To remember how much they harvested or how many stars in the sky, people came up with symbols. In different areas, these symbols were different.

But with the development of trade, in order to understand the designations of another people, people began to use the most convenient symbols. We, for example, use Arabic symbols. And they are called Arabic because the Europeans learned them from the Arabs. But the Arabs learned these symbols from the Indians.

The symbols used to write numbers are called figures .

The word digit comes from the Arabic name for the number 0 (sifr). This is a very interesting number. It is called insignificant and denotes the absence of something.

In the picture we see a plate with 3 apples on it and an empty plate with no apples on it. In the case of an empty plate, we can say that there are 0 apples on it.

The remaining numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9 are called meaningful .

Bit units

Notation which we use is called decimal. Because it is exactly ten units of one category that makes up one unit of the next category.

We count in units, tens, hundreds, thousands, and so on. These are the bit units of our number system.

10 ones - 1 ten (10)

10 tens - 1 hundred (100)

10 hundreds - 1 thousand (1000)

10 times 1 thousand - 1 ten thousand (10,000)

10 tens of thousands - 100 thousand (100,000) and so on ...

A digit is the place of a digit in a number notation.

For example, among 12 two digits: the units digit consists of 2 units, the tens digit consists of one dozen.

We talked about the fact that 0 is an insignificant number, which means the absence of something. In numbers, the number 0 means the absence of ones in the discharge.

In the number 190, the digit 0 indicates the absence of a units digit. In the number 208, the digit 0 indicates the absence of a tens digit. Such numbers are called incomplete .

And the numbers in the digits of which there are no zeros are called complete .

The digits are counted from right to left:

It will be clearer if you depict the bit grid as follows:

1. In list 2375 :

5 units of the first category, or 5 units

7 units of the second digit, or 7 tens

3 units of the third category, or 3 hundreds

2 units of the fourth category, or 2 thousand

This number is pronounced like this: two thousand three hundred seventy five

1. In list 1000462086432

2 pieces

3 dozen

8 tens of thousands

0 hundred thousand

2 units million

6 tens of millions

4 hundred million

0 units billion

0 tens of billions

0 hundred billion

1 unit trillion

This number is pronounced like this: one trillion four hundred sixty-two million eighty-six thousand four hundred thirty-two .

1. In list 83 :

3 units

8 tens

Pronounced like this: eighty three .

Bit , call numbers consisting of units of only one digit:

For example, numbers 1, 3, 40, 600, 8000 - bit, in such numbers of zeros (insignificant digits) there can be as many as you like or not at all, and there is only one significant digit.

Other numbers, for example: 34, 108, 756 and so on, non-digit , they are called algorithmic.

Non-bit numbers can be represented as a sum of bit terms.

For example, number 6734 can be represented like this:

6000 + 700 + 30 + 4 = 6734

Digits in the notation of multi-digit numbers are divided from right to left into groups of three digits each. These groups are called classes. In each class, the numbers from right to left represent the units, tens, and hundreds of that class:

The first class on the right is called unit class, second - thousand, third - million, fourth - billion, fifth - trillion, sixth - quadrillion, seventh - quintillion, eighth - sextillions.

For the convenience of reading the record of a multi-digit number, a small gap is left between the classes. For example, to read the number 148951784296, we select classes in it:

and read the number of units of each class from left to right:

148 billion 951 million 784 thousand 296.

When reading a class of units, the word units is usually not added at the end.

Each digit in the record of a multi-digit number occupies a certain place - a position. The place (position) in the record of the number on which the digit stands is called discharge.

The digits are counted from right to left. That is, the first digit on the right in the number entry is called the first digit, the second digit on the right is the second digit, etc. For example, in the first class of the number 148 951 784 296, the number 6 is the first digit, 9 is the second digit, 2 - digit of the third digit:

Units, tens, hundreds, thousands, etc. are also called bit units:
units are called units of the 1st category (or simple units)
tens are called units of the 2nd digit
hundreds are called units of the 3rd category, etc.

All units except simple units are called constituent units. So, a dozen, a hundred, a thousand, etc. are constituent units. Every 10 units of any rank is one unit of the next (higher) rank. For example, a hundred contains 10 tens, a dozen - 10 simple ones.

Any constituent unit compared to another unit smaller than it is called unit of the highest category, and in comparison with a unit greater than it is called lowest rank unit. For example, a hundred is a higher unit relative to ten and a lower unit relative to a thousand.

