The same game can be played on a computer.

1. wooden chest

Find how much money (in thousand rubles) is in this chest. To do this, you need to find the total number of the least significant bit units of the millions class for the number: 125308453231.

1. Tin chest

Find how much money (in thousand rubles) is in this chest. To do this, in the number 12530845323 find the number of the least significant bit units of the unit class and the number of the least significant bit units of the million class. Then find the sum of these numbers and on the right attribute the number in the tens of millions place.

1. Copper chest

To find the money of this chest (in thousands of rubles), in the number 751305432198203 find the number of the lowest digit units in the trillion class and the number of the lowest digit units in the billion class. Then find the sum of these numbers and on the right assign the natural numbers of the class of units of this number in the order of their arrangement.

1. Silver chest

The money of this chest (in million rubles) will be shown by the sum of two numbers: the number of the lowest digit units of the thousands class and the average digit units of the billion class for the number 481534185491502.

1. golden chest

Given the number 800123456789123456789. If we multiply the numbers in the highest digits of all classes of this number, we get the money of this chest in million rubles.

Block 1.12. Match

Write natural numbers. Representation of natural numbers as a sum of bit terms

For each task in the left column, choose a solution from the right column. Write down the answer in the form: 1a; 2g; 3b…

Option 2

Block 1.13. Facet test

The name of the test comes from the word "compound eye of insects." This is a compound eye, consisting of separate "eyes". The tasks of the faceted test are formed from separate elements, indicated by numbers. Usually faceted tests contain a large number of items. But there are only four tasks in this test, but they are made up of a large number of elements. This is done in order to teach you how to "collect" test problems. If you can compose them, then you can easily cope with other facet tests.

Let us explain how tasks are composed using the example of the third task. It is made up of test elements numbered: 1, 4, 7, 11, 1, 5, 7, 9, 10, 16, 17, 22, 21, 25

« If a» 1) take numbers from the table (number); 4) 7; 7) place it in a category; 11) billion; 1) take a number from the table; 5) 8; 7) place it in ranks; 9) tens of millions; 10) hundreds of millions; 16) hundreds of thousands; 17) tens of thousands; 22) place the numbers 9 and 6 in the thousands and hundreds places. 21) fill in the remaining digits with zeros; " THEN» 26) we get a number equal to the time (period) of the revolution of the planet Pluto around the Sun in seconds (s); " This number is»: 7880889600 s. In the answers, it is indicated by the letter "in".

When solving problems, write the numbers in the cells of the table with a pencil.

Facet test. Make up a number

The table contains the numbers:

If a

1) take the number (numbers) from the table:

2) 4; 3) 5; 4) 7; 5) 8; 6) 9;

7) place this figure (numbers) in the category (digits);

8) hundreds of quadrillions and tens of quadrillions;

9) tens of millions;

10) hundreds of millions;

11) billion;

12) quintillions;

13) tens of quintillions;

14) hundreds of quintillions;

15) trillion;

16) hundreds of thousands;

17) tens of thousands;

18) fill the class (classes) with her (them);

19) quintillions;

20) billion;

21) fill in the remaining digits with zeros;

22) place the numbers 9 and 6 in the thousands and hundreds places;

23) we get a number equal to the mass of the Earth in tens of tons;

24) we get a number approximately equal to the volume of the Earth in cubic meters;

25) we get a number equal to the distance (in meters) from the Sun to the farthest planet of the solar system Pluto;

26) we get a number equal to the time (period) of the revolution of the planet Pluto around the Sun in seconds (s);

This number is:

a) 5929000000000

b) 999990000000000000000

d) 598000000000000000000

Solve problems:

1, 3, 6, 5, 18, 19, 21, 23

1, 6, 7, 14, 13, 12, 8, 21, 24

1, 4, 7, 11, 1, 5, 7, 10, 9, 16, 17, 22, 21, 26

1, 3, 7, 15, 1, 6, 2, 6, 18, 20, 21, 25

Answers

1, 3, 6, 5, 18, 19, 21, 23 - g

1, 6, 7, 14, 13, 12, 8, 21, 24 - b

1, 4, 7, 11, 1, 5, 7, 10, 9, 16, 17, 22, 21, 26 - in

1, 3, 7, 15, 1, 6, 2, 6, 18, 20, 21, 25 - a

Numbers are an abstract concept. They are a quantitative characteristic of objects and are real, rational, negative, integer and fractional, as well as natural.

The natural series is usually used in counting, in which quantity designations naturally arise. Acquaintance with the account begins in early childhood. What kid has avoided funny counting rhymes, in which elements of natural counting were just used? "One, two, three, four, five ... The bunny came out for a walk!" or "1, 2, 3, 4, 5, 6, 7, 8, 9, 10, the king decided to hang me..."

For any natural number, you can find another, greater than it. This set is usually denoted by the letter N and should be considered infinite in the direction of increase. But this set has a beginning - this is a unit. Although there are French natural numbers, the set of which also includes zero. But the main distinguishing features of both sets is the fact that they do not include either fractional or negative numbers.

The need to count a variety of items arose in prehistoric times. Then the concept of "natural numbers" was supposedly formed. Its formation took place throughout the entire process of changing the worldview of a person, the development of science and technology.

However, they could not yet think abstractly. It was difficult for them to understand what is the commonality of the concepts of "three hunters" or "three trees". Therefore, when indicating the number of people, one definition was used, and when indicating the same number of objects of a different kind, a completely different definition was used.

And it was extremely short. Only the numbers 1 and 2 were present in it, and the count ended with the concept of “many”, “herd”, “crowd”, “heap”.

