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How to write the sum of bit terms. The sum of the bit terms of a natural number

The level of proficiency in the methods of oral and written calculations directly depends on the assimilation of questions of numbering by children. A certain number of hours are allotted for the study of this topic in each elementary school class. As practice shows, the time provided by the program is not always enough to develop skills.

Understanding the importance of the question, an experienced teacher will definitely include exercises related to the numbering of numbers in each lesson. In addition, he will take into account the types of these tasks and the sequence of their presentation to students.

Program Requirements

To understand what the teacher himself and his pupils should strive for, the first one must clearly know the requirements that the program puts forward in mathematics in general and in matters of numbering in particular.

  • The student must be able to form any numbers (understand how this is done) and name them - a requirement that applies to oral numbering.
  • When studying written numbering, children should learn not only to write down numbers, but also to compare them. At the same time, they rely on the knowledge of the local meaning of the digit in the notation of the number.
  • Children get acquainted with the concepts of "digit", "digit unit", "digit term" in the second grade. Starting from the same time, the terms are entered into the active dictionary of schoolchildren. But the teacher used them in mathematics lessons in the first grade, before learning the concepts.
  • To know the names of the digits, to write the number as a sum of digit terms, to use in practice such counting units as ten, one hundred, one thousand, to reproduce the sequence of any segment of the natural series of numbers - these are also the requirements of the program for the knowledge of elementary school students.

How to use assignments

The following groups of tasks will help the teacher to fully develop the skills that will eventually lead to the desired results in the development of students' computational skills.

Exercises can be used in the classroom during the repetition of the material covered, at the time of learning new things. They can be offered for homework, in extracurricular activities. Based on the material of the exercises, the teacher can organize group, frontal and individual forms of activity.

Much will depend on the arsenal of techniques and methods that the teacher owns. But the regularity of using tasks and the sequence of developing skills are the main conditions that will lead to success.

Forming numbers

Below are examples of exercises aimed at practicing understanding the formation of numbers. Their required number will depend on the level of development of the students in the class.


Name and write numbers

  1. Exercises of this type include tasks where you need to name the numbers represented by the geometric model.
  2. Name the numbers by typing them on the canvas: 967, 473, 285, 64, 3985. How many units of each category do they contain?

3. Read the text and write down each numeral in numbers: seven ... cars transported one thousand five hundred and twelve ... boxes of tomatoes. How many of these machines will be needed to transport two thousand eight hundred and eight ... of the same boxes?

4. Write the numbers in numbers. Express the values ​​in small units: 8 hundred. 4 units = …; 8 m 4 cm = ...; 4 hundred. 9 dec. =…; 4 m 9 dm = ...

Reading and comparing numbers

1. Read aloud the numbers that consist of: 41 dec. 8 units; 12 dec.; 8 dec. 8 units; 17 dec.

2. Read the numbers and select the appropriate image for them (different numbers are written on the board in one column, and models of these numbers are shown in random order in the other, students must match them.)

3. Compare the numbers: 416 ... 98; 199 ... 802; 375 ... 474.

4. 35 cm ... 3 m 6 cm; 7 m 9 cm ... 9 m 3 cm

Working with bit units

1. Express in different bit units: 3 hundred. 5 dec. 3 units = … cells. … units = … dec. … units

2. Fill in the table:

3. Write down the numbers, where the number 2 denotes the units of the first digit: 92; 502; 299; 263; 623; 872.

4. Write down a three-digit number, where the number of hundreds is three, and units - nine.

The sum of bit terms

Task examples:

  1. Read the notes on the board: 480; 700 + 70 + 7; 408; 108; 400+8; 777; 100+8; 400 + 80. Place three-digit numbers in the first column, the sum of the bit terms should be in the second column. Connect the sum with its value with an arrow.
  2. Read the numbers: 515; 84; 307; 781. Replace with the sum of bit terms.
  3. Write a 5 digit number with 3 digits.
  4. Write a six-digit number that contains one digit term.

Learning multi-digit numbers

  1. Find and underline three-digit numbers: 362, 7; 17; 107; 1001; 64; 204; 008.
  2. Write down the number that has 375 units of the first class and 79 units of the second class. Name the largest and smallest bit term.
  3. How are the numbers of each pair similar and different from each other: 8 and 708; 7 and 707; 12 and 112?

