Natural numbers– natural numbers are numbers that are used to count objects. Lots of everyone natural numbers sometimes called natural next to: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, etc.

To write natural numbers, ten digits are used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Using them, you can write any natural number. This notation of numbers is called decimal.

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

Places of natural numbers

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

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

Single and multi-digit numbers

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

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

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

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

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

Digit classes for reading large natural numbers

To read large natural numbers, the number is divided into groups of three digits, starting from the right edge. These groups are called classes.

The first three digits on the right edge make up the units class, the next three are the thousands class, and the next three are the millions class.

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

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

Example of writing and reading

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

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

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

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

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

The symbols that are used to write numbers are called in numbers .

The word number comes from the Arabic name for the number 0 (sifr). This is very interesting figure. It's 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 the one we use is called decimal. Because it is precisely ten units of one category that constitute one unit of the next category.

We count in units, tens, hundreds, thousands, and so on. These are the digit 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...

Place is the place of a digit in a number record.

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

We talked about how 0 is an insignificant number that means the absence of something. In numbers, the number 0 indicates the absence of ones in the digit.

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

And numbers whose digits do not have zeros are called full .

The digits are counted from right to left:

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

1. Among 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. Among 1000462086432

2 units

3 tens

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. Among 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 numbers, in such numbers there can be as many zeros (insignificant digits) as desired or not at all, but there is only one significant digit.

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

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

For example, number 6734 can be represented like this:

6000 + 700 + 30 + 4 = 6734

They are all different. For example, 2, 67, 354, 1009. Let's look at these numbers in detail.
2 consists of one digit, so this number is called single digit. Another example of single-digit numbers: 3, 5, 8.
67 consists of two digits, so this number is called double digit number. Example double digit numbers: 12, 35, 99.
Three digit numbers consist of three numbers, for example: 354, 444, 780.
Four digit numbers consist of four digits, for example: 1009, 2600, 5732.

Two digits, three digits, four digits, five digits, six digits, etc. numbers are called multi-digit numbers.

Number digits.

Consider the number 134. Each digit of this number has its own place. Such places are called discharges.

The number 4 takes the place or place of ones. The number 4 can also be called a number first category.
The number 3 occupies the place or tens place. Or the number 3 can be called a number second class.
And the number 1 occupies the hundreds place. In another way, the number 1 can be called the number third category. The number 1 is the last digit of the glory of the number 134, so the number 1 can be called the highest digit. The highest digit is always greater than 0.

Every 10 units of any rank form a new unit of a higher rank. 10 ones form one tens place, 10 tens form one hundreds place, ten hundreds form one thousand place, etc.
If there is no digit, then it will be replaced by 0.

For example: the number 208.
The number 8 is the first digit of units.
The number 0 is the second tens place. 0 means nothing in mathematics. From the record it follows that there are tens given number No.
The number 2 is the third hundreds place.

This parsing of a number is called digit composition of the number.

Classes.

Multi-digit numbers are divided into groups of three digits from right to left. Such groups of numbers are called classes. The first class on the right is called class of units, the second one is called class of thousands, third - million class, fourth - class of billions, fifth - trillion class, sixth – class quadrillion, seventh - class quintillions, eighth – class sextillion.

Unit class– the first class on the right from the end is three digits consisting of a units place, a tens place and a hundreds place.
Class of thousands– the second class consists of the category: units of thousands, tens of thousands and hundreds of thousands.
Million class– the third class consists of the category: units of millions, tens of millions and hundreds of millions.

Let's look at an example:
We have the number 13,562,006,891.
This number has 891 units in the units class, 6 units in the thousands class, 562 units in the millions class, and 13 units in the billions class.

13 billion 562 million 6 thousand 891.

Sum of bit terms.

Anything having different digits can be decomposed into sum of bit terms. Let's look at an example:
Let's write the number 4062 into digits.

4 thousand 0 hundreds 6 tens 2 units or in another way you can write

4062=4 ⋅1000+0 ⋅100+6 ⋅10+2

Next example:
26490=2 ⋅10000+6 ⋅1000+4 ⋅100+9 ⋅10+0

With this lesson we will study the digits of counting terms. First, let's repeat the ratio of counting units. Let us remember what digits are, what digits hundreds, tens and ones belong to. We will solve many different and interesting tasks to secure the material. After this lesson, you will easily determine which category the units, tens and hundreds belong to in a three-digit number. You will also easily convert length units into smaller or larger units. Don't waste a minute. Go ahead - learn 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 category is tens. And the third category is hundreds.

Write down the numbers on the abacus (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 deposited in the accounts. The result is the number three hundred twenty-seven.

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

3. In Figure 4 there are seven units, no tens and three hundreds. The result is the number three hundred and seven.

2. In the second magnitude, five hundred and forty centimeters. In this number, 5 hundreds are 5 m and 4 tens are 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.

86 mm = 8 cm 6 mm

4. B last date(42 dm) four tens are visible and it is known that there are 10 dm in 1 m.

42 dm = 4 m 2 dm

Express these quantities in smaller units:

2. 2 dm 8 mm

Solution: 1. To solve the problem, 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

References

1. Mathematics. 3rd grade. Textbook for general education institutions with adj. per electron carrier. At 2 hours Part 1 / [M.I. Moreau, 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-COUNT.
3. Peterson L.G. Mathematics, 3rd grade. - M.: Yuventa.

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

Homework

1. Mathematics. 3rd grade. Textbook for general education institutions with adj. per electron carrier. At 2 p.m. Part 2 / [M.I. Moreau, M.A. Bantova, G.V. Beltyukova and others] - 2nd ed. - M.: Education, 2012., pp. 44, 45 No. 1-7.
2. Express in millimeters

In titles 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 it easier to read. large numbers. Each class has its own name: the first three digits are the class of units, followed by the class of thousands, then millions, billions (or billions), and so on.

Since we use decimal system calculus, then the basic unit of measurement 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 are read by the last digit after the decimal point, for example 0.325 - three hundred twenty-five thousandths, where the thousandth is the place of the 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 unit of quadrillion 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