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) of writing numbers uses Arabic numerals. It's ten various characters-digits: 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. Integers written using decimal numbers.
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 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 mean different numbers. 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 ranks and classes have proper 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:
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Classes and ranks |
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quintillions |
hundreds of quintillions |
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tens of quintillions |
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quintillions |
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quadrillions |
hundreds of quadrillions |
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tens of quadrillions |
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quadrillions |
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trillions |
hundreds of trillions |
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tens of trillions |
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trillions |
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billions |
hundreds of billions |
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tens of billions |
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billions |
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millions |
hundreds of millions |
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tens of millions |
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millions |
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hundreds of thousands |
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tens of thousands |
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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 digit unit of tens of thousands place,
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 .
- Russia's budget for next year will be: 6328251684128 rubles.
- The planned expenses for this year are: 5124983252134 rubles.
- The country's income exceeded expenses by 1203268431094 rubles.
Questions and tasks
- Read all three numbers given
- Write the digits in the millions class for each of the three numbers.
- To which section in each of the numbers does the digit located in the seventh position from the end of the number record belong?
- 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?
- Name the digit 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's oceans (Mariana 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 _______
billions _______
hundreds of millions _______
- For the second number (in cm), write down the digit units corresponding to the numbers 4, 7, 5, 9 in the number notation
- Convert the third value to millimeters and read the resulting number.
- For all positions in the entry 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 quantities to square meters and read the resulting numbers.
- Name the classes and categories corresponding to non-zero digits in the recording of these numbers (in sq. m).
- In writing the third number (in sq. m), name the digit 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 the place value units for digit 2 in the second quantity notation.
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 the 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 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.
- 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 represent 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)
- 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)
- 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)
- 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)
- 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)
- 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)
- 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)
- 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)
- 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 various actions above 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 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.
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Roman numeral |
C |
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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.
- 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).
- 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. After completing all the tasks and finding 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.
- Mentally divide the number into triplets (classes) from right to left, from the end of the number.
- Beginning with junior class, from right to left (from the end of the number) write down the names of the classes: units, thousands, millions, billions, trillions, quadrillions, quintillions.
- They read the number starting in high school. In this case, the number of bit units and the name of the class are called.
- 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.
- 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.
- 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 ones. 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.
- Wooden chest
Find how much money (in thousand rubles) is in this chest. To do this you need to find total number the lowest digit units of the million 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 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.
- 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 bit units in the class of trillions and the number of the lowest bit 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.
- 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.
- 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…
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Write the number in numbers: five million twenty five thousand |
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Write the number in numbers: five billion twenty five million |
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Write the number in numbers: five trillion twenty five |
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Write the number in numbers: seventy-seven million seventy-seven thousand seven hundred seventy-seven |
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Write the number in numbers: seventy-seven trillion seven hundred seventy-seven thousand seven |
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Write the number in numbers: seventy-seven million seven hundred seventy-seven thousand seven |
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Write the number in numbers: one hundred twenty-three billion four hundred fifty-six million seven hundred eighty-nine thousand |
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Write the number in numbers: one hundred twenty-three million four hundred fifty-six thousand seven hundred eighty-nine |
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Write the number in numbers: three billion eleven |
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Write the number in numbers: three billion eleven million |
Option 2
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thirty-two billion one hundred seventy-five million two hundred ninety-eight thousand three hundred forty-one |
100000000 + 1000000 + 10000 + 100 + 1 |
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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 |
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Present the number as a sum of digit terms: 321000175298341 |
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Present the number as a sum of digit terms: 101010101 |
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Present the number as a sum of digit terms: 11111 |
300000000 + 20000000 + 1000000 + |
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5000000 + 300000 + 20000 + 1000 |
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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 |
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Write in decimal notation the number presented as a sum of digit terms: 10000000000 + 2000000000 + 100000 + 10 + 9 |
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Write in decimal notation the number presented as a sum of digit terms: 10000000000 + 2000000000 + 100000000 + 10000000 + 9000000 |
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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 there are only four problems in this test, but they are made up of large number 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 solar system Pluto;
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:
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 familiar to humans and intuitive, because they surround us since childhood. In the article below we will give a basic understanding of the meaning of natural numbers and describe the basic skills of writing and reading them. The entire theoretical part will be accompanied by examples.