Applying a new counting unit

  1. Read the numbers and say how many tens are in each of them: 571; 358; 508; 115.
  2. How many hundreds are in each written number?
  3. Break the numbers into several groups, justifying your choice: 10; 510; 940; 137; 860; 86; 832.

Local meaning of a digit

  1. From the numbers 3; five; 6 make up all possible variants of three-digit numbers.
  2. Read the numbers: 6; 16; 260; 600. What figure is repeated in each of them? What does she mean?
  3. Find similarities and differences by comparing the numbers with each other: 520; 526; 506.

We can count quickly and correctly

Tasks of this type should include exercises that require a certain number of numbers to be arranged in ascending or descending order. You can invite children to restore the broken order of numbers, insert missing ones, remove extra numbers.

Finding the values ​​of numerical expressions

Using the knowledge of numbering, students should easily find the values ​​of expressions like: 800 - 400; 500 - 1; 204 + 40. At the same time, it will be useful to constantly ask the children what they noticed while performing the action, ask them to name one or another bit term, draw their attention to the position of the same digit in the number, etc.

All exercises are divided into groups for ease of use. Each of them can be supplemented by the teacher at his discretion. The science of mathematics is very rich in tasks of this type. Bit terms, which help to master the composition of any multi-digit number, should take a special place in the selection of tasks.

If this approach to the study of the numbering of numbers and their digit composition is used by the teacher throughout all four years of study in elementary school, then a positive result will definitely appear. Children will easily and without errors perform arithmetic calculations of any level of complexity.

A number is a mathematical concept for a quantitative description of something or a part of it, it also serves to compare the whole and parts, arrange in order. The concept of number is represented by signs or numbers in various combinations. At present, numbers from 1 to 9 and 0 are used almost everywhere. Numbers in the form of seven Latin letters have almost no use and will not be considered here.

Integers

When counting: “one, two, three ... forty-four” or arranging in turn: “first, second, third ... forty-fourth”, natural numbers are used, which are called natural numbers. This whole set is called “a series of natural numbers” and is denoted by the Latin letter N and has no end, because there is always a number even more, and the largest simply does not exist.

Digits and classes of numbers

Discharges

dozens

  • 10…90;
  • 100…900.

This shows that the digit of a number is its position in digital notation, and any value can be represented through bit terms in the form nnn = n00 + n0 + n, where n is any digit from 0 to 9.

One ten is a unit of the second digit, and one hundred is a unit of the third. Units of the first category are called simple, all the rest are composite.

For the convenience of recording and transmission, a grouping of digits into classes of three in each is used. A space is allowed between classes for readability.

Classes

First - units, contains up to 3 characters:

  • 200 + 10 +3 = 213.

Two hundred and thirteen contains the following digit terms: two hundred, one ten and three simple ones.

  • 40 + 5 = 45;

Forty-five is made up of four tens and five primes.

Second - thousand, 4 to 6 characters:

  • 679 812 = 600 000 + 70 000 + 9 000 + 800 +10 + 2.

This sum consists of the following bit terms:

  1. six hundred thousand;
  2. seventy thousand;
  3. nine thousand;
  4. eight hundred;
  5. ten;
  • 3 456 = 3000 + 400 +50 +6.

There are no terms above the fourth category.

The third - million, 7 to 9 digits:

  • 887 213 644;

This number contains nine bit terms:

  1. 800 million;
  2. 80 million;
  3. 7 million;
  4. 200 thousand;
  5. 10 thousand;
  6. 3 thousand;
  7. 6 hundreds;
  8. 4 tens;
  9. 4 units;
  • 7 891 234.

There are no terms higher than 7 digits in this number.

The fourth is billions, from 10 to 12 digits:

  • 567 892 234 976;

Five hundred sixty-seven billion eight hundred ninety-two million two hundred thirty-four thousand nine hundred seventy-six.

Bit terms of class 4 are read from left to right:

  1. units of hundreds of billions;
  2. units of tens of billions;
  3. units of billions;
  4. hundreds of millions;
  5. tens of millions;
  6. million;
  7. hundreds of thousands;
  8. tens of thousands;
  9. thousand;
  10. simple hundreds;
  11. simple tens;
  12. simple units.

