Designation of natural numbers (Digits and classes in the notation of numbers). Name of numbers

1. Numbers of the second ten (twenties).

2. Numbers of the first hundred.

3. Numbers of the first thousand.

4. Multi-digit numbers.

5. Number systems.

1. Numbers of the second ten (twenties)

The second ten numbers (11, 12, 13, 14, 15, 16, 17, 18, 19, 20) are two-digit numbers.

To write a two-digit number, two digits are used. The first digit on the right in a two-digit number is called the first digit or units digit, the second digit on the right is called the second digit or tens digit.

Numbers of the second ten in all mathematics textbooks for primary classes are considered separately from other two-digit numbers. This is explained by the fact that the names of the numbers of the second ten contradict the way they are written. Therefore, many children for some time confuse the order of writing the numbers in the numbers of the second ten, although they can name them correctly.

For example, when writing the number 12 (twenty-twenty) by ear, the first word the child hears is “two(a),” so he can write the numbers in that order 21, but read this entry as “twelve.”

The formation of an idea of ​​two-digit numbers is based on the concept of “digit”.

The concept of place is basic in the decimal number system. A digit is a specific place in a number’s notation in a positional number system (a digit is the position of a digit in a number’s notation).

Each position in this system has its own name and its own conditional meaning: the number in the first position on the right means the number of units in the number; the number in the second position from the right means the number of tens in the number, etc.

The numbers 1 to 9 are called significant, and zero is an insignificant digit. At the same time, its role in writing two-digit and other multi-digit numbers is very important: zero in writing a two-digit (etc.) number means that the number contains the digit indicated by zero, but there are no significant digits in it, i.e. the presence of a zero on the right in number 20, means that the number 2 should be perceived as a tens symbol, and at the same time the number contains only two whole tens; the entry 23 will mean that in addition to 2 whole tens, the number contains 3 more units, in addition to the whole tens.

The concept of "discharge" plays big role in the system of studying numbering, and is also the basis for mastering the so-called “numerical” cases of addition and subtraction, in which actions are performed with whole digits:

27 - 20 365 - 300

The ability to recognize and identify digits in numbers is the basis of the ability to decompose numbers into digit terms: 34 = 30 + 4.

For numbers in the second ten, the concept of “bit composition” coincides with the concept of “decimal composition”. For two-digit numbers containing more than one ten, these concepts do not coincide. For the number 34, the decimal composition is 3 tens and 4 ones. For the number 340, the digit composition is 300 and 40, and the decimal is 34 tens.

It is convenient to start getting acquainted with the numbers of the second ten (11-20) with the method of their formation and the name of the numbers, accompanying it first with a model on sticks, and then reading the number using the model:

In this case, remembering the names of two-digit numbers will not be difficult for children with the entry contradicting the name: 11, 13,17. (After all, in accordance with the tradition of reading in European scripts from left to right, the names of these numbers should first have the tens digit, and then the units digits!) Due to this feature of the second ten numbers, many children in the first grade get confused for a long time when writing them down hearing and reading from notes. The early introduction of symbolism plays a negative role in this case both for memorizing the names of the numbers of the second ten and for understanding their structure. To form a correct idea of ​​the structure of a two-digit number, you should always put tens on the left and ones on the right. In this way, the child will fix in the internal plane the correct image of the concept, without special verbose explanations that are not always clear to him.

At the next stage, we offer the child a correlation between the material model and the symbolic notation:

one-on-twenty three-on-twenty seven-on-twenty

Then we move on to graphical models and read numbers using a graphical model:

and then a symbolic notation of the bit composition of the numbers of the second ten:

Subsequently, at school, the concept of digit is introduced and children are introduced to the concept of “digit terms”:

37 = 30 + 7; 624 = 600 + 20 + 4.

Using the decimal model instead of the digit model to get acquainted with all two-digit numbers allows, without introducing the concept of “digit,” to introduce the child both to the method of forming these numbers, and to teach him to read a number using a model (and vice versa, to build a model based on the name of the number), and then write it down :

When children study second-order numbers, we recommend that teachers use the following types of tasks:

1) on the method of forming the numbers of the second ten:

Show me thirteen sticks. How many tens are these and how many more individual sticks?