Later, a more progressive account was formed, already wider. An interesting fact is that there were only two numbers - 1 and 2, and the following numbers were already obtained by adding.

An example of this was the information that has come down to us about the number series of the Australian tribe. They 1 denoted the word "Enza", and 2 - the word "petcheval". The number 3 therefore sounded like "petcheval-Enza", and 4 - already like "petcheval-petcheval".

Most nations recognized the fingers as the standard for counting. Further, the development of the abstract concept of "natural numbers" went along the path of using notches on a stick. And then there was a need to designate a dozen with another sign. The ancient people, our way out, began to use another stick, on which notches were made, indicating tens.

The possibilities for reproducing numbers expanded enormously with the advent of writing. At first, numbers were depicted as dashes on clay tablets or papyrus, but gradually other signs began to be used for writing. This is how Roman numerals appeared.

Much later appeared which opened up the possibility of writing numbers with a relatively small set of characters. Today it is not difficult to write down such huge numbers as the distance between the planets and the number of stars. One has only to learn how to use the degrees.

Euclid in the 3rd century BC in the book "Beginnings" establishes the infinity of the numerical set. And Archimedes in "Psamit" reveals the principles for constructing the names of arbitrarily large numbers. Almost until the middle of the 19th century, people did not face the need for a clear formulation of the concept of "natural numbers". The definition was required with the advent of the axiomatic mathematical method.

And in the 70s of the 19th century he formulated a clear definition of natural numbers based on the concept of a set. And today we already know that natural numbers are all integers, ranging from 1 to infinity. Little children, taking their first step in getting to know the queen of all sciences - mathematics - begin to study these numbers.

Integers- natural numbers are numbers that are used to count objects. The set of all natural numbers is sometimes called the natural series: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, etc.

To write natural numbers, ten digits are used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. With the help of them, you can write any natural number. This notation is called decimal.

The natural series of numbers can be continued indefinitely. There is no number that would be the last, because one can always be added to the last number and one will get a number that is already greater than the desired one. In this case, we say that there is no greatest number in the natural series.

Digits of natural numbers

In writing any number using numbers, the place on which the number stands in the number is crucial. For example, the number 3 means: 3 units if it comes last in the number; 3 tens if it will be in the number in the penultimate place; 4 hundreds, if she will be in the number in third place from the end.

The last digit means the units digit, the penultimate one - the tens digit, 3 from the end - the hundreds digit.

Single and multiple digits

If there is a 0 in any digit of the number, this means that there are no units in this digit.

The number 0 stands for zero. Zero is "none".

Zero is not a natural number. Although some mathematicians think otherwise.

If a number consists of one digit, it is called single-digit, two - two-digit, three - three-digit, etc.

Numbers that are not single digits are also called multiple digits.

Digit classes for reading large natural numbers

To read large natural numbers, the number is divided into groups of three digits, starting from the right edge. These groups are called classes.

The first three digits from the right edge make up the units class, the next three the thousands class, the next three the millions class.

A million is a thousand thousand, for the record they use the abbreviation million 1 million = 1,000,000.

A billion = a thousand million. For recording, the abbreviation billion 1 billion = 1,000,000,000 is used.

Write and Read Example

This number has 15 units in the billions class, 389 units in the millions class, zero units in the thousands class, and 286 units in the units class.

This number reads like this: 15 billion 389 million 286.

Read numbers from left to right. In turn, the number of units of each class is called and then the name of the class is added.

Natural numbers are one of the oldest mathematical concepts.

In the distant past, people did not know numbers, and when they needed to count objects (animals, fish, etc.), they did it differently than we do now.

The number of objects was compared with parts of the body, for example, with the fingers on the hand, and they said: "I have as many nuts as there are fingers on the hand."

Over time, people realized that five nuts, five goats and five hares have a common property - their number is five.

Remember!

Integers are numbers, starting with 1, obtained when counting objects.

1, 2, 3, 4, 5…

smallest natural number — 1 .

largest natural number does not exist.

When counting, the number zero is not used. Therefore, zero is not considered a natural number.

People learned to write numbers much later than to count. First of all, they began to represent the unit with one stick, then with two sticks - the number 2, with three - the number 3.

| — 1, || — 2, ||| — 3, ||||| — 5 …

Then special signs appeared for designating numbers - the forerunners of modern numbers. The numbers we use to write numbers originated in India about 1,500 years ago. The Arabs brought them to Europe, so they are called Arabic numerals.

There are ten digits in total: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These digits can be used to write any natural number.

Remember!

natural series is the sequence of all natural numbers:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 …

In the natural series, each number is greater than the previous one by 1.

The natural series is infinite, there is no largest natural number in it.

The counting system we use is called decimal positional.

Decimal because 10 units of each digit form 1 unit of the most significant digit. Positional because the value of a digit depends on its place in the notation of a number, that is, on the digit in which it is written.

Important!

The classes following the billion are named according to the Latin names of numbers. Each next unit contains a thousand previous ones.

• 1,000 billion = 1,000,000,000,000 = 1 trillion (“three” is Latin for “three”)
• 1,000 trillion = 1,000,000,000,000,000 = 1 quadrillion (“quadra” is Latin for “four”)
• 1,000 quadrillion = 1,000,000,000,000,000,000 = 1 quintillion (“quinta” is Latin for “five”)

However, physicists have found a number that surpasses the number of all atoms (the smallest particles of matter) in the entire universe.

This number has a special name - googol. A googol is a number that has 100 zeros.