Yandex.RTB R-A-339285-1
General understanding of natural numbers
At a certain stage in the development of mankind, the task of counting certain objects and designating their quantity arose, which, in turn, required finding a tool to solve this problem. Natural numbers became such a tool. It is also clear that the main purpose of natural numbers is to give an idea of the number of objects or the serial number of a specific object, if we are talking about a set.
It is logical that for a person to use natural numbers, it is necessary to have a way to perceive and reproduce them. Thus, a natural number can be voiced or depicted, which are natural ways of transmitting information.
Let's look at the basic skills of voicing (reading) and representing (writing) natural numbers.
Decimal notation of a natural number
Let us remember how the following characters are represented (we will indicate them separated by commas): 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . We call these signs numbers.
Now let’s take it as a rule that when depicting (recording) any natural number, only the indicated numbers are used without the participation of any other symbols. Let the digits when writing a natural number have the same height, are written one after another in a line and there is always a digit other than zero on the left.
Let us indicate examples of the correct recording of natural numbers: 703, 881, 13, 333, 1,023, 7, 500,001. The spacing between numbers is not always the same; this will be discussed in more detail below when studying the classes of numbers. The given examples show that when writing a natural number, all the digits from the above series do not have to be present. Some or all of them may be repeated.
Definition 1
Records of the form: 065, 0, 003, 0791 are not records of natural numbers, because On the left is the number 0.
The correct recording of a natural number, made taking into account all the described requirements, is called decimal notation of a natural number.
Quantitative meaning of natural numbers
As already mentioned, natural numbers initially carry a quantitative meaning, among other things. Natural numbers, as a numbering tool, are discussed in the topic on comparing natural numbers.
Let's proceed to natural numbers, the entries of which coincide with the entries of digits, i.e.: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 .
Let's imagine a certain object, for example, like this: Ψ. We can write down what we see 1 item. The natural number 1 is read as "one" or "one". The term "unit" also has another meaning: something that can be considered as a single whole. If there is a set, then any element of it can be designated as one. For example, out of a set of mice, any mouse is one; any flower from a set of flowers is one.
Now imagine: Ψ Ψ . We see one object and another object, i.e. in the recording it will be 2 items. The natural number 2 is read as “two”.
Further, by analogy: Ψ Ψ Ψ – 3 items (“three”), Ψ Ψ Ψ Ψ – 4 (“four”), Ψ Ψ Ψ Ψ Ψ – 5 (“five”), Ψ Ψ Ψ Ψ Ψ Ψ – 6 (“six”), Ψ Ψ Ψ Ψ Ψ Ψ Ψ – 7 (“seven”), Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ – 8 (“eight”), Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ – 9 (“ nine").
From the indicated position, the function of a natural number is to indicate quantities items.
Definition 1
If the record of a number coincides with the record of the number 0, then such a number is called "zero". Zero is not a natural number, but it is considered along with other natural numbers. Zero denotes absence, i.e. zero items means none.
Single digit natural numbers
It is an obvious fact that when writing each of the natural numbers discussed above (1, 2, 3, 4, 5, 6, 7, 8, 9), we use one sign - one digit.
Definition 2
Single digit natural number– a natural number, which is written using one sign – one digit.
There are nine single-digit natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9.
Two-digit and three-digit natural numbers
Definition 3Two-digit natural numbers- natural numbers, when writing which two signs are used - two digits. In this case, the numbers used can be either the same or different.
For example, the natural numbers 71, 64, 11 are two-digit.
Let's consider what meaning is contained in two-digit numbers. We will rely on the quantitative meaning of single-digit natural numbers that is already known to us.
Let's introduce such a concept as “ten”.
Let's imagine a set of objects that consists of nine and one more. In this case, we can talk about 1 ten (“one dozen”) objects. If you imagine one ten and one more, then we are talking about 2 tens (“two tens”). Adding one more to two tens, we get three tens. And so on: continuing to add one ten at a time, we will get four tens, five tens, six tens, seven tens, eight tens and, finally, nine tens.