2) on the principle of formation of a natural series of numbers:

Make a drawing for the problem and solve it orally. “There were 10 cinemas in the city. We built 1 more. How many cinemas are there in the city?”

Decrease by 1: 16, 11, 13, 20

Increase by 1:19, 18, 14, 17

Find the value of the expression: 10+ 1; 14+ 1; 18- 1;20- 1.

(In all cases, we can refer to the fact that adding 1 leads to obtaining the number of the subsequent one, and decreasing by 1 leads to obtaining the number of the previous one.)

3) to the place value of a digit in a number notation:

What does each digit in the number represent: 15, 13, 18, 11, 10,20?

(In writing the number 15, the number 1 indicates the number of tens, and the number 5 indicates the number of ones. In writing the number 20, the number 2 means that there are 2 tens in the number, and the number 0 means that there are no units in the first digit.)

4) in place of a number in a series of numbers:

Fill in the missing numbers: 12.........16 17 ... 19 20

Fill in the missing numbers: 20 ... 18 17.........13 ... 11

(When completing the task, they refer to the order of numbers when counting.)

5) for digit (decimal) composition:

10 + 3 = ... 13-3 = ... 13-10 = ...

12=10 + ... 15 = ... + 5

When performing a task, they refer to the digit (decimal) model of a number made of ten (a bunch of sticks) and units (individual sticks),

6) to compare the numbers of the second ten:

Which number is greater: 13 or 15? 14 or 17? 18 or 14? 20 or 12?

When performing a task, you can compare two models of numbers from sticks (quantitative model), or refer to the order of numbers when counting (the smaller number is called earlier when counting), or rely on the process of counting and counting (counting two units to 13 we get 15, which means 15 more than 13).

When comparing the numbers of the second ten with single-digit numbers, one should refer to the fact that all single-digit numbers are smaller than double-digit numbers:

Name the largest and smallest of these numbers: 12 6 18 10 7 20.

When comparing numbers in the second ten, it is convenient to use a ruler.

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

By comparing the lengths of the corresponding segments, the child clearly determines the placement of the comparison sign: 17< 19.

This lesson will help you get an idea of ​​the topic “Reading Multi-Digit Numbers,” which is included in the 4th grade school mathematics course. The teacher will talk about how to correctly read multi-digit numbers consisting of thousands, and how to correctly write such numbers using digits.

Introduction, acquaintance with the new class - the class of thousands

If there are a lot of objects, then when counting they use not only the counting units you know: units, tens, hundreds - but also larger ones, for example you-sya-chi. You count in the same way as simple units: one you, two you, three you, three you-sya-chi and so on.

Ten thousand is one ten thousand.

Ten ten thousand is one hundred thousand.

Ten hundred thousand is a thousand thousand, or a million.

We create a table of classes and ranks (Fig. 1).

Rice. 1. Table of classes and categories

You know that units, dozens, hundreds constitute the class of units, or first class. Units of thousands, tens of thousands and hundreds of thousands constitute the class of thousands, or the second class. Look at the table again: how many rows are there in each class? Check it out: three times in a row. Numbers of the first class: units, tens, hundreds. Second class ranks: units of thousands, tens of thousands and hundreds of thousands.

To read a multi-digit number, it is divided into classes, starting from the right by three digits, then counting as many -to units of each class, starting from the highest.

Example

2nd class - thousand class			1st class - class of units
	Tens of thousands	One thousand		Tens	One thing

Three zeros in the record indicate the presence of first-class units. The name of the class of units is not about it. Read the number from the highest class: “three hundred seventy two thousand.”

In this number we see 145 units of the second class and 312 units of the first class. We read the number from the highest class: “one hundred and forty-five thousand three-hundred and two-twenty.”