The numbering of the digit of the number is made starting from the smallest, and reading - from the largest.

If there are no intermediate values ​​in the number of terms, zeros are put during recording, when pronouncing the name of the missing bits, as well as the class of units, it is not pronounced:

  • 400 000 000 004;

Four hundred billion four. Here, due to lack, the following names of ranks are not pronounced: tenth and eleventh fourth grade; ninth, eighth and seventh third and third class itself; the names of the second class and its categories, as well as hundreds and tens of units, are also not voiced.

Fifth - trillion, from 13 to 15 characters.

  • 487 789 654 427 241.

Reading on the left:

Four hundred eighty-seven trillion seven hundred eighty-nine billion six hundred fifty-four million four hundred twenty-seven two hundred and forty-one.

Sixth - quadrillion, 16-18 digits.

  • 321 546 818 492 395 953;

Three hundred twenty one quadrillion five hundred forty six trillion eight hundred eighteen billion four hundred ninety two million three hundred ninety five thousand nine hundred fifty three.

Seventh - quintillion, 19-21 signs.

  • 771 642 962 921 398 634 389.

Seven hundred seventy one quintillion six hundred forty two quadrillion nine hundred sixty two trillion nine hundred twenty one billion three hundred ninety eight million six hundred thirty four thousand three hundred eighty nine.

Eighth - sextillions, 22-24 digits.

  • 842 527 342 458 752 468 359 173

Eight hundred and forty-two sextillion five hundred twenty-seven quintillion three hundred and forty-two quadrillion four hundred and fifty-eight trillion seven hundred and fifty-two billion four hundred and sixty-eight million three hundred and fifty-nine thousand one hundred and seventy-three.

You can simply distinguish between classes by numbering, for example, the number 11 of the class contains from 31 to 33 characters when written.

But in practice, writing such a number of characters is inconvenient and most often leads to errors. Therefore, during operations with such values, the number of zeros is reduced by raising to a power. After all, it is much easier to write 10 31 than to attribute thirty-one zeros to one.


To perform some operations on natural numbers, one has to represent these natural numbers in the form sums of bit terms or, as they say, sort natural numbers into digits. No less important is the reverse process - writing a natural number by the sum of the bit terms.

In this article, we will understand in great detail, using examples, the representation of natural numbers as a sum of bit terms, and also learn how to write a natural number according to its known expansion into bits.

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Representation of a natural number as a sum of bit terms.

As you can see, the words “sum” and “terms” appear in the title of the article, therefore, for starters, we recommend that you understand the information in the article well, a general idea of ​​the addition of natural numbers. It also does not hurt to repeat the material from the discharge section, the value of the discharge of a natural number.

Let's take on faith the following statements, which will help us define bit terms.

Bit terms can only be natural numbers, the entries of which contain a single digit that is different from a digit 0 . For example, natural numbers 5 , 10 , 400 , 20 000 etc. can be bit terms, and the numbers 14 , 201 , 5 500 , 15 321 etc. - can not.

The number of bit terms of a given natural number must be equal to the number of digits in the record of this number that are different from a digit 0 . For example, a natural number 59 can be represented as the sum of two bit terms, since two digits are involved in writing this number ( 5 And 9 ) different from 0 . And the sum of the bit terms of a natural number 44 003 will consist of three terms, since the notation of a number contains three digits 4 , 4 And 3 , which are different from the number 0 .

All bit terms of a given natural number in their record contain a different number of characters.

The sum of the bit terms of a given natural number must be equal to the given number.

Now we can define bit terms.

Definition.

Discharge terms given natural number are such natural numbers,

  • in the record of which there is only one digit, different from the digit 0 ;
  • the number of which is equal to the number of digits in a given natural number that are different from the digit 0 ;
  • records of which consist of a different number of characters;
  • the sum of which is equal to the given natural number.

From the above definition it follows that single-digit natural numbers, as well as multi-digit natural numbers, the entries of which consist entirely of digits 0 , with the exception of the first digit on the left, do not decompose into a sum of bit terms, since they themselves are bit terms of some natural numbers. The remaining natural numbers can be represented as the sum of bit terms.

It remains to deal with the representation of natural numbers as a sum of bit terms.