Let's look at two-digit number, as a set of single-digit numbers, one of which is written on the right, the other on the left. The number on the left will indicate the number of tens in a natural number, and the number on the right will indicate the number of units. In the case where the number 0 is located on the right, then we are talking about the absence of units. The above is the quantitative meaning of two-digit natural numbers. There are 90 of them in total.
Definition 4
Three-digit natural numbers- natural numbers, when writing which three signs are used - three digits. The numbers can be different or repeated in any combination.
For example, 413, 222, 818, 750 are three-digit natural numbers.
To understand the quantitative meaning of three-digit natural numbers, we introduce the concept "a hundred".
Definition 5
One hundred (1 hundred) is a set consisting of ten tens. A hundred and another hundred make 2 hundreds. Add one more hundred and get 3 hundreds. By gradually adding one hundred at a time, we get: four hundred, five hundred, six hundred, seven hundred, eight hundred, nine hundred.
Let's consider the notation of a three-digit number itself: the single-digit natural numbers included in it are written one after another from left to right. The rightmost single digit number indicates the number of units; the next single-digit number to the left is by the number of tens; the leftmost single digit number is in the number of hundreds. If the entry contains the number 0, it indicates the absence of units and/or tens.
Thus, the three-digit natural number 402 means: 2 units, 0 tens (there are no tens that are not combined into hundreds) and 4 hundreds.
By analogy, the definition of four-digit, five-digit, and so on natural numbers is given.
Multi-digit natural numbers
From all of the above, it is now possible to move on to the definition of multi-valued natural numbers.
Definition 6
Multi-digit natural numbers– natural numbers, when writing which two or more characters are used. Multi-digit natural numbers are two-digit, three-digit, and so on numbers.
One thousand is a set that includes ten hundred; one million consists of a thousand thousand; one billion – one thousand million; one trillion – one thousand billion. Even larger sets also have names, but their use is rare.
Similar to the principle above, we can consider any multi-digit natural number as a set of single-digit natural numbers, each of which, being in a certain place, indicates the presence and number of units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions , hundreds of millions, billions and so on (from right to left, respectively).
For example, the multi-digit number 4,912,305 contains: 5 units, 0 tens, three hundreds, 2 thousand, 1 ten thousand, 9 hundred thousand and 4 million.
To summarize, we looked at the skill of grouping units into various sets (tens, hundreds, etc.) and saw that the numbers in the notation of a multi-digit natural number indicate the number of units in each of such sets.
Reading natural numbers, classes
In the theory above, we indicated the names of natural numbers. In Table 1 we indicate how to correctly use the names of single-digit natural numbers in speech and in letter writing:
| Number | Masculine | Feminine | Neuter gender |
|
1 |
One |
One |
One |
| Number | Nominative case | Genitive | Dative | Accusative | Instrumental case | Prepositional |
| 1 2 3 4 5 6 7 8 9 |
One Two Three Four Five Six Seven Eight Nine |
One Two Three Four Five Six Semi Eight Nine |
Alone Two Three Four Five Six Semi Eight Nine |
One Two Three Four Five Six Seven Eight Nine |
One Two Three Four Five Six Family Eight Nine |
About one thing About two About three About four Again About six About seven About eight About nine |
To correctly read and write two-digit numbers, you need to memorize the data in Table 2:
| Number |
Masculine, feminine and neuter gender |
| 10 11 12 13 14 15 16 17 18 19 20 30 40 50 60 70 80 90 |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Fourty Fifty Sixty Seventy Eighty Ninety |
| Number | Nominative case | Genitive | Dative | Accusative | Instrumental case | Prepositional |
| 10 11 12 13 14 15 16 17 18 19 20 30 40 50 60 70 80 90 |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Fourty Fifty Sixty Seventy Eighty Ninety |
Ten |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Magpie Fifty Sixty Seventy Eighty Ninety |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Fourty Fifty Sixty Seventy Eighty Ninety |
Ten Eleven twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Magpie Fifty sixty Seventy Eighty nineteen |
About ten About eleven About twelve About thirteen About fourteen About fifteen About sixteen About seventeen About eighteen About nineteen About twenty About thirty Oh magpie About fifty About sixty About seventy About eighty Oh ninety |
To read other two-digit natural numbers, we will use the data from both tables; we will consider this with an example. Let's say we need to read the two-digit natural number 21. This number contains 1 unit and 2 tens, i.e. 20 and 1. Turning to the tables, we read the indicated number as “twenty-one”, while the conjunction “and” between the words does not need to be pronounced. Let's say we need to use the specified number 21 in a certain sentence, indicating the number of objects in genitive case: “there are no 21 apples.” sound in in this case the pronunciation will be as follows: “there are not twenty-one apples.”