This includes 528 second-class units and 609 first-class units. Read the number: “five hundred twenty eight thousand six hundred ten.”

This number includes 60 units of the second class and 500 units of the first class. This is “sixty thousand five hundred.”

The last number includes 7 second class units and 4 first class units. The number "seven thousand wh-re."

Task 1

Divide the number into classes. Tell me how many units of each class there are in it.

Counting from the right, each number has three digits.

Among them are 5 second-class units and 400 first-class units. Chi-ta-eat: “five thousand che-re-hundred.”

There are 5 second class units and 432 first class units. I read: “five thousand four-hundred thirty-two.”

Among them are 61 second-class units and 209 first-class units. Read: “six-de-ten one thou-sha-cha two-hundred-and-nine.”

Among them are 61 second-class units and 290 first-class units. Chi-ta-eat: “six-de-syat one thou-sha-cha two-hundred de-vya-no-hundred.”

Among 500 units of the second class and 500 units of the first class. Chi-ta-eat: “five hundred thousand five hundred.”

Among 500 units of the second class and 5 units of the first class. Chi-ta-eat: “five hundred thousand five.”

Task 2

Write down the numbers:

1. One hundred eight thousand three hundred and nine

2. Thirty thousand seven hundred nine

3. Eight thousand six hundred

Solution

Multi-digit numbers are written according to class, starting from the highest. To write down a number, for example, “one hundred and eight thousand three-hundred and nine”, you need to write how many total units of the second, highest, class in number - 108, then they write down how many units of the first class there are in total in number.

For the number “thirty thousand seven hundred seven ten” we write down the number of units of the second highest class in number, there are three of them tsat, and the number of units of the first class in number, seven hundred seventy.

Among the “eight thousand six hundred” there are 8 units of the second class and six hundred units of the first class.

Task 3

Read about the numbers differently: 3754, 2900, 3970.

Solution

3754. This number can be read in different ways:

A) 3 thousand 754 units.

The name of the class of units is usually not pro-situated, so we say it like this: three thousand seven hundred five ten wh-re.

B) 3 thousand 7 hundred. 5 dec. 4 units

We named a number of units every time.

B) 37 hundred. 5 dec. 4 units

D) 37 hundred. 54 units

D) 375 des. 4 units

E) 3 thousand 75 des. 4 units

A) 2 thousand 9 hundred.

B) 2 thousand 90 des.

A) 3 thousand 9 hundred. 7 dec.

B) 3 thousand 97 des.

B) 3 thousand 9 hundred. 70 units

D) 39 hundred. 7 dec.

D) 39 hundred. 70 units

Property

A number in which there are units of different ranks can be replaced by the sum of ranks of slurs.

Task 4

For the sum of the weak numbers:

1903: 1 thousand 9 hundred. 3 units

407 020: 4 cells. thousand 0 des. thousand 7 units thousand 0 hundred 2 dec. 0 units

300 206: 3 cells. thousand 0 des. thousand 0 units thousand 2 hundred 0 dec. 6 units

164,800: 1 hundred. thousand 6 des. thousand 4 units thousand 8 hundred 0 dec. 0 units

Note: if there is a zero in the row, you don’t have to write it, since adding zero gives the same number.

If a natural number consists of one sign - one digit, then it is called single-digit, for example, the numbers 3, 5, 9 are single-digit.

If a number consists of two characters - two digits, then it is called two-digit. For example, the numbers 10, 23, 75 are two-digit.

Also, based on the number of characters in a given number, names are given to other numbers. For example: 145, 809 are three-digit numbers.

There are four-digit numbers, five-digit numbers, and so on.

To read, a multi-digit natural number is divided from right to left into groups of three digits each (the leftmost group can consist of one or two digits). These groups are called classes. Each of the three digits of the class represents a place: the ones place, the tens place, the hundreds place.

Classification starts on the right. The first three digits on the right constitute the class of units, the next three are the class of thousands, then the class of millions, then the class of billions. (see Fig.). Since the series natural numbers is infinite, then billions are followed by trillions, trillions are followed by trillions, etc.