To do this, you need to remember that natural numbers are inherently related to the number of certain objects, while in the record of the number, the values ​​\u200b\u200bof the digits set the corresponding numbers of ones, tens, hundreds, thousands, tens of thousands, and so on. For example, a natural number 48 answers 4 dozens and 8 units, and the number 105 070 corresponds 1 a hundred thousand 5 thousands and 7 dozens. Then, by virtue of the sense of addition of natural numbers, the following equalities hold true 48=40+8 And 105 070=100 000+5 000+70 . This is how we represent natural numbers 48 And 105 070 as a sum of bit terms.

Arguing in a similar way, we can expand any natural number into digits.

Let's take another example. Imagine a natural number 17 as a sum of bit terms. Number 17 corresponds 1 top ten and 7 units, so 17=10+7 . This is the expansion of the number 17 by ranks.

And here is the amount 9+8 is not the sum of the bit terms of a natural number 17 , since the sum of bit terms cannot contain two numbers whose records consist of the same number of characters.

Now it became clear why the bit terms are called bit terms. This is due to the fact that each bit term is a "representative" of its bit of a given natural number.

Finding a natural number from a known sum of bit terms.

Let's consider the inverse problem. We will assume that we are given the sum of the bit terms of some natural number, and we need to find this number. To do this, one can imagine that each of the bit terms is written on a transparent film, but the areas with numbers other than the number 0 are not transparent. To get the desired natural number, it is necessary, as it were, to “superpose” all the bit terms on top of each other, combining their right edges.

For example, the amount 300+20+9 is a digit expansion of a number 329 , and the sum of bit terms of the form 2 000 000+30 000+3 000+400 corresponds to natural number 2 033 400 . I.e, 300+20+9=329 , but 2 000 000+30 000+3 000+400=2 033 400 .

To find a natural number by a known sum of bit terms, you can add these bit terms in a column (if necessary, refer to the material of the article column addition of natural numbers). Let's take a look at an example solution.

Find a natural number if the sum of bit terms of the form 200 000+40 000+50+5 . Write down the numbers 200 000 , 40 000 , 50 And 5 as required by the column addition method:

It remains to add the numbers in columns. To do this, remember that the sum of zeros is equal to zero, and the sum of zeros and a natural number is equal to this natural number. We get

Under the horizontal line, we got the desired natural number 240 055 , the sum of bit terms of which has the form 200 000+40 000+50+5 .

In conclusion, I would like to draw your attention to one more point. The skills of decomposing natural numbers into bits and the ability to perform the reverse action allow you to represent natural numbers as a sum of terms that are not bits. For example, the expansion in digits of a natural number 725 has the following form 725=700+20+5 , and the sum of bit terms 700+20+5 due to the properties of addition of natural numbers, it can be represented as (700+20)+5=720+5 or 700+(20+5)=700+25 , or (700+5)+20=705+20 .

A logical question arises: “What is it for?” The answer is simple: in some cases it can simplify calculations. Let's take an example. Let's subtract natural numbers 5 677 And 670 . First, we represent the reduced as a sum of bit terms: 5 677=5 000+600+70+7 . It is easy to see that the resulting sum of bit terms is equal to the sum (5000+7)+(600+70)=5007+670 . Then
5 677−670=(5 007+670)−670= 5 007+(670−670)=5 007+0=5 007 .

Bibliography.

  • Maths. Any textbooks for grades 1, 2, 3, 4 of educational institutions.
  • Maths. Any textbooks for 5 classes of educational institutions.

The presented article is devoted to an interesting topic about natural numbers. In order to perform some actions, it is necessary to represent the original expressions as the addition of several numbers - in a different language, to decompose the numbers into digits. The reverse process is also very important for solving exercises and problems.

In this section, we will consider typical examples in detail for better assimilation of information. We will also learn how to convert natural numbers and write them in a different form.

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How can you split a number into digits?

Based on the title of the article, we can conclude that this paragraph is devoted to such mathematical terms as "sum" and "terms". Before proceeding to the study of this information, you should study the topic in detail in order to have an understanding of natural numbers.

Let's get down to work and consider the basic concepts of bit terms.