To find out how many units of any digit are in a number, you must discard all the digits that mean the units of the lower digits and read the number expressed by the remaining digits.

For example, you want to know how many hundreds are in the number 6284, i.e. how many hundreds are in thousands and hundreds of this number together.

In the number 6284, the number 2 is in third place in the class of units, which means that there are two simple hundreds in the number. The next number to the left is 6, meaning thousands. Since every thousand contains 10 hundreds, there are 60 of them in 6 thousand. In total, therefore, this number contains 62 hundreds.

The number 0 in any category means the absence of units in this category. For example, the number 0 in the tens place means the absence of tens, in the hundreds place - the absence of hundreds, etc. In the place where 0 stands, nothing is pronounced when reading the number:

172 526 - one hundred seventy-two thousand five hundred twenty-six.
102026 - one hundred two thousand twenty-six.

Numbers greater than a thousand are considered multivalued. Multi-digit numbers are numbers in the thousands class and the million class. Multi-valued numbers are formed, named, written based not only on the concept of a category, but also on the concept of a class.

The class combines three categories.

The class of units is units, tens and hundreds. This is first class.

The thousands class is units of thousands, tens of thousands, hundreds of thousands. This is second class. The unit of this class is the thousand.

Class of millions - units of millions, tens of millions, hundreds of millions. This is third grade. The unit of this class is the million.

Table of ranks of class I:

The table contains the number 257. Table of digits of class II:

The table contains the number 275,000,000.

Multi-digit numbers form the second class - the class of thousands and the third class - the class of millions.

Ten hundred is a thousand. Numbers from 1001 to 1,000,000 are called numbers in the thousands class.

Numbers in the thousands class are four-, five-, and six-digit numbers.

Four-digit numbers are written in four digits: 1537, 7455, 3164, 3401. The first digit on the right in a four-digit number is called the first digit or units digit, the second digit on the right is the second digit or tens digit, the third digit on the right is the third digit or hundreds digit , the fourth digit from the right - the digit of the fourth digit or thousand digit.

The fifth digit is the tens of thousands, the sixth digit is the hundreds of thousands.

The table contains the number 257,000. Class III rank table:

Integer thousands: 1000,2000,3000,4000,5000,6000,7000,8000,9000.

Read multi-digit numbers from left to right. For numbers 1001 and further, the order of naming their bit numbers and the order of recording is the same: 4321 - four thousand three hundred and twenty one; 346 456 - three hundred forty-six thousand four hundred fifty-six.

Rule for reading multi-digit numbers: multi-digit numbers are read from left to right. First, the number is divided into classes, counting three digits from the right. Reading begins with units of the senior classes (on the left). Units of the senior classes are read immediately as a three-digit number, then adding the name of the class. Class I units are read without adding the class name.

For example: 1 234 456 - one million two hundred thirty-four thousand four hundred fifty-six.

If some class in the number entry does not contain significant digits, it is skipped when reading.

For example: 123 000 324 - one hundred twenty-three million three hundred twenty-four.

The concept of "class" is basic for the formation of multi-valued numbers. All multi-digit numbers contain two or more classes.

The class combines three digits (ones, tens and hundreds).

In writing, when writing a multi-digit number, it is customary to make a detente between classes: 345 674, 23 456, 101 405.12 345 567.

The rule for writing multi-digit numbers: multi-digit numbers are written by class, starting with the highest. To write down a number in numbers, for example, twelve million four hundred and fifty thousand seven hundred and forty-two, they do this: they write down the units of each named class in groups, separating one class from another with a small gap (discharge): 12 450 742.

Class composition - the allocation of "class numbers" (class components) in a multi-valued number.

For example: 123,456 = 123,000 + 456

34 123 345 - 34 000 000 + 123 000 + 345

Bit composition - selection of bit numbers in a multi-digit number: _____

On the basis of the discharge composition, the cases of discharge addition and subtraction are considered:

400 000 + 3 000 20 534 - 34 340 000 - 40 000

534 000 - 30 000 672 000 - 600 000 24 000 + 300

When finding the values ​​of these expressions, they refer to the bit composition of three-digit numbers: the number 340,000 consists of 300,000 and 40,000. Subtracting 40,000 we get 300,000.

Bit terms - the sum of the bit numbers of a multi-digit number:

247 000 - 200 000 + 40 000 + 7 000

968 460 - 900 000 + 60 000 + 8 000 + 400 + 60

Decimal composition - highlighting tens and ones in a multi-digit number: 234,000 is 23,400 dess. or 2,340 cells.

