>

Notation of natural numbers - Knowledge Hypermarket. Integers. Natural number series

Where does learning mathematics begin? Yes, that's right, from studying natural numbers and operations with them.Integers (fromlat. naturalis- natural; natural numbers) -numbers that occur naturally when counting (for example, 1, 2, 3, 4, 5, 6, 7, 8, 9...). The sequence of all natural numbers arranged in ascending order is called a natural series.

There are two approaches to defining natural numbers:

  1. counting (numbering) items ( first, second, third, fourth, fifth"…);
  2. natural numbers are numbers that arise when quantity designation items ( 0 items, 1 item, 2 items, 3 items, 4 items, 5 items ).

In the first case, the series of natural numbers begins with one, in the second - with zero. There is no consensus among most mathematicians on whether the first or second approach is preferable (that is, whether zero should be considered a natural number or not). The overwhelming majority of Russian sources traditionally adopt the first approach. The second approach, for example, is used in the worksNicolas Bourbaki , where the natural numbers are defined aspower finite sets .

Negative and integer (rational , real ,...) numbers are not considered natural numbers.

The set of all natural numbers usually denoted by the symbol N (fromlat. naturalis- natural). The set of natural numbers is infinite, since for any natural number n there is a natural number greater than n.

The presence of zero makes it easier to formulate and prove many theorems in natural number arithmetic, so the first approach introduces the useful concept expanded natural series , including zero. The extended series is designated N 0 or Z 0 .

TOclosed operations (operations that do not derive a result from the set of natural numbers) on natural numbers include the following arithmetic operations:

  • addition: term + term = sum;
  • multiplication: factor × factor = product;
  • exponentiation: a b , where a is the base of the degree, b is the exponent. If a and b are natural numbers, then the result will be a natural number.

Additionally, two more operations are considered (from a formal point of view, they are not operations on natural numbers, since they are not defined for allpairs of numbers (sometimes exist, sometimes not)):

  • subtraction: minuend - subtrahend = difference. In this case, the minuend must be greater than the subtrahend (or equal to it, if we consider zero to be a natural number)
  • division with remainder: dividend / divisor = (quotient, remainder). The quotient p and the remainder r from dividing a by b are defined as follows: a=p*r+b, with 0<=r

It should be noted that the operations of addition and multiplication are fundamental. In particular,

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

The natural series is usually used when counting, in which quantity notations naturally arise. Acquaintance with counting begins in early childhood. What kid avoided funny rhymes that used elements of natural counting? "One, two, three, four, five... The bunny went 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 one 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 - it is one. 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 objects arose in prehistoric times. Then the concept of “natural numbers” was supposedly formed. Its formation occurred throughout the entire process of changing a person’s worldview and the development of science and technology.

However, they could not yet think abstractly. It was difficult for them to understand what the commonality of the concepts of “three hunters” or “three trees” was. 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. It contained only the numbers 1 and 2, and the count ended with the concepts of “many”, “herd”, “crowd”, “heap”.

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

An example of this was the information that has reached us about the numerical series of the Australian tribe. They had 1 for the word “Enza”, and 2 for the word “petcheval”. The number 3 therefore sounded like “petcheval-Enza”, and 4 sounded like “petcheval-petcheval”.

Most peoples recognized fingers as the standard of counting. Further development of the abstract concept of “natural numbers” followed the path of using notches on a stick. And then it became necessary to designate a dozen with another sign. The ancient people found our way out - they began to use another stick, on which notches were made to indicate tens.

The ability to reproduce numbers expanded enormously with the advent of writing. At first, numbers were depicted as lines on clay tablets or papyrus, but gradually other writing icons began to be used. This is how Roman numerals appeared.

Much later, they appeared that 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 planets and the number of stars. You just have to learn to use degrees.

Euclid in the 3rd century BC in the book “Elements” establishes the infinity of the numerical set, and Archimedes in “Psamita” 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 set. And today we already know that natural numbers are all integers, starting from 1 to infinity. Young children, taking their first step in becoming acquainted with the queen of all sciences - mathematics - begin to study these very numbers.

1.1.Definition

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

Nowadays it is accepted decimal number system. The decimal system (or method) for writing numbers uses Arabic numerals. These are ten different numeric characters: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 .

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

1.2. Number "zero"

To indicate the absence of something, use the number " zero" or " zero". It is written using numbers 0 (zero). For example, in a box all the balls are red. How many of them are green? - Answer: zero . This means there are no green balls in the box! The number 0 may mean that something has ended. For example, Masha had 3 apples. She shared two with friends and ate one herself. So she has left 0 (zero) apples, i.e. there is not a single one left. The number 0 may mean that something did not happen. For example, the hockey match Team Russia - Team Canada ended with the score 3:0 (we read “three - zero”) in favor of the Russian team. This means that the Russian team scored 3 goals, and the Canadian team scored 0 goals and could not score a single goal. We must remember that the number zero is not a natural number.