Definition 1

Discharge terms are certain numbers that consist of zeros and a single non-zero digit. Natural numbers 5 , 10 , 400 , 200 belong to this category, and the numbers 144, 321, 5540, 16441 do not.

The number of bit terms for the presented number is equal to the number of non-zero digits contained in the record. If we represent the number 61 as the sum of bit terms, since 6 and 1 differ from 0 . If we expand the number 55050 as the sum of bit terms, then it is represented as the sum of 3 terms. The three fives represented in the entry are non-zero.

Definition 2

It should be remembered that all bit terms of a number contain a different number of characters in their record.

Definition 3

Sum bit terms of a natural number is equal to this number.

Let's move on to the concept of bit terms.

Definition 4

Discharge terms are natural numbers that contain a digit other than zero. The number of numbers must be equal to the number of non-zero digits. All terms of a number can be written with a different number of characters. If we decompose a number into digits, then the sum of the terms of the number will always be equal to this number.

After analyzing the concept, we can conclude that single-digit and multi-digit numbers (consisting entirely of zeros with the exception of the first digit) cannot be represented as a sum. This is because these numbers themselves will be bit terms for some numbers. With the exception of these numbers, all other examples can be decomposed into terms.

How to split numbers?

To decompose a number as a sum of digit terms, it is necessary to remember that natural numbers are associated with the number of certain objects. In the notation of a number, the digits depend on the number of units, tens, hundreds, thousands, and so on. If you take, for example, the number 58, then you can note that he answers 5 dozens and 8 units. Number 134 400 corresponds 1 hundred thousand, 3 tens of thousands, 4 thousand and 4 hundreds. You can represent these numbers in the form of equalities - 50 + 8 \u003d 58 and 134,400 \u003d 100,000 + 30,000 + 4,000 + 400. In these examples, we clearly saw how you can decompose a number in the form of bit terms.

Looking at this example, we can represent any natural number as a sum of bit terms.

Let's take another example. Let's represent the natural number 25 as a sum of digit terms. Number 25 corresponds 2 dozens and 5 units, so 25 = 20 + 5 . And here is the amount 17 + 8 is not the sum of the bit terms of the number 25 , since it cannot contain two numbers consisting of the same number of characters.

We have covered the basic concepts. Bit terms got their name due to the fact that each belongs to a certain category.

In order to analyze this example, let's analyze the inverse problem. Imagine that we know the sum of the bit terms. We need to find this natural number.

For example, the amount 200 + 30 + 8 decomposed into digits of the number 238, and the sum 3 000 000 + 20 000 + 2 000 + 500 corresponds to natural number 3 022 500 . Thus, we can easily determine a natural number if we know its sum of reserve terms.

Another way to find a natural number is to add the bit terms in columns. This example should not cause you any difficulty at run time. Let's talk about this in more detail.

Example 1

It is necessary to determine the original number if the sum of the bit terms is known 200 000 + 40 000 + 50 + 5 . Let's move on to the solution. It is necessary to write down the numbers 200,000, 40,000, 50 and 5 for stacking:

It remains to add the numbers in columns. To do this, remember that the sum of zeros is equal to zero, and the sum of zeros and a natural number is equal to this natural number.

We get:

After adding, we get a natural number 240 055 , the sum of bit terms of which has the form 200 000 + 40 000 + 50 + 5 .

Let's talk about one more thing. If we learn to decompose numbers and represent them as a sum of bit terms, then we can also represent natural numbers as a sum of terms that are not bit terms.

Example 2

Decomposition by digits of a number 725 will be presented as 725 = 700 + 20 + 5 , and the sum of bit terms 700 + 20 + 5 can be imagined as (700 + 20) + 5 = 720 + 5 or 700 + (20 + 5) = 700 + 25 , or (700 + 5) + 20 = 705 + 20 .

Sometimes complex calculations can be simplified a bit. Consider another small example to consolidate information.

Example 3

Let's subtract numbers 5 677 And 670 . First, let's represent the number 5677 as a sum of bit terms: 5 677 = 5 000 + 600 + 70 + 7 . After performing the action, we can conclude that. sum ( 5000 + 7) + (600 + 70) = 5007 + 670 . Then 5 677 − 670 = (5 007 + 670) − 670 = 5 007 + (670 − 670) = 5 007 + 0 = 5 007 .

