What numbers are included in natural numbers. Integers

25.09.2019

Integers- natural numbers are numbers that are used to count objects. The set of all natural numbers is sometimes called natural side by side: 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 one, 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.

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 various 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). Various ranks and classes have 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 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. Vocabulary

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 .

Russian budget for 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 text word 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.

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, 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). AT binary system entry 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.

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- 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 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.

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 given class 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 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.

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 tasks. But there are only four tasks in this test, but they are made up of a large number 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 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 the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:

Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet been able to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.

If we turn the logic we are used to, everything falls into place. Achilles runs with constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly overtake the tortoise."

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. During the next time interval, equal to the first, Achilles will run another thousand steps, and the tortoise will crawl one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia of Zeno tells of a flying arrow:

A flying arrow is motionless, since at each moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow is at rest at different points in space, which, in fact, is movement. There is another point to be noted here. From one photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (naturally, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.

Wednesday, July 4, 2018

Very well the differences between set and multiset are described in Wikipedia. We look.

As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.

Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.

No matter how mathematicians hide behind the phrase "mind me, I'm in the house", or rather "mathematics studies abstract concepts", there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.

We studied mathematics very well and now we are sitting at the cash desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.

First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will begin to convulsively recall physics: different coins there is a different amount of dirt, the crystal structure and the arrangement of atoms of each coin is unique...

And now I have the most interest Ask: where is the boundary beyond which elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science here is not even close.

Look here. We select football stadiums with the same field area. The area of the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.

To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."

Sunday, March 18, 2018

The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but they are shamans for that, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.

Do you need proof? Open Wikipedia and try to find the "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics, the task sounds like this: "Find the sum of graphic symbols representing any number." Mathematicians cannot solve this problem, but shamans can do it elementarily.

Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let's say we have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.

1. Write down the number on a piece of paper. What have we done? We have converted the number to a number graphic symbol. This is not a mathematical operation.

2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.

3. Convert individual graphic characters to numbers. This is not a mathematical operation.

4. Add up the resulting numbers. Now that's mathematics.

The sum of the digits of the number 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that is not all.

From the point of view of mathematics, it does not matter in which number system we write the number. So, in different systems reckoning, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With a large number of 12345, I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have already done that. Let's look at the result.

As you can see, in different number systems, the sum of the digits of the same number is different. This result has nothing to do with mathematics. It's like finding the area of a rectangle in meters and centimeters would give you completely different results.

Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.

The result obtained should be considered as proof that number systems are units of measurement of numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.

What is real mathematics? This is when the result of a mathematical action does not depend on the value of the number, the unit of measure used, and on who performs this action.

Sign on the door Opens the door and says:

Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?

Female... A halo on top and an arrow down is male.

If you have such a work of design art flashing before your eyes several times a day,

Then it is not surprising that you suddenly find a strange icon in your car:

Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.

1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.

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 …

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.

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.