1.3. Writing natural numbers

In the decimal way of writing a natural number, each digit can represent a different number. It depends on the place of this digit in the number record. A certain place in the notation of a natural number is called position. Therefore, the decimal number system is called positional. Consider the decimal notation of 7777 seven thousand seven hundred seventy seven. This entry contains seven thousand, seven hundred, seven tens and seven ones.

Each of the places (positions) in the decimal notation of a number is called discharge. Every three digits are combined into Class. This merging is done from right to left (from the end of the number record). Various categories and classes have their own names. The range of natural numbers is unlimited. Therefore, the number of ranks and classes is also not limited ( endlessly). Let's look at 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, the 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 digit units. The following numbers are called digit units:

1 - digit unit of units digit,

10-digit unit of tens place,

100 - hundreds digit unit,

1 000 - thousand digit unit,

10 000 is a place unit of tens of thousands,

100,000 is a place unit for hundreds of thousands,

1,000,000 is the million digit unit, etc.

A number in any of the digits shows the number of units of this digit. Thus, the number 9, in the hundreds of billions place, means that the number 38,001,102,987,000 128,425 includes nine billion (i.e., 9 times 1,000,000,000 or 9 digit units of the billions place). An empty hundreds of quintillion place means that there are no hundreds of quintillion in the given number or their number is 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. Thus, three zeros in the millions class means that the hundreds of millions, tens of millions, and units of millions are empty.

1.5. Abbreviations for 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 three trillion)

107,000,000,000,000,000 = 107 square meters. (one hundred seven quadrillion)

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

Block 1.1. Dictionary

Compile a dictionary of new terms and definitions from §1. To do this, write words from the list of terms below in the empty cells. 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-preparation

In the world of big numbers

Economy .

  1. Russia's budget for next year will be: 6328251684128 rubles.
  2. The planned expenses for this year are: 5124983252134 rubles.
  3. The country's income exceeded expenses by 1203268431094 rubles.

Questions and tasks

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

  1. To which section in each of the numbers does the digit located in the seventh position from the end of the number record belong?
  2. What number of digit units does the number 2 indicate in the entry of the first number?... in the entry of the second and third number?
  3. Name the digit 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's oceans (Mariana 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 _______

billions _______

hundreds of millions _______

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

  1. Convert the third value to millimeters and read the resulting number.
  2. For all positions in the entry 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 categories corresponding to non-zero digits in the recording of these numbers (in sq. m).
  3. In writing the third number (in sq. m), name the digit 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 the place value units for digit 2 in the second quantity notations.

Block 1.3. Dialogue with the 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 149,504 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 entry 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 (use the cursor to set “1”) → number of columns (calculate yourself). Create tables for other numbers (in the “Self-preparation” block).

Block 1.4. Big Numbers Relay


The first row of the table contains a large number. Read it. Then complete the tasks: by moving the numbers in the number record 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 classroom, 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 next number. Fill 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 with the following numbers. Fill empty cells with zeros. Read the number.

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

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

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

Line 7 . Move all digits of the number in line 6 to the right one cell. Swap the numbers in the tens of quadrillions and tens of billions 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 numbers in the quintillion and quadrillion places. Fill 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 digits from the millions and trillions classes in a number line. Read the resulting number.

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

Block 1.5. let's play

Light the flame

The playing field is a drawing of a Christmas tree. It has 24 light bulbs. But only 12 of them are connected to the power grid. To select connected lamps, you must answer the questions correctly with “Yes” or “No”. The same game can be played on a computer; the correct answer “lights” 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 represent 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. The number 543,384 is given. Is it true that the number of the highest digit units in it is 543, and the lowest digits are 384? (9 - yes, 10 - no)
  6. Is it true that in the class of billions, the highest digit unit is one hundred billion, and the lowest 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 highest digit units and the number of the lowest ones is 5? (13 - yes, 14 - no)
  8. Is it true that the highest digit unit in the trillion class is a million times larger than the highest digit unit in the million class? (15 - yes, 16 - no)
  9. Given two numbers 637,508 and 831. Is it true that the highest digit unit of the first number is 1000 times greater than the highest digit unit of the second number? (17 - yes, 18 - no)
  10. Given the number 432. Is it true that the highest digit unit of this number is 2 times larger than the lowest? (19 - yes, 20 - no)
  11. The number 100,000,000 is given. Is it true that the number of digit units in it that make up 10,000 is equal to 1000? (21 - yes, 22 - no)
  12. Is it true that before the class of trillions there is a class of quadrillions, and before this class there is a class of quintillions? (23 - yes, 24 - no)

1.6. From the history of numbers

Since ancient times, people have been faced with the need to count the number of things, compare the quantities 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, etc. Numbers were used for these purposes. Special names were invented for them. In speech they are called numerals: one, two, three, etc. are cardinal numerals, and the first, second, third are ordinal numerals. Numbers were written using special characters - numbers.