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To write numbers, people came up with ten characters, which are called numbers. They are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

With ten digits, you can write any natural number.

Its name depends on the number of characters (digits) in the number.

A number consisting of one sign (digit) is called a single digit. The smallest single natural number is 1, the largest is 9.

A number consisting of two characters (digits) is called a two-digit number. The smallest two-digit number is 10, the largest is 99.

Numbers written with two, three, four or more digits are called two-digit, three-digit, four-digit or multi-digit. The smallest three-digit number is 100, the largest is 999.

Each digit in the record of a multi-digit number occupies a certain place - a position.

Discharge- this is the place (position) at which the digit stands in the notation of the number.

The same digit in a number entry can have different meanings depending on which digit it is in.

The digits are counted from the end of the number.

Units digit is the least significant digit that ends any number.

The number 5 - means 5 units, if the five is in last place in the number entry (in the units place).

Tens place is the digit that comes before the units digit.

The number 5 means 5 tens if it is in the penultimate place (in the tens place).

Hundreds place is the digit that comes before the tens digit. The number 5 means 5 hundreds if it is in the third place from the end of the number (in the hundreds place).

If there is no digit in the number, then the digit 0 (zero) will be in its place in the number entry.

Example. The number 807 contains 8 hundreds, 0 tens and 7 units - such an entry is called bit composition of the number.

807 = 8 hundreds 0 tens 7 units

Every 10 units of any rank form a new unit of a higher rank. For example, 10 ones make 1 tens, and 10 tens make 1 hundred.

Thus, the value of a digit from digit to digit (from ones to tens, from tens to hundreds) increases 10 times. Therefore, the counting system (calculus) that we use is called the decimal number system.

Classes and ranks

In the notation of a number, the digits, starting from the right, are grouped into classes of three digits each.

Unit class or the first class is the class that the first three digits form (to the right of the end of the number): units place, tens place and hundreds place.

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Bit terms of a number

The sum of bit terms

Any natural number can be written as a sum of bit terms.

How this is done can be seen from the following example: the number 999 consists of 9 hundreds, 9 tens and 9 ones, so:

999 = 9 hundreds + 9 tens + 9 units = 900 + 90 + 9

The numbers 900, 90 and 9 are bit terms. Discharge term is simply the number of 1s in the given digit.

The sum of the bit terms can also be written as follows:

999 = 9 100 + 9 10 + 9 1

The numbers that are multiplied by (1, 10, 100, 1000, etc.) are called bit units. So, 1 is the unit of the digit of units, 10 is the unit of the digit of tens, 100 is the unit of the digit of hundreds, etc. Numbers that are multiplied by bit units express number of bit units.

Write any number in the form:

12 = 1 10 + 2 1 or 12 = 10 + 2

called decomposing a number into bit terms(or the sum of bit terms).

3278 = 3 1000 + 2 100 + 7 10 + 8 1 = 3000 + 200 + 70 + 8
5031 = 5 1000 + 0 100 + 3 10 + 1 1 = 5000 + 30 + 1
3700 = 3 1000 + 7 100 + 0 10 + 0 1 = 3000 + 700

Calculator for decomposing a number into bit terms

To represent a number as a sum of digit terms, this calculator will help you. Just enter the desired number and click the Decompose button.

Bit terms in mathematics

A number is a mathematical concept for a quantitative description of something or a part of it, it also serves to compare the whole and parts, arrange in order. The concept of number is represented by signs or numbers in various combinations. At present, numbers from 1 to 9 and 0 are used almost everywhere. Numbers in the form of seven Latin letters have almost no use and will not be considered here.

Integers

When counting: “one, two, three ... forty-four” or arranging in turn: “first, second, third ... forty-fourth”, natural numbers are used, which are called natural numbers. This whole set is called “a series of natural numbers” and is denoted by the Latin letter N and has no end, because there is always a number even more, and the largest simply does not exist.

Digits and classes of numbers

This shows that the digit of a number is its position in digital notation, and any value can be represented through bit terms in the form nnn = n00 + n0 + n, where n is any digit from 0 to 9.

One ten is a unit of the second digit, and one hundred is a unit of the third. Units of the first category are called simple, all the rest are composite.

