The first sixteen numbers of the natural series. Notation of natural numbers
Numbers intended for counting objects and answering the question “how many?” ("How many
balls?", "How many apples?", "How many soldiers?"), are called natural.
If you write them in order, from smallest number to largest, you get a natural series of numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 99, 100, 101, …, 999, 1000, 1001 …
The natural series of numbers begins with the number 1.
Each next natural number is 1 greater than the previous one.
The natural series of numbers is infinite.
Numbers can be even or odd. Even numbers are divisible by two, but odd numbers are not divisible by two.
Series of odd numbers:
1, 3, 5, 7, 9, 11, 13, …, 99, 101, …, 999, 1001, 1003 …
Series of even numbers:
2, 4, 6, 8, 10, 12, 14, …, 98, 100, …, 998, 1000, 1002 …
In the natural series, odd and even numbers alternate:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …, 99, 100, …, 999, 1000 …
How to compare natural numbers
When comparing two natural numbers, the one to the right in the natural series is greater:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 …
So, seven is more than three, and five is more than one.
In mathematics, the word “less” is written using the sign “<», а для записи слова «больше» - знак « > ».
The sharp corner of the greater than and less than symbols always points towards the smaller of the two numbers.
The entry 7 > 3 is read as “seven over three.”
Entry 3< 7 читается как «три меньше семи».
The entry 5 > 1 is read as “five over one.”
Entry 1< 5 читается как «один меньше пяти».
The word “equal” in mathematics is replaced with the sign “=”:
When the numbers are large, it is difficult to immediately say which one is to the right in the natural series.
When comparing two natural numbers with different numbers of digits, the one with the most digits is greater.
For example, 233,000< 1 000 000, потому что в первом числе шесть цифр, а во втором - семь.
Multi-digit natural numbers with the same number of digits are compared bitwise, starting with the most significant digit.
First, the units of the most significant digit are compared, then the next one, the next one, and so on. For example, let's compare the numbers 5401 and 5430:
5401 = 5 thousand 4 hundreds 0 tens 1 unit;
5430 = 5 thousand 4 hundreds 3 tens 0 units.
Comparing units of thousands. In the place of units of thousands of the number 5401 there are 5 units, in the place of units of thousands of the number 5430 there are 5 units. By comparing units of thousands, it is still impossible to say which number is larger.
Comparing hundreds. In the hundreds place of the number 5401 there are 4 units, in the hundreds place the number 5430 is also 4 units. We must continue the comparison.
Comparing tens. In the tens place of the number 5401 there are 0 units, in the tens place of the number 5430 there are 3 units.
Comparing, we get 0< 3, поэтому 5401 < 5430.
Numbers can be arranged in descending or ascending order.
If in a record of several natural numbers each next number is less than the previous one, then the numbers are said to be written in descending order.
Let's write down the numbers 5, 22, 13, 800 in descending order.
Let's find a larger number. The number 5 is a single digit number, 13 and 22 are two digit numbers, 800 is a three digit number and therefore the largest. We write 800 in the first place.
Of the two-digit numbers 13 and 22, the greater is 22. After the number 800 we write the number 22, and then 13.
The smallest number is the single-digit number 5. We write it last.
800, 22, 13, 5 - recording these numbers in descending order.
If in a record of several natural numbers each next number is greater than the previous one, then the numbers are said to be written in ascending order.
How to write the numbers 15, 2, 31, 278, 298 in ascending order?
Among the numbers 15, 2, 31, 278, 298 we will find the smaller one.
This is a single-digit number 2. Let's write it in first place.
From the two-digit numbers 15 and 31, choose the smaller one - 15, write it in second place, and after it - 31.
Of the three-digit numbers, 278 is the smallest, we write it after the number 31, and the last we write the number 298.
2, 15, 21, 278, 298 - writing these numbers in ascending order
The lesson “Denotation of natural numbers” is the first lesson in the fifth-grade mathematics course and is a continuation, and at some points, a repetition of a similar topic that was studied in the elementary school course. As a result, students often do not perceive the educational material very carefully. Therefore, to achieve maximum interest and concentration, it is necessary to introduce new methods of explanation, for example, use the presentation “Designation of natural numbers.”