Over time there appeared number systems. These are systems that include ways to write numbers and perform various operations on them. The most ancient known number systems are the Egyptian, Babylonian, and Roman number systems. In ancient times, in Rus', letters of the alphabet with a special sign ~ (title) were used to write numbers. Currently, the decimal number system is most widespread. Binary, octal and hexadecimal number systems are widely used, especially in the computer world.

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

This is the Egyptian way of writing numbers. This is the same number in Roman numerals: CDXXV(Roman way of writing numbers) or decimal digits 425 (decimal number system). In binary notation it looks like this: 110101001 (binary or binary number system), and in octal - 651 (octal number system). In the hexadecimal number system it will be written: 1A9(hexadecimal number system). You can do it quite simply: make, like Robinson Crusoe, four hundred and twenty-five notches (or strokes) on a wooden post - 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 symbols - 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 numbers came to European countries from the Middle East and 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 number system

One of the ancient number systems that is used today is the Roman system. We present in the table the main numbers of the Roman number system and the corresponding numbers of the decimal system.

Roman numeral
C

50 fifty

500 five hundred

1000 thousand

The Roman number system is addition system. In it, unlike positional systems (for example, decimal), each digit represents 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 lower number is after larger, then it is added to the larger: 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 fifty (1000 + 100 + 50 = 1150).
  2. If the lower number is before larger, then it is subtracted from the larger: IX- number nine (9 = 10 - 1), L.M.- number nine hundred and fifty (1000 - 50 = 950).

To write large numbers, you have to use (invent) new symbols - numbers. At the same time, recording numbers turns out to be cumbersome, and it is very difficult to perform calculations with Roman numerals. Thus, the year of launch of the first artificial Earth satellite (1957) in Roman records has the form MCMLVII .

Block 1. 8. Punched card

Reading natural numbers

These tasks are checked using a map with circles. Let us explain its application. Having completed all the tasks and found the correct answers (they are indicated by the letters A, B, C, etc.), place a sheet of transparent paper on the map. Use “X” signs to mark the correct answers on it, as well as the matching mark “+”. Then lay the clear sheet over the page so that the registration marks line up. If all the “X” marks are in the gray circles on this page, then the tasks were completed correctly.

1.9. Order of reading natural numbers

When reading a natural number, proceed as follows.

  1. Mentally divide the number into triplets (classes) from right to left, from the end of the number.
  1. Starting from the junior class, from right to left (from the end of the number) write down the names of classes: units, thousands, millions, billions, trillions, quadrillions, quintillions.
  2. They read the number starting in high school. In this case, the number of bit units and the name of the class are called.
  3. If the bit contains a zero (the bit is empty), then it is not called. If all three digits of the named 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. We indicate the names of the classes in this number, starting from the end his records: 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 single-digit numbers, adding the name of the corresponding class. We do not name empty classes. We get the following number:

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

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

1.9. The order of writing natural numbers

Natural numbers are written in the following order.

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

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

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

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

Block 1.11. let's play

Dungeon Treasures

On the playing field is a drawing from Kipling's fairy tale "Mowgli". Five chests have padlocks. To open them, you need to solve problems. At the same time, by opening a wooden chest, you get one point. Opening a tin chest gives you two points, a copper chest gets three points, a silver chest gets four points, and a gold chest gets five points. The one who opens all the chests the fastest wins. 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 lowest digit units of the million 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 lowest digit units of the class of units and the number of the lowest digit units of the class of millions. Then find the sum of these numbers and add the number in the tens of millions place to the right.

  1. Copper chest

To find the money in this chest (in thousands of rubles), you need to find in the number 751305432198203 the number of the lowest digit units in the class of trillions and the number of the lowest units in the class of billions. Then find the sum of these numbers and on the right write the natural numbers of the class of units of this number in the order of their location.

