natural value. Natural numbers - basics

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 acquainted with the queen of all sciences - mathematics - begin to study these numbers.

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:

Classes and ranks
quintillions		hundreds of quintillions
		tens of quintillions
		quintillions
quadrillions		hundreds of quadrillions
		tens of quadrillions
		quadrillions
trillions		hundreds of trillions
		tens of trillions
		trillions
billions		hundreds of billions
		tens of billions
		billions
millions		hundreds of millions
		tens of millions
		millions
		hundreds of thousands
		tens of thousands

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 units digit,

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 .

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

Questions and tasks

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

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

Geography (length)

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

Questions and tasks

Convert all three values to centimeters and read the resulting numbers.
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 _______

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

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

Geography (square)

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

Questions and tasks

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

Is it true that numbers are special signs for writing natural numbers? (1 - yes, 2 - no)
Is it true that 0 is the smallest natural number? (3 - yes, 4 - no)
Is it true that in the positional number system the same digit can denote different numbers? (5 - yes, 6 - no)
Is it true that a certain place in the decimal notation of numbers is called a place? (7 - yes, 8 - no)
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)
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)
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)
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)
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)
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)
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)
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 second is the strongest warrior of the tribe, etc. 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 Rus' 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 that is used 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.

Roman numeral					C
				50 fifty		500 five hundred	1000 thousand

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.

If the smaller number is after larger, then it is added to the larger one: VII- the 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).
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 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.

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

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.
Read the number, starting with high school. In this case, the number of bit units and the name of the class are called.
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.

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.
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.

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.

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.

Copper chest

To find the money of this chest (in thousand 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.

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.

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…


	Write down the numbers: five million twenty five thousand
	Write down the numbers: five billion twenty five million
	Write down the numbers: five trillion twenty five
	Write down the numbers: seventy-seven million seventy-seven thousand seven hundred seventy-seven
	Write down the numbers: seventy-seven trillion seven hundred seventy-seven thousand seven
	Write down the numbers: seventy-seven million seven hundred seventy-seven thousand seven
	Write down the numbers: one hundred twenty-three billion four hundred fifty-six million seven hundred eighty-nine thousand
	Write down the numbers: one hundred twenty-three million four hundred fifty-six thousand seven hundred eighty-nine
	Write down the numbers: three billion eleven
	Write down the numbers: three billion eleven million

Option 2


	thirty-two billion one hundred seventy-five million two hundred ninety-eight thousand three hundred forty-one		100000000 + 1000000 + 10000 + 100 + 1
	Express the number as a sum of bit terms: three hundred twenty one million forty one		30000000000 + 2000000000 + 100000000 + 70000000 + 5000000 + 200000 + 90000 + 8000 + 300 + 40 + 1
	Express the number as a sum of bit terms: 321000175298341
	Express the number as a sum of bit terms: 101010101
	Express the number as a sum of bit terms: 11111		300000000 + 20000000 + 1000000 +
	5000000 + 300000 + 20000 + 1000
	Write in decimal notation the number represented as the sum of the bit terms: 5000000 + 300 + 20 + 1		30000000000000 + 2000000000000 + 1000000000000 + 100000000 + 70000000 + 5000000 + 200000 + 90000 + 8000 + 300 + 40 + 1
	Write in decimal notation the number represented as the sum of the bit terms: 10000000000 + 2000000000 + 100000 + 10 + 9
	Write in decimal notation the number represented as the sum of the bit terms: 10000000000 + 2000000000 + 100000000 + 10000000 + 9000000
	Write in decimal notation the number represented as the sum of the bit terms: 9000000000000 + 9000000000 + 9000000 + 9000 + 9		10000 + 1000 + 100 + 10 + 1

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» 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; " THAT» 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 "V".

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:

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

In mathematics, there are several different sets of numbers: real, complex, integer, rational, irrational, ... In our Everyday life we most often use natural numbers, as we encounter them when counting and when searching, indicating the number of objects.

In contact with

What numbers are called natural

From ten digits, you can write down absolutely any existing sum of classes and ranks. Natural values are those which are used:

When counting any items (first, second, third, ... fifth, ... tenth).
When indicating the number of items (one, two, three ...)

N values are always integer and positive. There is no largest N, since the set of integer values is not limited.

Attention! Natural numbers are obtained by counting objects or by designating their quantity.

Absolutely any number can be decomposed and represented as bit terms, for example: 8.346.809=8 million+346 thousand+809 units.

