Because decimal number system local, 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 local, or digit, notation system allows you to make three-digit numbers from these three digits: 103, 130, 301, 310 and two-digit numbers: 013, 031. The given numbers are arranged in order ascending: each previous number is less than the next.
Therefore, the numbers that are used to write a number do not fully determine this number, but only serve as a tool for writing it.
The number itself is built taking into account discharges, in which this or that digit is written, i.e. the desired digit must also occupy the right place in the notation of the number.
Rule. Digits of natural numbers are named from right to left from 1 to a larger number, each bit has its own number and place in the notation of the number.
The most used numbers have up to 12 digits. Numbers with more than 12 digits belong to the group of large numbers.
The number of places occupied by digits, provided that the digit of the largest digit is not 0, determines the capacity of the number. We can say about a number that it is: single-valued (single-digit), for example 5; two-digit (two-digit), for example 15; three-digit (three-digit), such as 551, etc.
In addition to the serial number, each of the digits has its own name: the digit of units (1st), the digit of tens (2nd), the digit of hundreds (3rd), the digit of thousands of units (4th), the digit of tens of thousands (5th ), etc. Every three digits, starting from the first, are combined into classes. Each Class also has its own serial number and name.
For example, the first 3 discharge(from 1st to 3rd inclusive) is Class units with serial number 1; third Class- this is Class million, it includes the 7th, 8th and 9th ranks.
Let us give the structure of the bit construction of a number, or a table of bits 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 multi-digit number of the fourth class. 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. The name of the class is added to each class of a three-digit number: "billions", "millions", "thousands".
For a class of units, the name is omitted (meaning "units").
Numbers from the 5th grade and above are large numbers. Large numbers are used only in specific branches of Knowledge (astronomy, physics, electronics, etc.).
Let us give an introductory name of the classes from the fifth to the ninth: units of the 5th class - trillions, 6th class - quadrillions, 7th class - quintillions, 8th class - sextillions, 9th class - septillions.
In the names of Arabic numbers, each digit belongs to its 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 place of units. The next, second from the end, digit indicates tens (the tens digit), and the third digit from the end indicates the number of hundreds in the number - the hundreds digit. Further, the digits are repeated in the same way in turn in each class, denoting units, tens and hundreds in the classes of thousands, millions, and so on. If the number is small and does not contain a tens or hundreds digit, it is customary to take them as zero. Classes group numbers in numbers of three, often in computing devices or records a period or space is placed between classes 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 the decimal system, the basic unit of quantity is the ten, or 10 1 . Accordingly, with an increase in the number of digits in a number, the number of tens of 10 2, 10 3, 10 4, etc. also increases. Knowing the number of tens, you can easily determine the class and category 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 as follows - each digit is displayed in a separate term, multiplied by the required coefficient 10 n, where n is the position of the digit in the count 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
Also, the power of 10 is also used in writing decimals: 10 (-1) is 0.1 or one tenth. Similarly with the previous paragraph, a decimal number can also be decomposed, in which case n will indicate the position of the digit from the comma from right to left, for example: 0.347629= 3x10 (-1) +4x10 (-2) +7x10 (-3) +6x10 (-4) +2x10 (-5) +9x10 (-6) )
Names of decimal numbers. Decimal numbers are read by the last digit after the decimal point, for example 0.325 - three hundred and twenty-five thousandths, where thousandths are the digit of the last digit 5.
Table of names of large numbers, digits and classes
| 1st class unit | 1st unit digit 2nd place ten 3rd rank hundreds |
1 = 10 0 10 = 10 1 100 = 10 2 |
| 2nd class thousand | 1st digit units of thousands 2nd digit tens of thousands 3rd rank hundreds of thousands |
1 000 = 10 3 10 000 = 10 4 100 000 = 10 5 |
| 3rd grade millions | 1st digit units million 2nd digit tens of millions 3rd digit hundreds of millions |
1 000 000 = 10 6 10 000 000 = 10 7 100 000 000 = 10 8 |
| 4th grade billions | 1st digit units billion 2nd digit tens of billions 3rd digit 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 trillion units 2nd digit tens of trillions 3rd digit hundred trillion |
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 quadrillion units 2nd digit 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 units of quintillions 2nd digit tens of quintillions 3rd rank 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 sextillion units 2nd digit tens of sextillions 3rd rank hundred sextillions |
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 septillion | 1st digit units of septillion 2nd digit tens of septillions 3rd rank 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 octillion units 2nd digit ten octillion 3rd rank 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 |
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-digit 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.