A million is a thousand thousand, it is written using one and six zeros.

A billion is a thousand million. It is written using one and 9 zeros.

How to correctly read a multi-digit number? They begin to read a multi-digit number from left to right, take turns calling the number of units of each class and adding the name of the class. At the same time, the name of the class of units is not named, as well as the class in which all three digits are zeros.

For example, this number (42,135,308) is divided into classes like this: it has 308 units, 135 units in the class of thousands, 42 units in the class of millions. Therefore, they read it like this: 42 million 135 thousand 308.

Any natural number can be represented as a sum of digit units.

For example:

32 537 = 30 000 + 2 000 + 500 + 30 + 7

Thus, in this lesson you became acquainted with the concept of a natural number and a natural series, learned to read and classify natural multi-digit numbers, as well as sort them into ranks.

Source of abstract:: http://interneturok.ru/ru/school/matematika/4-klass/tema-3/chtenie-mnogoznachnyh-chisel?konspekt

http://znaika.ru/catalog/5-klass/matematika/Naturalnye-chisla.-Chtenie-i-zapis

Video source: http://www.youtube.com/watch?v=frHwo0rvmvM

Because decimal number system place number, then the number depends not only on the digits written in it, but also on the place where each digit is written.

Definition: The place where a digit is written in a number is called the digit of the number.

For example, a number consists of three digits: 1, 0 and 3. The place, or digit, notation system allows you to create three-digit numbers from these three digits: 103, 130, 301, 310 and two-digit numbers: 013, 031. The given numbers are arranged in order increasing: each previous number is less than the next one.

Consequently, the numbers that are used to write a number do not completely define this number, but only serve as a tool for writing it.

The number itself is constructed taking into account ranks, in which this or that digit is written, i.e. required numbers must also occupy the right place in the number record.

Rule. Places of natural numbers are named from right to left from 1 to the larger number, each digit has its own number and place in the number record.

The most commonly used numbers have up to 12 digits. Numbers with more than 12 digits belong to the group large numbers.

The number of places occupied by digits, provided that the largest digit is not 0, determines the digit capacity of the number. We can say about a number that it is: single-digit (single-digit), for example 5; two-digit (two-digit), for example 15; three-digit (three-digit), for example 551, etc.

In addition to the serial number, each of the digits has its own name: the units digit (1st), the tens digit (2nd), the hundreds digit (3rd), the units of thousands digit (4th), the tens of thousands digit (5th ) etc. Every three digits, starting from the first, are combined into classes. Every Class also has its own serial number and name.

For example, the first 3 category(from 1st to 3rd inclusive) - this is Class units with serial number 1; third Class- This Class million, it includes the 7th, 8th and 9th ranks.

Let us present the structure of the digit construction of a number, or a table of digits and classes.

The number 127 432 706 408 is twelve-digit and reads like this: one hundred twenty-seven billion four hundred thirty-two million seven hundred six thousand four hundred eight. This is a fourth grade multi-digit number. The three digits of each class are read as three-digit numbers: one hundred twenty-seven, four hundred thirty-two, seven hundred six, four hundred eight. To each class of a three-digit number the name of the class is added: “billions”, “millions”, “thousands”.

For the class of units, the name is omitted (implying “units”).

Numbers from 5th grade and above are considered large numbers. Large numbers are used only in specific branches of Knowledge (astronomy, physics, electronics, etc.).

Let us give an introduction to the names of the classes from the fifth to the ninth: the units of the 5th class are trillions, the 6th class are quadrillions, the 7th class are quintillions, the 8th class are sextillions, the 9th class are septillions.