For the convenience of recording and transmission, a grouping of digits into classes of three in each is used. A space is allowed between classes for readability.

First - units, contains up to 3 characters:

Two hundred and thirteen contains the following digit terms: two hundred, one ten and three simple ones.

Forty-five is made up of four tens and five primes.

Second - thousand, 4 to 6 characters:

  • 679 812 = 600 000 + 70 000 + 9 000 + 800 +10 + 2.

This sum consists of the following bit terms:

  1. six hundred thousand;
  2. seventy thousand;
  3. nine thousand;
  4. eight hundred;
  5. ten;
  • 3 456 = 3000 + 400 +50 +6.

There are no terms above the fourth category.

The third - million, 7 to 9 digits:

This number contains nine bit terms:

  1. 800 million;
  2. 80 million;
  3. 7 million;
  4. 200 thousand;
  5. 10 thousand;
  6. 3 thousand;
  7. 6 hundreds;
  8. 4 tens;
  9. 4 units;
  • 7 891 234.

There are no terms higher than 7 digits in this number.

The fourth is billions, from 10 to 12 digits:

Five hundred sixty-seven billion eight hundred ninety-two million two hundred thirty-four thousand nine hundred seventy-six.

Bit terms of class 4 are read from left to right:

  1. units of hundreds of billions;
  2. units of tens of billions;
  3. units of billions;
  4. hundreds of millions;
  5. tens of millions;
  6. million;
  7. hundreds of thousands;
  8. tens of thousands;
  9. thousand;
  10. simple hundreds;
  11. simple tens;
  12. simple units.

The numbering of the digit of the number is made starting from the smallest, and reading - from the largest.

If there are no intermediate values ​​in the number of terms, zeros are put during recording, when pronouncing the name of the missing bits, as well as the class of units, it is not pronounced:

Four hundred billion four. Here, due to lack, the following names of ranks are not pronounced: tenth and eleventh fourth grade; ninth, eighth and seventh third and most? third class; the names of the second class and its categories, as well as hundreds and tens of units, are also not voiced.

Fifth - trillion, from 13 to 15 characters.

Four hundred eighty-seven trillion seven hundred eighty-nine billion six hundred fifty-four million four hundred twenty-seven two hundred and forty-one.

Sixth - quadrillion, 16-18 digits.

  • 321 546 818 492 395 953;

Three hundred twenty one quadrillion five hundred forty six trillion eight hundred eighteen billion four hundred ninety two million three hundred ninety five thousand nine hundred fifty three.

Seventh - quintillion, 19-21 signs.

  • 771 642 962 921 398 634 389.

Seven hundred seventy one quintillion six hundred forty two quadrillion nine hundred sixty two trillion nine hundred twenty one billion three hundred ninety eight million six hundred thirty four thousand three hundred eighty nine.

Eighth - sextillions, 22-24 digits.

  • 842 527 342 458 752 468 359 173

Eight hundred and forty-two sextillion five hundred twenty-seven quintillion three hundred and forty-two quadrillion four hundred and fifty-eight trillion seven hundred and fifty-two billion four hundred and sixty-eight million three hundred and fifty-nine thousand one hundred and seventy-three.

You can simply distinguish between classes by numbering, for example, the number 11 of the class contains from 31 to 33 characters when written.

But in practice, writing such a number of characters is inconvenient and most often leads to errors. Therefore, during operations with such values, the number of zeros is reduced by raising to a power. After all, it is much easier to write 10 31 than to attribute thirty-one zeros to one.

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What are bit terms

Answers and explanations

For example: 5679=5000+600+70+9
That is, the number of units in the discharge

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the sum of the bit terms of the number 526 is 500+20+6

The "sum of bit terms" is the representation of a two (or more) digit number as the sum of its bits.

Bit terms are the addition of numbers with different bit depths. For example, the number 17.890 is divided into bit terms: 17.890=10.000+7.000+800+90+0

Rule for multiplying any number by zero

Even at school, teachers tried to hammer the simplest rule into our heads: "Any number multiplied by zero equals zero!", - but still a lot of controversy constantly arises around him. Someone just memorized the rule and does not bother with the question “why?”. “You can’t do everything here, because at school they said so, the rule is the rule!” Someone can fill half a notebook with formulas, proving this rule or, conversely, its illogicality.