Remember!
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 category of tens).
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).
Remember!
If there is no digit in the number, then the number “0” (zero) will be in its place in the record of the number.
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 unitsEvery 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.
Thousand class or the second class is the class which is formed by the following three digits: units of thousands, tens of thousands, and hundreds of thousands.
| Numbers | Thousand class (second class) | Unit class (first class) | ||||
|---|---|---|---|---|---|---|
| hundreds of thousands | tens of thousands | units of thousands | hundreds | dozens | units | |
| 5 234 | - | - | 5 | 2 | 3 | 4 |
| 12 803 | - | 1 | 2 | 8 | 0 | 3 |
| 356 149 | 3 | 5 | 6 | 1 | 4 | 9 |
We remind you that 10 units of the hundreds place (from the class of units) form one thousand (the unit of the next place: the unit of thousands in the class of thousands).
10 hundreds = 1 thousandMillion class or the third class is the class which is formed by the following three digits: units of millions, tens of millions, and hundreds of millions.
The million place unit is one million or one thousand thousand (1,000 thousand). One million can be written as the number "1,000,000".
Ten such units form a new bit unit - ten million "10,000,000"
Ten tens of millions form a new digit unit - one hundred million or in the notation in numbers " 100 000 000".
| Numbers | Thousand class (second class) | Unit class (first class) | |||||||
|---|---|---|---|---|---|---|---|---|---|
| hundreds of millions | tens of millions | units million | hundreds of thousands | tens of thousands | units of thousands | hundreds | dozens | units | |
| 8 345 216 | - | - | 8 | 3 | 4 | 5 | 2 | 1 | 6 |
| 93 785 342 | - | 9 | 3 | 7 | 8 | 5 | 3 | 4 | 2 |
| 134 590 720 | 1 | 3 | 4 | 5 | 9 | 0 | 7 | 2 | 0 |
| Numbers | Million class (third class) | Thousand class (second class) | Unit class (first class) | ||||||
|---|---|---|---|---|---|---|---|---|---|
| hundreds of millions | tens of millions | units million | hundreds of thousands | tens of thousands | units of thousands | hundreds | dozens | units | |
| 8 345 216 | - | - | 8 | 3 | 4 | 5 | 2 | 1 | 6 |
| 93 785 342 | - | 9 | 3 | 7 | 8 | 5 | 3 | 4 | 2 |
| 134 590 720 | 1 | 3 | 4 | 5 | 9 | 0 | 7 | 2 | 0 |
How to read a multi-digit number
Remember!
Do not pronounce the name of the class of units, as well as the name of the class, all three digits of which are zeros.
For example, the number " 134 590 720"We read: one hundred thirty-four million five hundred ninety thousand seven hundred twenty.
The number " 418 000 547"We read: four hundred and eighteen million five hundred and forty-seven.
On our website, to check your results, you can use the calculator for decomposing a number into digits online.
Important!
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 numbers of the second ten (11, 12, 13, 14, 15, 16, 17, 18, 19, 20) are two-digit numbers.
Two digits are used to write a two-digit number. The first digit on the right in a two-digit number is called the digit of the first digit or units digit, the second digit on the right is called the digit of the second digit or tens digit.
The numbers of the second ten in all mathematics textbooks for elementary grades are considered separately from other two-digit numbers. This is because 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 numbers in the numbers of the second ten, although they can name them correctly.
For example, when recording the number 12 (two-twenty) by ear, the child hears “two (a)” as the first word, so he can write the numbers in this order 21, but read this entry as “twelve”.
The formation of the concept of two-digit numbers is based on the concept of "digit".
The concept of a digit is basic in the decimal number system. A digit is understood as a certain place in a number entry in a positional number system (a digit is the position of a digit in a number entry).
Each position in this system has its own name and its conventional meaning: the number in the first position on the right means the number of units in the number; the figure in the second position from the right means the number of tens in the number, etc.