Let us give another example for clarity: the number 76, which is read as “seventy-six” and, for example, “seventy-six tons.”
| Number | Nominative | Genitive | Dative | Accusative | Instrumental case | Prepositional |
| 100 200 300 400 500 600 700 800 900 |
One hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
hundred Two hundred Three hundred Four hundred Five hundred Six hundred Semistam Eight hundred Nine hundred |
One hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
Oh hundred About two hundred About three hundred About four hundred About five hundred About six hundred About the seven hundred About eight hundred About nine hundred |
To fully read a three-digit number, we also use the data from all of the indicated tables. For example, given the natural number 305. This number corresponds to 5 units, 0 tens and 3 hundreds: 300 and 5. Taking the table as a basis, we read: “three hundred and five” or in declension by case, for example, like this: “three hundred and five meters.”
Let's read one more number: 543. According to the rules of the tables, the indicated number will sound like this: “five hundred forty-three” or in declension according to cases, for example, like this: “there are no five hundred forty-three rubles.”
Let's move on to general principle reading multi-digit natural numbers: to read a multi-digit number, you need to divide it from right to left into groups of three digits, and the leftmost group can have 1, 2 or 3 digits. Such groups are called classes.
The rightmost class is the class of units; then the next class, to the left - the class of thousands; further – the class of millions; then comes the class of billions, followed by the class of trillions. The following classes also have a name, but the natural numbers consisting of large quantity characters (16, 17 or more) are rarely used in reading; it is quite difficult to perceive them by ear.
To make the recording easier to read, classes are separated from each other by a small indentation. For example, 31,013,736, 134,678, 23,476,009,434, 2,533,467,001,222.
| Class trillion |
Class billions |
Class millions |
Class of thousands | Unit class |
| 134 | 678 | |||
| 31 | 013 | 736 | ||
| 23 | 476 | 009 | 434 | |
| 2 | 533 | 467 | 001 | 222 |
To read a multi-digit number, we call the numbers that make it up one by one (from left to right by class, adding the name of the class). The name of the class of units is not pronounced, and those classes that make up three digits 0 are also not pronounced. If one or two digits 0 are present on the left in one class, then they are not used in any way when reading. For example, 054 will be read as “fifty-four” or 001 as “one”.
Example 1
Let's look at the reading of the number 2,533,467,001,222 in detail:
We read the number 2 as a component of the class of trillions - “two”;
By adding the name of the class, we get: “two trillion”;
We read the next number, adding the name of the corresponding class: “five hundred thirty-three billion”;
We continue by analogy, reading the next class to the right: “four hundred sixty-seven million”;
In the next class we see two digits 0 located on the left. According to the above reading rules, digits 0 are discarded and do not participate in reading the record. Then we get: “one thousand”;
We read the last class of units without adding its name - “two hundred twenty-two”.
Thus, the number 2 533 467 001 222 will sound like this: two trillion five hundred thirty-three billion four hundred sixty-seven million one thousand two hundred twenty-two. Using this principle, we will read the other given numbers:
31,013,736 – thirty-one million thirteen thousand seven hundred thirty-six;
134 678 – one hundred thirty-four thousand six hundred seventy-eight;
23 476 009 434 – twenty-three billion four hundred seventy-six million nine thousand four hundred thirty-four.
Thus, the basis for correctly reading multi-digit numbers is the skill of dividing a multi-digit number into classes, knowledge of the corresponding names and understanding of the principle of reading two- and three-digit numbers.