The lesson begins with a review of a series of numbers, as well as the concept of a natural number and its decimal notation. It is explained that the sequence of all natural numbers is called a natural series and an example of its first twenty elements is given. During the presentation, special attention is paid to the meaning of the number depending on its place in the number record. To do this, we consider writing a number by digits. Using effective and non-intrusive animation, students are shown what the same number means depending on where it is located: in the ones place, in the tens place, etc.
It is not uncommon to see that, along with the fact that the number zero is often used both in everyday life and in mathematics courses, schoolchildren experience difficulty when they need to explain what kind of number it is. To increase the effectiveness of understanding the concept of zero, an example of a score in a football match is given. Students’ attention is also focused on the fact that 0 are not classified as natural numbers.
The presentation examines in detail, using examples, the concepts of single-digit, two-digit, three-digit and four-digit numbers. Records of one million and one billion were considered. Special attention is paid to the correct reading of multi-digit numbers and their breakdown into classes. Using a table for writing a multi-digit number highlighting classes and ranks, it is demonstrated that the left class, unlike all the others, can have less than three digits.
In order to be able to check the results of students’ assimilation of new material, this presentation development contains a list of questions that fully cover the material presented. This will allow the teacher to react as quickly as possible to points that are not fully understood by schoolchildren. as a result of studying this topic.
Since the presentation “Designation of Natural Numbers” presents the topic at a clear and accessible level, the presentation of educational material is logical and consistent, it can be successfully used not only during a classroom explanation of this topic, but also during independent or distance learning by schoolchildren.
The simplest number is natural number. They are used in everyday life for counting objects, i.e. to calculate their number and order.
What is a natural number: natural numbers name the numbers that are used to counting items or to indicate the serial number of any item from all homogeneous items.
Integers- these are numbers starting from one. They are formed naturally when counting.For example, 1,2,3,4,5... -first natural numbers.
Smallest natural number- one. There is no greatest natural number. When counting the number Zero is not used, so zero is a natural number.
Natural number series is the sequence of all natural numbers. Writing natural numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ...
In the natural series, each number is greater than the previous one by one.
How many numbers are there in the natural series? The natural series is infinite; the largest natural number does not exist.
Decimal since 10 units of any digit form 1 unit of the highest digit. Positionally so how the meaning of a digit depends on its place in the number, i.e. from the category where it is written.
Classes of natural numbers.
Any natural number can be written using 10 Arabic numerals:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
To read natural numbers, they are divided, starting from the right, into groups of 3 digits each. 3 first the numbers on the right are the class of units, the next 3 are the class of thousands, then the classes of millions, billions andetc. Each of the class digits is called itsdischarge.
Comparison of natural numbers.
Of 2 natural numbers, the smaller is the number that is called earlier when counting. For example, number 7 less 11 (written like this:7 < 11 ). When one number is greater than the second, it is written like this:386 > 99 .
Table of digits and classes of numbers.
|
1st class unit |
1st digit of the unit 2nd digit tens 3rd place hundreds |
|
2nd class thousand |
1st digit of unit of thousands 2nd digit tens of thousands 3rd category hundreds of thousands |
|
3rd class millions |
1st digit of unit of millions 2nd category tens of millions 3rd category hundreds of millions |
|
4th class billions |
1st digit of unit of billions 2nd category tens of billions 3rd category hundreds of billions |
|
Numbers from 5th grade and above are considered large numbers. Units of the 5th class are trillions, 6th class - quadrillions, 7th class - quintillions, 8th class - sextillions, 9th class - eptillions. Basic properties of natural numbers.