When studying the numbering of multi-valued numbers, the cases of addition and subtraction are also considered, based on the principle of constructing a sequence of natural numbers:

443 999 +1 20 443 - 1 640 000 + 1 640 000 - 1

10599+1 700000-1 99999 + 1 100000-1

When finding the meaning of these expressions, they refer to the principle of constructing a natural series of numbers: adding 1 to the number, we get the next (subsequent) number. Subtracting from the number 1, we get the previous number.

Here are the main types of tasks performed by children in the study of multi-digit numbers:

1) for reading and writing multi-digit numbers:

Break the number into classes, say how many units of each class are in it, and then read the number:

7300 29608 305220 400400 90060

7340 29680 305020 400004 60090

When completing the task, you should use the rule for reading multi-digit numbers.

Write and read the numbers in which: a) 30 units. second class and 870 units. first class; 6) 8 units second class and 600 units. first class; c) 4 units. second class and 0 units. first class.

When completing the task, you should use the table of ranks and classes.

Write the numbers in numbers: "The smallest distance from the Earth to the Moon is three hundred and fifty-six thousand four hundred and ten kilometers, and the largest is four hundred and six thousand seven hundred and forty kilometers."

The students wrote down the number nine thousand forty like this: 940, 900040, 9040. Explain which entry is correct.

When performing tasks, you should use the rule for writing multi-digit numbers.

2) on the bit and class composition of multi-digit numbers:

Replace these numbers with the sum according to the sample: 108201 = 108000 + 201

360 400 = ... + ... 50070 = ... + ... 9007 = ... + ... Task for the class composition of a multi-digit number.

Replace each number with the sum of the bit terms:

205 000 = ... + ... 640 000 = ... + ...

200 000 + 90 000 + 9 000 299 000 - 200 000

4 000 + 8 000 408 000 - 8 000

How many units of each category in the number 395 028, in the number 602 023? How many units of each class are in these numbers?

When performing tasks, a scheme of bit composition of multi-digit numbers is used.

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

Find the values ​​of the expressions: 99 999 +1 30 000 - 1

100000-1 699999 + 1

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

4) on the order of the numbers in the natural series:

Three tractors have the following serial numbers: 250,000,249,999, 250,001. Which of them rolled off the assembly line first? Second? Third?

Write down all six-digit numbers that are greater than 999996.

5) on the local value of the digit in the notation of the number:

What does the number 2 mean in the entry of each number: 2, 20, 200, 2,000, 20,000, 200,000? Explain how the value of the number 2 in the notation of a number changes when its place changes.

What does each digit in the number entry mean: 140,401, 308,000, 70,050?

(In the entry of the number 140401, the number 4, which is in the third place from the right, indicates the number of hundreds, the number 4, which is in the fifth place from the right, indicates the number

tens of thousands. The number 1, which is in the first place from the right, indicates the number of units in the number, and the number 1, which is in the sixth place from the right, indicates the number of hundreds of thousands. The number 0, which is second from the right and fourth from the right, means that there are no ones in the second and fourth digits.)

Use the numbers 9 and 0 to write one five-digit number and one six-digit number. Use the same numbers to write other multi-digit numbers.

6) for comparison of multi-digit numbers:

Check if the equalities are correct:

5 312 < 5 320 900 001 > 901 000

Compare numbers:

a) 999 ... 1000 b) 9 999 ... 999 c) 415 760 ... 415 670

d) 200 030 ... 200 003 e) 94 875 ... 94 895

When comparing the first pair of numbers, they refer to the order of the numbers in the natural series: the next number is greater than the previous number.

When comparing the second pair of numbers, they refer to the number of characters in the number entry: a three-digit number is always less than a four-digit number.

When comparing the third, fourth and fifth pairs of numbers, the multi-digit comparison rule is used: To find out which of the two multi-digit numbers is greater and which is less, do this:

Compare numbers bit by bit, starting with the highest digits.

For example, of the two numbers 34567 and 43567, the second one is larger, since it contains 4 ones in the tens of thousands place, and the first one contains three ones in the same place.

Of the two numbers 415,760 and 415,670, the first is greater, since the class of thousands in both numbers contains the same number of units - 415 units. thousand, but in the discharge of hundreds of thousands, the first number contains 7 units, and the second - 6 units.

Of the two numbers 200,030 and 200,003, the first is larger, since the class of thousands in both numbers contains the same number of units - 200 units. thousand, in the hundreds place both numbers contain zeros, in the tens place the first number contains 3 units, and the second number in the tens place has no significant digits (contains zero), so the first number is larger.