Who is right in the end

During these disputes, both people, having opposite points of view, look at each other like a ram, and prove with all their might that they are right. Although, if you look at them from the side, you can see not one, but two rams resting against each other with their horns. The only difference between them is that one is slightly less educated than the other. Most often, those who consider this rule to be wrong try to call for logic in this way:

I have two apples on my table, if I put zero apples to them, that is, I don’t put a single one, then my two apples will not disappear from this! The rule is illogical!

Indeed, apples will not disappear anywhere, but not because the rule is illogical, but because a slightly different equation is used here: 2 + 0 \u003d 2. So let's discard this conclusion right away - it is illogical, although it has the opposite goal - to call to logic.

This is interesting: How to find the difference of numbers in mathematics?

What is multiplication

The original multiplication rule was defined only for natural numbers: multiplication is a number added to itself a certain number of times, which implies the naturalness of the number. Thus, any number with multiplication can be reduced to this equation:

  1. 25?3 = 75
  2. 25 + 25 + 25 = 75
  3. 25?3 = 25 + 25 + 25

From this equation follows the conclusion, that multiplication is a simplified addition.

What is zero

Any person from childhood knows: zero is emptiness. Despite the fact that this emptiness has a designation, it does not carry anything at all. Ancient Eastern scientists thought differently - they approached the issue philosophically and drew some parallels between emptiness and infinity and saw a deep meaning in this number. After all, zero, which has the value of emptiness, standing next to any natural number, multiplies it ten times. Hence all the controversy over multiplication - this number carries so much inconsistency that it becomes difficult not to get confused. In addition, zero is constantly used to determine empty digits in decimal fractions, this is done both before and after the decimal point.

Is it possible to multiply by emptiness

It is possible to multiply by zero, but it is useless, because, whatever one may say, but even when multiplying negative numbers, zero will still be obtained. It is enough just to remember this simplest rule and never ask this question again. In fact, everything is simpler than it seems at first glance. There are no hidden meanings and secrets, as ancient scientists believed. The most logical explanation will be given below that this multiplication is useless, because when multiplying a number by it, the same thing will still be obtained - zero.

Going back to the very beginning, the argument about two apples, 2 times 0 looks like this:

  • If you eat two apples five times, then eaten 2 × 5 = 2+2+2+2+2 = 10 apples
  • If you eat two of them three times, then eaten 2? 3 = 2 + 2 + 2 = 6 apples
  • If you eat two apples zero times, then nothing will be eaten - 2?0 = 0?2 = 0+0 = 0

After all, eating an apple 0 times means not eating a single one. This will be clear even to the smallest child. Like it or not, 0 will come out, two or three can be replaced with absolutely any number and absolutely the same thing will come out. And to put it simply, zero is nothing and when you have there is nothing, then no matter how much you multiply - it's all the same will be zero. There is no magic, and nothing will make an apple, even if you multiply 0 by a million. This is the simplest, most understandable and logical explanation of the rule of multiplication by zero. For a person who is far from all formulas and mathematics, such an explanation will be enough for the dissonance in the head to resolve and everything to fall into place.

From all of the above follows another important rule:

You can't divide by zero!

This rule, too, has been stubbornly hammered into our heads since childhood. We just know that it is impossible and that's it, without stuffing our heads with unnecessary information. If you are suddenly asked the question, for what reason it is forbidden to divide by zero, then the majority will be confused and will not be able to clearly answer the simplest question from the school curriculum, because there are not so many disputes and contradictions around this rule.

Everyone just memorized the rule and does not divide by zero, not suspecting that the answer lies on the surface. Addition, multiplication, division and subtraction are unequal, only multiplication and addition are full of the above, and all other manipulations with numbers are built from them. That is, the entry 10: 2 is an abbreviation of the equation 2 * x = 10. Therefore, the entry 10: 0 is the same abbreviation for 0 * x = 10. It turns out that division by zero is a task to find a number, multiplying by 0, you get 10 And we have already figured out that such a number does not exist, which means that this equation has no solution, and it will be a priori incorrect.

Let me tell you

To not divide by 0!

Cut 1 as you like, along,

Just don't divide by 0!

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