Set N

The set N is in the set real, integer and positive. In the set diagram, they would be in each other, since the set of naturals is part of them.

The set of natural numbers is denoted by the letter N. This set has a beginning but no end.

There is also an extended set N, where zero is included.

smallest natural number

In most mathematical schools, the smallest value of N counted as a unit, since the absence of objects is considered empty.

But in foreign mathematical schools, for example, in French, it is considered natural. The presence of zero in the series facilitates the proof some theorems.

A set of values N that includes zero is called extended and is denoted by the symbol N0 (zero index).

Series of natural numbers

An N row is a sequence of all N sets of digits. This sequence has no end.

The peculiarity of the natural series is that the next number will differ by one from the previous one, that is, it will increase. But the meanings cannot be negative.

Attention! For the convenience of counting, there are classes and categories:

Units (1, 2, 3),
Tens (10, 20, 30),
Hundreds (100, 200, 300),
Thousands (1000, 2000, 3000),
Tens of thousands (30.000),
Hundreds of thousands (800.000),
Millions (4000000) etc.

All N

All N are in the set of real, integer, non-negative values. They are theirs integral part.

These values go to infinity, they can belong to the classes of millions, billions, quintillions, etc.

For example:

Five apples, three kittens,
Ten rubles, thirty pencils,
One hundred kilograms, three hundred books,
A million stars, three million people, etc.

Sequence in N

In different mathematical schools, one can find two intervals to which the sequence N belongs:

from zero to plus infinity, including the ends, and from one to plus infinity, including the ends, that is, all positive whole answers.

N sets of digits can be either even or odd. Consider the concept of oddness.

Odd (any odd ones end in the numbers 1, 3, 5, 7, 9.) with two have a remainder. For example, 7:2=3.5, 11:2=5.5, 23:2=11.5.

What does even N mean?

Any even sums of classes end in numbers: 0, 2, 4, 6, 8. When dividing even N by 2, there will be no remainder, that is, the result is a whole answer. For example, 50:2=25, 100:2=50, 3456:2=1728.

Important! A numerical series of N cannot consist only of even or odd values, since they must alternate: an even number is always followed by an odd number, then an even number again, and so on.

N properties

Like all other sets, N has its own special properties. Consider the properties of the N series (not extended).

The value that is the smallest and that does not follow any other is one.
N are a sequence, i.e. one natural value follows another(except for one - it is the first).
When we perform computational operations on N sums of digits and classes (add, multiply), then the answer always comes out natural meaning.
In calculations, you can use permutation and combination.
Each subsequent value cannot be less than the previous one. Also in the N series, the following law will operate: if the number A is less than B, then in the number series there will always be a C, for which the equality is true: A + C \u003d B.
If we take two natural expressions, for example, A and B, then one of the expressions will be true for them: A \u003d B, A is greater than B, A is less than B.
If A is less than B and B is less than C, then it follows that that A is less than C.
If A is less than B, then it follows that: if we add the same expression (C) to them, then A + C is less than B + C. It is also true that if these values are multiplied by C, then AC is less than AB.
If B is greater than A but less than C, then B-A is less than C-A.

Attention! All of the above inequalities are also valid in the opposite direction.

What are the components of a multiplication called?

In many simple and even complex tasks, finding the answer depends on the ability of schoolchildren.

In order to quickly and correctly multiply and be able to solve inverse problems, you need to know the components of multiplication.

15. 10=150. In this expression, 15 and 10 are factors, and 150 is a product.

Multiplication has properties that are necessary when solving problems, equations and inequalities:

Rearranging the factors does not change the final product.
To find the unknown factor, you need to divide the product by the known factor (valid for all factors).

For example: 15 . X=150. Divide the product by a known factor. 150:15=10. Let's do a check. 15 . 10=150. According to this principle, even complex linear equations(if you simplify them).

Important! The product can consist of more than just two factors. For example: 840=2 . 5. 7. 3. 4

What are natural numbers in mathematics?

Discharges and classes of natural numbers

Conclusion

Let's summarize. N is used when counting or indicating the number of items. The number of natural sets of digits is infinite, but it includes only integer and positive sums of digits and classes. Multiplication is also necessary for to count things, as well as for solving problems, equations and various inequalities.

Mathematics emerged from general philosophy around the sixth century BC. e., and from that moment began her victorious march around the world. Each stage of development introduced something new - elementary counting evolved, transformed into differential and integral calculus, centuries changed, formulas became more and more confusing, and the moment came when "the most complex mathematics began - all numbers disappeared from it." But what was the basis?