For greater clarity, when completing a task, you can compare two models of numbers from the bones in the accounts (quantitative model).

Comparing multi-digit numbers, you can refer to the fact that a number containing more characters in the record will always be greater than a number containing fewer characters.

When comparing numbers of the form:

99 999 ... 100 000 989 000 ... 989 001

567 999 ... 568 000 599 999 ... 600 000

you should refer to the order of the numbers when counting: the next number is always greater than the previous one.

7) on the decimal composition of multi-digit numbers:

Write down the numbers: 376, 6517, 85742, 375264. How many tens are in each of them? Highlight them.

To determine the number of tens in a multi-digit number, you can cover the last digit (first from the right) with your hand. The remaining numbers will show the number of tens.

To determine the number of hundreds in a number, you can cover the last two digits in the number entry (first and second from the right) with your hand. The remaining digits will show the number of hundreds in the number.

For example, in the number 2 846 - tens 284, hundreds - 28. In the number 375 264 - tens 37 526, hundreds - 3 752.

Consider the numbers: 3849. 56018. 370843. Which of the underlined numbers shows how many tens are in the number? Hundreds? Thousand?

How many hundreds are there in 6800?

Write down 5 numbers, each containing 370 tens.

8) on the relationship between the categories:

Write by filling in the blanks:

1 thousand = ... hundred. 1 hundred = ... des. 1 thousand = ... dec.

How will the numbers 3,000, 8,000, 17,000 change if one zero is discarded in their entries on the right? Two zeros? Three zeros?

Compare the numbers in each column. How many times does a number increase when one zero is added to its right side? Two zeros? Three zeros?

17 170 1 700 17000

Numbers 57, 90, 300 increase 10 times, 1,000 times.

Decrease the numbers 3,000, 60,000, 152,000 by 10 times, by 100 times, by 1,000 times.

When completing the last two tasks, they refer to the fact that increasing the number by 10 times transfers it to the next digit on the left (tens to hundreds, hundreds to thousands, etc.), and decreasing the number to. 10 times transfers it to the next category on the right (tens to units, hundreds to tens).

When the number is increased by a factor of 10 (100.1000) in this way, you can simply assign zero to the right (two zeros, three zeros). When the number is reduced by 10 times (100, 1000), one zero can be discarded on the right in the number entry (two zeros, three zeros).

Acquaintance with the number 1,000,000 (million) completes the study of the class of thousands.

Ten hundred thousand is a million. A thousand thousand is a million.

A million is written like this: 1,000,000.

The number 1,000,000 completes the study of numbers in the class of thousands.

A million (1000,000) is a unit of a new class - the class of millions.

A million (1,000,000) is the first seven-digit number in the series of natural numbers.

A million is the smallest seven-digit number.

A million is a new counting unit in the decimal number system.

In the entry of the number 1,000,000, the number 1 means that in the VII digit (million digit) there is one unit, and in the digits of hundreds of thousands, tens of thousands, units of thousands, etc., zeros mean that there are no significant digits in these digits.

The class of millions contains three digits of units of millions, tens of millions and hundreds of millions (VII, VIII and IX digits).

The millions class ends with the number billion.

A billion is 1000 million.

1000 billion is a trillion.

1000 trillion is a quadrillion.

1000 quadrillion is a quintillion.

It is impossible to imagine such a quantity of something. AND I. Depman in The History of Arithmetic gives the following example to illustrate large numbers: “A heavy rail car can hold 50 million rubles in ten-rouble tickets (bills). It would take 20,000 wagons to transport a trillion rubles.”

Visual class table model:

The number is read like this: 412 million 163 thousand 539

They write it down like this: 412 163 539

For numbers of the million class, the reading rule, the writing rule, and the multi-digit comparison rule apply (see above).

In a stable textbook of mathematics for elementary grades, numbers over a million are not considered.

An easy way to explain the digits and classes of a number to a child. Even a preschooler understands. The method of adding and subtracting multi-digit numbers by children without problems and clearly. Teaching mathematics in a playful way. Simple and fun math for kids.

How easy it is to explain to a child the digits and classes of a number.

My son has been able to count to 10 since he was 2.5 years old, he mastered tens and counting up to 20 at 3, and hundreds at 4. Board, mathematical and logic games helped us a lot in this. But, it's only verbal. Visually, he always confused the numbers 43 and 34. I could say that he has “two hundred hundred toys”, that is, the names of the classes, he knew, but the composition of the number for a long time was a mystery to him. Started looking how to explain simply and intelligibly, I found several methods, but we liked the most and this one came up.