  1. Silver chest

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

  1. Golden chest

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

Block 1.12. Match

Writing natural numbers. Representation of natural numbers as a sum of digit terms

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

Write the number in numbers: five million twenty five thousand

Write the number in numbers: five billion twenty five million

Write the number in numbers: five trillion twenty five

Write the number in numbers: seventy-seven million seventy-seven thousand seven hundred seventy-seven

Write the number in numbers: seventy-seven trillion seven hundred seventy-seven thousand seven

Write the number in numbers: seventy-seven million seven hundred seventy-seven thousand seven

Write the number in numbers: one hundred twenty-three billion four hundred fifty-six million seven hundred eighty-nine thousand

Write the number in numbers: one hundred twenty-three million four hundred fifty-six thousand seven hundred eighty-nine

Write the number in numbers: three billion eleven

Write the number in 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

Present the number as a sum of digit terms: three hundred twenty-one million forty-one

30000000000 + 2000000000 +

100000000 + 70000000 + 5000000 +

200000 + 90000 + 8000 + 300 + 40 + 1

Present the number as a sum of digit terms: 321000175298341

Present the number as a sum of digit terms: 101010101

Present the number as a sum of digit terms: 11111

300000000 + 20000000 + 1000000 +

5000000 + 300000 + 20000 + 1000

Write in decimal notation the number presented as a sum of digit terms: 5000000 + 300 + 20 + 1

30000000000000 + 2000000000000 + 1000000000000 + 100000000 + 70000000 + 5000000 + 200000 + 90000 + 8000 + 300 + 40 + 1

Write in decimal notation the number presented as a sum of digit terms:

10000000000 + 2000000000 + 100000 + 10 + 9

Write in decimal notation the number presented as a sum of digit terms:

10000000000 + 2000000000 + 100000000 +

10000000 + 9000000

Write in decimal notation the number presented as a sum of digit 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 “insect compound eye.” This is a complex eye consisting of individual “ocelli”. Facet test tasks are formed from individual elements indicated by numbers. Typically, facet tests contain a large number of tasks. But in this test there are only four tasks, but they are made up of a large number of elements. This is designed to teach you how to “assemble” test problems. If you can create them, you can easily cope with other facet tests.

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

« If» 1) take numbers (digit) from the table; 4) 7; 7) place it in a category; 11) billions; 1) take a number from the table; 5) 8; 7) place it in categories; 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 the remaining bits with zeros; " THAT» 26) we obtain a number equal to the time (period) of revolution of the planet Pluto around the Sun in seconds (s); " This number is equal to": 7880889600 p. In the answers it is indicated by the letter "V".

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

Facet test. Make up a number

The table contains the numbers:

If

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

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

7) place this digit(s) in the digit(s);

8) hundreds of quadrillions and tens of quadrillions;

9) tens of millions;

10) hundreds of millions;

11) billions;

12) quintillions;

13) tens of quintillions;

14) hundreds of quintillions;

15) trillion;

16) hundreds of thousands;

17) tens of thousands;

18) fill the class(es) with it (them);

19) quintillions;

20) billion;

21) fill the remaining bits with zeros;

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

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

This number is equal to:

a) 5929000000000

b) 9999900000000000000000

d) 5980000000000000000000

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

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 fingers on a hand, and they said: “I have as many nuts as there are fingers on my hand.”

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

Remember!

Integers- these are numbers, starting from 1, obtained by 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 depict one with one stick, then with two sticks - the number 2, with three - the number 3.

| — 1, || — 2, ||| — 3, ||||| — 5 …

Then special signs appeared to designate numbers - the predecessors of modern numbers. The numerals we use to write numbers originated in India approximately 1,500 years ago. The Arabs brought them to Europe, which is why they are called Arabic numerals.

There are ten numbers in total: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Using these numbers you can write any natural number.

Remember!

Natural series is a 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 greatest 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 meaning of a digit depends on its place in the number record, 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 subsequent 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 exceeds the number of all atoms (the smallest particles of matter) in the entire Universe.

This number received a special name - googol. Googol is a number with 100 zeros.

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. Using them, you can write any natural number. This notation of numbers is called decimal.

The natural series of numbers can be continued indefinitely. There is no such number that would be the last, because you can always add one to the last number and you will get a number that is already greater than the one you are looking for. In this case, they say that there is no greatest number in the natural series.

Places of natural numbers

When writing any number using digits, the place in which the digit appears in the number is critical. For example, the number 3 means: 3 units, if it appears in the last place in the number; 3 tens, if she is in the penultimate place in the number; 4 hundred if she is in third place from the end.

The last digit means the units place, the penultimate digit means the tens place, and the 3 from the end means the hundreds place.

Single and multi-digit numbers

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

The number 0 is used to denote the number zero. Zero is “not one”.

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

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

Numbers that are not single-digit are also called multi-digit.

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 on the right side make up the units class, the next three are the thousands class, and the next three are the millions class.

Million – one thousand thousand; the abbreviation million is used for recording. 1 million = 1,000,000.

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

Example of writing and reading

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

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

Read numbers from left to right. Take turns calling the number of units of each class and then adding the name of the class.