The beginning of time

Natural numbers appeared along with the first mathematical operations. Once a spine, two spines, three spines ... They appeared thanks to Indian scientists who deduced the first positional

The word "positionality" means that the location of each digit in a number is strictly defined and corresponds to its category. For example, the numbers 784 and 487 are the same numbers, but the numbers are not equivalent, since the first includes 7 hundreds, while the second only 4. The Arabs picked up the innovation of the Indians, who brought the numbers to the form that we know Now.

In ancient times, numbers were given a mystical meaning, Pythagoras believed that the number underlies the creation of the world along with the main elements - fire, water, earth, air. If we consider everything only from the mathematical side, then what is a natural number? The field of natural numbers is denoted as N and is an infinite series of numbers that are integer and positive: 1, 2, 3, … + ∞. Zero is excluded. It is mainly used for counting items and indicating order.

What is in mathematics? Peano's axioms

The field N is the base field on which elementary mathematics relies. Over time, the fields of integers, rational,

The work of the Italian mathematician Giuseppe Peano made possible the further structuring of arithmetic, achieved its formality and paved the way for further conclusions that went beyond the field N.

What is a natural number was clarified earlier in simple language, below we will consider a mathematical definition based on Peano's axioms.

One is considered a natural number.
The number that follows a natural number is a natural number.
There is no natural number before one.
If the number b follows both the number c and the number d, then c=d.
The axiom of induction, which in turn shows what a natural number is: if some statement that depends on a parameter is true for the number 1, then we assume that it also works for the number n from the field of natural numbers N. Then the statement is true for n =1 from the field of natural numbers N.

Basic operations for the field of natural numbers

Since the field N became the first for mathematical calculations, both the domains of definition and the ranges of values of a number of operations below refer to it. They are closed and not. The main difference is that closed operations are guaranteed to leave a result within the set N, no matter what numbers are involved. It is enough that they are natural. The outcome of the remaining numerical interactions is no longer so unambiguous and directly depends on what kind of numbers are involved in the expression, since it may contradict the main definition. So, closed operations:

addition - x + y = z, where x, y, z are included in the field N;
multiplication - x * y = z, where x, y, z are included in the N field;
exponentiation - x y , where x, y are included in the N field.

The remaining operations, the result of which may not exist in the context of the definition "what is a natural number", are the following:

Properties of numbers belonging to the field N

All further mathematical reasoning will be based on the following properties, the most trivial, but no less important.

The commutative property of addition is x + y = y + x, where the numbers x, y are included in the field N. Or the well-known "the sum does not change from a change in the places of the terms."
The commutative property of multiplication is x * y = y * x, where the numbers x, y are included in the field N.
The associative property of addition is (x + y) + z = x + (y + z), where x, y, z are included in the field N.
The associative property of multiplication is (x * y) * z = x * (y * z), where the numbers x, y, z are included in the field N.
distribution property - x (y + z) = x * y + x * z, where the numbers x, y, z are included in the field N.

Pythagorean table

One of the first steps in the knowledge of the entire structure of elementary mathematics by schoolchildren, after they have understood for themselves which numbers are called natural, is the Pythagorean table. It can be considered not only from the point of view of science, but also as a valuable scientific monument.

This multiplication table has undergone a number of changes over time: zero has been removed from it, and numbers from 1 to 10 denote themselves, without taking into account orders (hundreds, thousands ...). It is a table in which the headings of rows and columns are numbers, and the contents of the cells of their intersection is equal to their product.

In the practice of teaching in recent decades, there has been a need to memorize the Pythagorean table "in order", that is, memorization went first. Multiplication by 1 was excluded because the result was 1 or greater. Meanwhile, in the table with the naked eye, you can see a pattern: the product of numbers grows by one step, which is equal to the title of the line. Thus, the second factor shows us how many times we need to take the first one in order to get the desired product. This system is much more convenient than the one practiced in the Middle Ages: even understanding what a natural number is and how trivial it is, people managed to complicate their everyday counting using a system based on powers of two.

Subset as the cradle of mathematics

At the moment, the field of natural numbers N is considered only as one of the subsets of complex numbers, but this does not make them less valuable in science. A natural number is the first thing a child learns by studying himself and the world around him. One finger, two fingers ... Thanks to him, a person develops logical thinking, as well as the ability to determine the cause and deduce the effect, paving the way for great discoveries.

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.