The numbers from 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 the notation of a two-digit (etc.) number means that the number contains a bit designated by zero, but there are no significant digits in it, i.e. the presence of 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; writing 23 will mean that in addition to 2 integer tens, the number contains 3 more units, in addition to integer tens.
The concept of "digit" plays a big role in the system of studying numbering, and is also the basis for mastering the so-called "numbering" cases of addition and subtraction, in which actions are performed by whole digits:
27 - 20 365 - 300
The ability to recognize and distinguish digits in numbers is the basis for the ability to decompose numbers into bit terms: 34 \u003d 30 + 4.
For numbers of the second ten, the concept of "digit composition" coincides with the concept of "decimal composition". For two-digit numbers containing more than one ten - these concepts do not match. For the number 34, the decimal composition is 3 tens and 4 ones. For the number 340, the bit composition is 300 and 40, and the decimal is 34 tens.
Acquaintance with the numbers of the second ten (11-20) is convenient to start with the way they are formed and the names of the numbers, accompanying it first with a model on sticks, and then reading the number according to the model:
Remembering the names of two-digit numbers in this case will not be difficult for children with a record that contradicts the name: 11, 13.17. (After all, in accordance with the tradition of reading in European scripts from left to right, in the name of these numbers, first the tens digit, and then the units digits!) hearing and reading by writing. The early introduction of symbolism plays a negative role in this case, both for remembering 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 units on the right. Thus, the child will fix the correct image of the concept in the internal plan, without special verbose explanations that are not always clear to him.
At the next stage, we offer the child the correlation of the real model and the symbolic notation:
one-on-twenty three-on-twenty seven-on-twenty
Then we move on to graphical models and to reading numbers according to the graphical model:
and then a symbolic notation of the bit composition of the numbers of the second ten:
Later, the concept of a category is introduced at school and children are introduced to the concept of "bit terms":
37 = 30 + 7; 624 = 600 + 20 + 4.
Using a decimal model instead of a bit 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 from the model (and vice versa, build a model from the name of the number), and then write :
When children study second-order numbers, we recommend that the teacher use the following types of tasks:
1) on the method of forming the numbers of the second ten:
Show thirteen sticks. How many tens and how many more individual sticks?
2) on the principle of formation of a natural series of numbers:
Draw a picture for the problem and solve it orally. “There were 10 cinemas in the city. They built 1 more. How many cinemas are there in the city?”
Decrease by 1: 16, 11, 13, 20
Zoom in 1:19, 18, 14, 17
Find the value of the expression: 10+ 1; 14+1; 18-1; 20-1.
(In all cases, one can refer to the fact that adding 1 leads to the next number, and decreasing by 1 leads to the previous number.)
3) on the local value of the digit in the notation of the number:
What does each digit in the number entry mean: 15, 13, 18, 11, 10.20?
(In the entry for the number 15, the number 1 indicates the number of tens, and the number 5 indicates the number of ones. In the entry for the number 20, the number 2 indicates that there are 2 tens in the number, and the number 0 indicates that there are no ones 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 a task, they refer to the order of numbers when counting.)
5) for the digit (decimal) composition:
10 + 3 = ... 13-3 = ... 13-10 = ...
12=10 + ... 15 = ... + 5
When performing a task, they refer to a bit (decimal) model of a number from a dozen (a bunch of sticks) and units (individual sticks),
6) to compare the numbers of the second ten:
Which number is larger: 13 or 15? 14 or 17? 18 or 14? 20 or 12?
When completing a task, you can compare two models of numbers from sticks (a quantitative model), or refer to the order of the numbers when counting (the smaller number is called when counting earlier), or rely on the process of counting and counting (counting two units to 13 we get 15, which means 15 more than 13).
Comparing the numbers of the second ten with single-digit numbers, one should refer to the fact that all single-digit numbers are less than two-digit ones:
What is the largest and smallest of these numbers: 12 6 18 10 7 20.
When comparing the numbers of 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
Comparing the lengths of the corresponding segments, the child clearly determines the setting of the comparison sign: 17< 19.
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:
- 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
- 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 .
- 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 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