As is already clear from all of the above, its value depends on the position at which the digit appears in the notation of a number. That is, for example, the number 3 in the natural number 314 indicates the number of hundreds, namely 3 hundreds. The number 2 is the number of tens (1 ten), and the number 4 is the number of units (4 units). In this case, we will say that the number 4 is in the ones place and is the value of the ones place in the given number. The number 1 is in the tens place and serves as the value of the tens place. The number 3 is located in the hundreds place and is the value of the hundreds place.
Definition 7
Discharge- this is the position of a digit in the notation of a natural number, as well as the value of this digit, which is determined by its position in a given number.
The categories have their own names, we have already used them above. From right to left there are digits: units, tens, hundreds, thousands, tens of thousands, etc.
For ease of remembering, you can use the following table (we indicate 15 digits):
Let us clarify this detail: the number of digits in a given multi-digit number the same as the number of characters in the number record. For example, this table contains the names of all digits for a number with 15 digits. Subsequent discharges also have names, but are used extremely rarely and are very inconvenient to hear.
With the help of such a table, it is possible to develop the skill of determining the digit by writing a given natural number into the table so that the rightmost digit is written in the units digit and then in each digit one by one. For example, let’s write the multi-digit natural number 56,402,513,674 like this:
Pay attention to the number 0, located in the tens of millions digit - it means the absence of units of this digit.
Let us also introduce the concepts of the lowest and highest digits of a multi-digit number.
Definition 8
Lowest (junior) rank of any multi-digit natural number – the units digit.
Highest (senior) category of any multi-digit natural number – the digit corresponding to the leftmost digit in the notation of a given number.
So, for example, in the number 41,781: the lowest digit is the ones digit; The highest rank is the rank of tens of thousands.
Logically it follows that it is possible to talk about the seniority of the digits relative to each other. Each subsequent digit, when moving from left to right, is lower (younger) than the previous one. And vice versa: when moving from right to left, each next digit is higher (older) than the previous one. For example, the thousands place is older than the hundreds place, but younger than the millions place.
Let us clarify that when solving some practical examples, it is not the natural number itself that is used, but the sum of the digit terms of a given number.
Briefly about the decimal number system
Definition 9Notation– a method of writing numbers using signs.
Positional number systems– those in which the meaning of a digit in a number depends on its position in the number record.
According to this definition, we can say that, while studying natural numbers and the way they are written above, we used the positional number system. The number 10 plays a special place here. We count in tens: ten units make a ten, ten tens will unite into a hundred, etc. The number 10 serves as the base of this number system, and the system itself is also called decimal.
In addition to it, there are other number systems. For example, computer science uses the binary system. When we keep track of time, we use the sexagesimal number system.
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Natural numbers and their properties
Natural numbers are used to count objects in life. When writing any natural number, the numbers $0,1,2,3,4,5,6,7,8,9$ are used.
A sequence of natural numbers, each next number in which is $1$ greater than the previous one, forms a natural series, which begins with one (since one is the smallest natural number) and has no highest value, i.e. infinite.
Zero is not considered a natural number.
Properties of the succession relation
All properties of natural numbers and operations on them follow from four properties of succession relations, which were formulated in 1891 by D. Peano:
One is a natural number that does not follow any natural number.
Each natural number is followed by one and only one number
Every natural number other than $1$ follows one and only one natural number
The subset of natural numbers containing the number $1$, and together with each number the number following it, contains all natural numbers.
If the entry of a natural number consists of one digit, it is called single-digit (for example, $2,6.9$, etc.), if the entry consists of two digits, it is called double-digit (for example, $12,18,45$), etc. Similarly. Two-digit, three-digit, four-digit, etc. In mathematics, numbers are called multi-valued.
Property of addition of natural numbers
Commutative property: $a+b=b+a$
The sum does not change when the terms are rearranged
Combinative property: $a+ (b+c) =(a+b) +c$
To add the sum of two numbers to a number, you can first add the first term, and then, to the resulting sum, add the second term
Adding zero does not change the number, and if you add any number to zero, you get the added number.
Properties of Subtraction
Property of subtracting a sum from a number $a-(b+c) =a-b-c$ if $b+c ≤ a$
In order to subtract a sum from a number, you can first subtract the first term from this number, and then the second term from the resulting difference.