On the sheet I drew a table like this

The child already knew the names of tens and hundreds in turn. I just reminded that one zero is ten, two zeros is a hundred, three zeros is a thousand, and if two zeros and three more zeros, then this is respectively ten thousand.

She gave the child buttons and offered to arrange them in columns as he wanted.

It turned out like this.

She asked me to count the buttons in the column, and put the desired number below. (we have a set of wooden numbers, but just drawn numbers on cardboard squares will do).

And then we just read what happened TWO THOUSAND (first by 2, and then by 1000, then I say that zero is empty, which means we just miss it, 13. Here, with 13, they messed around a little, 23, 33, 59 was easier to understand. Together they voiced that it turned out, then it helped a little, and then the child began to cope on his own.When I began to read the number correctly, I wrote the number on a sheet, and he laid it out in columns from buttons, the next step I just called the number, slowly, pausing between digits, and with each time it got better.

Simple addition and subtraction with transition through the category for children.

After playing like this for half a year, we moved on to addition and subtraction using the same tablet. For example 2013+224=2234 . Blue buttons put then purple

There were no problems with the transition through the category, by that time we had long and successfully played “Superfarmer” from Granna. She simply explained that as we changed 6 hares for a sheep, we also change 10 buttons in a column for one more button. The child understood. And at the age of 5, he successfully adds and subtracts arbitrarily significant numbers, and sometimes even in his mind. As he explained to me, he simply presents a sign in front of his eyes. I hope our experience will be useful.

Try it and write your impressions in the reviews.

With the help of this lesson, we will study the digits of countable terms. First, let's repeat the ratio of counting units. Recall what digits are, what category hundreds, tens and ones belong to. We will solve many different and interesting tasks to consolidate the material. After this lesson, you can easily determine what category the units, tens and hundreds belong to. three-digit number. You will also easily convert length units to smaller or larger units. Don't waste a minute. Forward - to study and comprehend new horizons!

When writing a number, each counting unit is written in its place (Table 1).

Table 1. Writing three-digit numbers

The digits are counted from right to left, starting with the first digit - one. The second digit is tens. And the third digit is hundreds.

Write down the numbers set aside on the accounts (Fig. 2, 3, 4) and read them.

Rice. 2. Numbers

Rice. 4. Numbers

Rice. 3. Numbers

Solution: 1. Seven units, two tens and three hundreds are set aside on the accounts. It turns out the number three hundred twenty-seven.

2. There are no units in the next number (Fig. 3). If there is no digit, you can put zero. The whole number is three hundred and twenty.

3. In Figure 4, there are seven units, no tens and three hundreds. It turns out the number three hundred and seven.

2. In the second magnitude, five hundred and forty centimeters. In this number, 5 hundreds - 5 m and 4 tens - 4 dm, and there are no units, therefore, there will be no centimeters.

540 cm = 5 m 4 dm

3. Eighty-six millimeters. There are ten millimeters in one centimeter, which means that this value will be eight centimeters and six millimeters.

86mm = 8cm 6mm

4. In the last number (42 dm), four tens are visible and it is known that in 1 m - 10 dm.

42 dm = 4 m 2 dm

Express these quantities in smaller units:

2. 2 dm 8 mm

Solution: 1. To solve the task, we will use Figure 5, which shows the relationship between units of length.

1 m 75 cm = 175 cm

2. Let's translate the second number.

2 dm 8 mm = 208 mm

Bibliography

1. Maths. Grade 3 Proc. for general education institutions with adj. to an electron. carrier. At 2 h. Part 1 / [M.I. Moro, M.A. Bantova, G.V. Beltyukova and others] - 2nd ed. - M.: Education, 2012. - 112 p.: ill. - (School of Russia).
2. Rudnitskaya V.N., Yudacheva T.V. Mathematics, 3rd grade. - M.: VENTANA-GRAF.
3. Peterson L.G. Mathematics, 3rd grade. - M.: Juventa.

1. All-schools.pp.ua ().
2. Urokonline.com ().
3. Uchu24.ru ().

Homework

1. Maths. Grade 3 Proc. for general education institutions with adj. to an electron. carrier. At 2 h. Part 2 / [M.I. Moro, M.A. Bantova, G.V. Beltyukova and others] - 2nd ed. - M.: Education, 2012., pp. 44, 45 No. 1-7.
2. Express in millimeters