In the names of Arabic numbers, each digit belongs to its own category, and every three digits form a class. Thus, the last digit in a number indicates the number of units in it and is called, accordingly, the ones place. The next, second from the end, digit indicates the tens (tens place), and the third from the end digit indicates the number of hundreds in the number - the hundreds place. Further, the digits are repeated in the same way in turn in each class, already denoting units, tens and hundreds in the classes of thousands, millions, and so on. If the number is small and does not have a tens or hundreds digit, it is customary to take them as zero. Classes group digits in numbers of three, often placing a period or space between classes in computing devices or records to visually separate them. This is done to make large numbers easier to read. Each class has its own name: the first three digits are the class of units, then the class of thousands, then millions, billions (or billions) and so on.

Since we use the decimal system, the basic unit of quantity is ten, or 10 1. Accordingly, as the number of digits in a number increases, the number of tens also increases: 10 2, 10 3, 10 4, etc. Knowing the number of tens, you can easily determine the class and rank of the number, for example, 10 16 is tens of quadrillions, and 3 × 10 16 is three tens of quadrillions. The decomposition of numbers into decimal components occurs in the following way - each digit is displayed in a separate term, multiplied by the required coefficient 10 n, where n is the position of the digit from left to right.
For example: 253 981=2×10 6 +5×10 5 +3×10 4 +9×10 3 +8×10 2 +1×10 1

The power of 10 is also used in writing decimal fractions: 10 (-1) is 0.1 or one tenth. In a similar way to the previous paragraph, you can also expand a decimal number, n in this case will indicate the position of the digit from the decimal point from right to left, for example: 0.347629= 3×10 (-1) +4×10 (-2) +7×10 (-3) +6×10 (-4) +2×10 (-5) +9×10 (-6 )

Names of decimal numbers. Decimal numbers read according to the last digit after the decimal point, for example 0.325 – three hundred twenty-five thousandths, where thousandths is the digit last digit 5 .

Table of names of large numbers, digits and classes

1st class unit	1st digit of the unit 2nd digit tens 3rd place hundreds	1 = 10 0 10 = 10 1 100 = 10 2
2nd class thousand	1st digit of unit of thousands 2nd digit tens of thousands 3rd category hundreds of thousands	1 000 = 10 3 10 000 = 10 4 100 000 = 10 5
3rd class millions	1st digit of unit of millions 2nd category tens of millions 3rd category hundreds of millions	1 000 000 = 10 6 10 000 000 = 10 7 100 000 000 = 10 8
4th class billions	1st digit of unit of billions 2nd category tens of billions 3rd category hundreds of billions	1 000 000 000 = 10 9 10 000 000 000 = 10 10 100 000 000 000 = 10 11
5th grade trillions	1st digit unit of trillions 2nd category tens of trillions 3rd category hundreds of trillions	1 000 000 000 000 = 10 12 10 000 000 000 000 = 10 13 100 000 000 000 000 = 10 14
6th grade quadrillions	1st digit of the quadrillion unit 2nd rank tens of quadrillions 3rd digit tens of quadrillions	1 000 000 000 000 000 = 10 15 10 000 000 000 000 000 = 10 16 100 000 000 000 000 000 = 10 17
7th grade quintillions	1st digit of quintillion unit 2nd category tens of quintillions 3rd digit hundred quintillion	1 000 000 000 000 000 000 = 10 18 10 000 000 000 000 000 000 = 10 19 100 000 000 000 000 000 000 = 10 20
8th grade sextillions	1st digit of the sextillion unit 2nd rank tens of sextillions 3rd rank hundred sextillion	1 000 000 000 000 000 000 000 = 10 21 10 000 000 000 000 000 000 000 = 10 22 1 00 000 000 000 000 000 000 000 = 10 23
9th grade septillions	1st digit of septillion unit 2nd category tens of septillions 3rd digit hundred septillion	1 000 000 000 000 000 000 000 000 = 10 24 10 000 000 000 000 000 000 000 000 = 10 25 100 000 000 000 000 000 000 000 000 = 10 26
10th grade octillion	1st digit of the octillion unit 2nd digit tens of octillions 3rd digit hundred octillion	1 000 000 000 000 000 000 000 000 000 = 10 27 10 000 000 000 000 000 000 000 000 000 = 10 28 100 000 000 000 000 000 000 000 000 000 = 10 29