The property of subtracting a number from the sum $(a+b) -c=a+(b-c)$ if $c ≤ b$
To subtract a number from a sum, you can subtract it from one term and add another term to the resulting difference.
If you subtract zero from a number, the number will not change
If you subtract it from the number itself, you get zero
Properties of Multiplication
Communicative $a\cdot b=b\cdot a$
The product of two numbers does not change when the factors are rearranged
Conjunctive $a\cdot (b\cdot c)=(a\cdot b)\cdot c$
To multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor
When multiplied by one, the product does not change $m\cdot 1=m$
When multiplied by zero, the product is zero
When there are no parentheses in the product notation, multiplication is performed in order from left to right
Properties of multiplication relative to addition and subtraction
Distributive property of multiplication relative to addition
$(a+b)\cdot c=ac+bc$
In order to multiply a sum by a number, you can multiply each term by this number and add the resulting products
For example, $5(x+y)=5x+5y$
Distributive property of multiplication relative to subtraction
$(a-b)\cdot c=ac-bc$
In order to multiply the difference by a number, multiply the minuend and subtrahend by this number and subtract the second from the first product
For example, $5(x-y)=5x-5y$
Comparison of natural numbers
For any natural numbers $a$ and $b$, only one of three relations can be satisfied: $a=b$, $a
The number that appears earlier in the natural series is considered smaller, and the number that appears later is larger. Zero is less than any natural number.
Example 1
Compare the numbers $a$ and $555$, if it is known that there is a certain number $b$, and the following relations hold: $a
Solution: Based on the specified property, because by condition $a
in any subset of natural numbers containing at least one number there is a smallest number
In mathematics, a subset is a part of a set. A set is said to be a subset of another if each element of the subset is also an element of the larger set
Often, to compare numbers, they find their difference and compare it with zero. If the difference is greater than $0$, but the first number is greater than the second, if the difference is less than $0$, then the first number is less than the second.
Rounding natural numbers
When full precision is not needed or is not possible, numbers are rounded, that is, they are replaced by close numbers with zeros at the end.
Natural numbers are rounded to tens, hundreds, thousands, etc.
When rounding a number to tens, it is replaced by the nearest number consisting of whole tens; such a number has the digit $0$ in the units place
When rounding a number to the nearest hundred, it is replaced by the nearest number consisting of whole hundreds; such a number must have the digit $0$ in the tens and ones place. Etc
The numbers to which this is rounded are called the approximate value of the number with an accuracy of the indicated digits. For example, if you round the number $564$ to tens, we find that you can round it down and get $560$, or with an excess and get $570$.
Rule for rounding natural numbers
If to the right of the digit to which the number is rounded there is a digit $5$ or a digit greater than $5$, then $1$ is added to the digit of this digit; otherwise, this figure is left unchanged
All digits located to the right of the digit to which the number is rounded are replaced with zeros
Integers
Natural numbers definition are positive integers. Natural numbers are used to count objects and for many other purposes. These are the numbers:
This is a natural series of numbers.
Is zero a natural number? No, zero is not a natural number.
How many natural numbers are there? There is an infinite number of natural numbers.
What is the smallest natural number? One is the smallest natural number.
What is the largest natural number? It is impossible to specify it, because there is an infinite number of natural numbers.
The sum of natural numbers is a natural number. So, adding natural numbers a and b:
The product of natural numbers is a natural number. So, the product of natural numbers a and b:
c is always a natural number.
Difference of natural numbers There is not always a natural number. If the minuend is greater than the subtrahend, then the difference of the natural numbers is a natural number, otherwise it is not.
The quotient of natural numbers is not always a natural number. If for natural numbers a and b
where c is a natural number, this means that a is divisible by b. In this example, a is the dividend, b is the divisor, c is the quotient.
The divisor of a natural number is a natural number by which the first number is divisible by a whole.
Every natural number is divisible by one and itself.
Prime natural numbers are divisible only by one and themselves. Here we mean divided entirely. Example, numbers 2; 3; 5; 7 is only divisible by one and itself. These are simple natural numbers.
One is not considered a prime number.
Numbers that are greater than one and that are not prime are called composite numbers. Examples of composite numbers:
One is not considered a composite number.
The set of natural numbers is one, prime numbers and composite numbers.
The set of natural numbers is denoted by the Latin letter N.
Properties of addition and multiplication of natural numbers:
commutative property of addition
associative property of addition
(a + b) + c = a + (b + c);
commutative property of multiplication
associative property of multiplication
(ab) c = a (bc);
distributive property of multiplication
A (b + c) = ab + ac;
Whole numbers
Integers are the natural numbers, zero and the opposites of the natural numbers.
The opposite of natural numbers are negative integers, for example:
1; -2; -3; -4;...
The set of integers is denoted by the Latin letter Z.
Rational numbers
Rational numbers are whole numbers and fractions.
Any rational number can be represented as a periodic fraction. Examples:
1,(0); 3,(6); 0,(0);...
From the examples it is clear that any integer is a periodic fraction with period zero.
Any rational number can be represented as a fraction m/n, where m is an integer number,n natural number. Let's imagine the number 3,(6) from the previous example as such a fraction.
The simplest number is natural number. They are used in Everyday life for counting objects, i.e. to calculate their number and order.
What is a natural number: natural numbers name the numbers that are used to counting items or to indicate the serial number of any item from all homogeneous items.
Integers- these are numbers starting from one. They are formed naturally when counting.For example, 1,2,3,4,5... -first natural numbers.
Smallest natural number- one. There is no greatest natural number. When counting the number Zero is not used, so zero is a natural number.
Natural number series is the sequence of all natural numbers. Writing 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 one.
How many numbers are there in the natural series? The natural series is infinite; the largest natural number does not exist.
Decimal since 10 units of any digit form 1 unit of the highest digit. Positionally so how the meaning of a digit depends on its place in the number, i.e. from the category where it is written.
Classes of natural numbers.
Any natural number can be written using 10 Arabic numerals:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
To read natural numbers, they are divided, starting from the right, into groups of 3 digits each. 3 first the numbers on the right are the class of units, the next 3 are the class of thousands, then the classes of millions, billions andetc. Each of the digits of a class is called itsdischarge.
Comparison of natural numbers.
Of 2 natural numbers, the smaller is the number that is called earlier when counting. For example, number 7 less 11 (written like this:7 < 11 ). When one number is greater than the second, it is written like this:386 > 99 .
Table of digits and classes of numbers.
|
1st class unit |
1st digit of the unit 2nd digit tens 3rd place hundreds |
|
2nd class thousand |
1st digit of unit of thousands 2nd digit tens of thousands 3rd category hundreds of thousands |
|
3rd class millions |
1st digit of unit of millions 2nd category tens of millions 3rd category hundreds of millions |
|
4th class billions |
1st digit of unit of billions 2nd category tens of billions 3rd category hundreds of billions |
|
Numbers from 5th grade and above refer to large numbers. Units of the 5th class are trillions, 6th class - quadrillions, 7th class - quintillions, 8th class - sextillions, 9th class - eptillions. Basic properties of natural numbers.
Operations on natural numbers. 4. Division of natural numbers is the inverse operation of multiplication. If b ∙ c = a, That
Formulas for division: a: 1 = a a: a = 1, a ≠ 0 0: a = 0, a ≠ 0 (A∙ b) : c = (a:c) ∙ b (A∙ b) : c = (b:c) ∙ a Numerical expressions and numerical equalities. A notation where numbers are connected by action signs is numerical expression. For example, 10∙3+4; (60-2∙5):10. Records where 2 numeric expressions are combined with an equal sign are numerical equalities. Equality has left and right sides. The order of performing arithmetic operations. Adding and subtracting numbers are operations of the first degree, while multiplication and division are operations of the second degree. When numeric expression consists of actions of only one degree, they are performed sequentially from left to right. When expressions consist of actions of only the first and second degrees, then the actions are performed first second degree, and then - actions of the first degree. When there are parentheses in an expression, the actions in the parentheses are performed first. For example, 36:(10-4)+3∙5= 36:6+15 = 6+15 = 21. |