Operations on natural numbers. 4. Division of natural numbers is the inverse operation of multiplication. If b ∙ c = a, That
Formulas for division: a: 1 = a a: a = 1, a ≠ 0 0: a = 0, a ≠ 0 (A∙ b) : c = (a:c) ∙ b (A∙ b) : c = (b:c) ∙ a Numerical expressions and numerical equalities. A notation where numbers are connected by action signs is numerical expression. For example, 10∙3+4; (60-2∙5):10. Records where 2 numeric expressions are combined with an equal sign are numerical equalities. Equality has left and right sides. The order of performing arithmetic operations. Adding and subtracting numbers are operations of the first degree, while multiplication and division are operations of the second degree. When a numerical expression consists of actions of only one degree, they are performed sequentially from left to right. When expressions consist of actions of only the first and second degrees, then the actions are performed first second degree, and then - actions of the first degree. When there are parentheses in an expression, the actions in the parentheses are performed first. For example, 36:(10-4)+3∙5= 36:6+15 = 6+15 = 21. |
Integers– numbers that are used to count objects . Any natural number can be written using ten numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This type of number is called decimal
The sequence of all natural numbers is called natural next to .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ...
The most small natural number is one (1). In the natural series, each next number is 1 greater than the previous one. Natural series endless, there is no largest number in it.
The meaning of a digit depends on its place in the number record. For example, the number 4 means: 4 units if it is in the last place in the number record (in units place); 4 ten, if she is in second to last place (in the tens place); 4 hundreds, if she is in third place from the end (V hundreds place).
The number 0 means absence of units of this category in the decimal notation of a number. It also serves to designate the number “ zero" This number means "none". The score 0:3 in a football match means that the first team did not score a single goal against the opponent.
Zero do not include to natural numbers. And indeed, counting objects never starts from scratch.
If the notation of a natural number consists of one sign – one digit, then it is called unambiguous. Those. unambiguousnatural number– a natural number, the notation of which consists of one sign – one digit. For example, the numbers 1, 6, 8 are single digits.
Double digitnatural number– a natural number, the notation of which consists of two characters – two digits.
For example, the numbers 12, 47, 24, 99 are two-digit numbers.
Also, based on the number of characters in a given number, they give names to other numbers:
numbers 326, 532, 893 – three-digit;
numbers 1126, 4268, 9999 – four-digit etc.
Two-digit, three-digit, four-digit, five-digit, etc. numbers are called multi-digit numbers .
To read multi-digit numbers, they are divided, starting from the right, into groups of three digits each (the leftmost group may consist of one or two digits). These groups are called classes.
Million– this is a thousand thousand (1000 thousand), it is written 1 million or 1,000,000.
Billion- that's 1000 million. It is written as 1 billion or 1,000,000,000.
The first three digits on the right make up the class of units, the next three – the class of thousands, then come the classes of millions, billions, etc. (Fig. 1).
Rice. 1. Millions class, thousands class and units class (from left to right)
The number 15389000286 is written in the bit grid (Fig. 2).
Rice. 2. Bit grid: number 15 billion 389 million 286
This number has 286 units in the units class, zero units in the thousands class, 389 units in the millions class, and 15 units in the billions class.
The history of natural numbers began in primitive times. Since ancient times, people have counted objects. For example, in trade you needed an account of goods or in construction an account of materials. Yes, even in everyday life I also had to count things, food, livestock. At first, numbers were used only for counting in life, in practice, but later, with the development of mathematics, they became part of science.
Integers- these are the numbers we use when counting objects.
For example: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ….
Zero is not a natural number.
All natural numbers, or let's say the set of natural numbers, are denoted by the symbol N.
Table of natural numbers.
Natural series.
Natural numbers written in a row in ascending order form natural series or a series of natural numbers.
Properties of the natural series:
- The smallest natural number is one.
- In a natural series, the next number is greater than the previous one by one. (1, 2, 3, ...) Three dots or ellipses are placed if it is impossible to complete the sequence of numbers.
- The natural series does not have a greatest number, it is infinite.
Example #1:
Write the first 5 natural numbers.
Solution:
Natural numbers start from one.
1, 2, 3, 4, 5
Example #2:
Is zero a natural number?
Answer: no.
Example #3:
What is the first number in the natural series?
Answer: The natural series starts from one.
Example #4:
What is the last number in the natural series? What is the largest natural number?
Answer: The natural series begins with one. Each next number is greater than the previous one by one, so the last number does not exist. There is no largest number.
Example #5:
Does one in the natural series have a previous number?
Answer: no, because one is the first number in the natural series.
Example #6:
Name the next number in the natural series: a)5, b)67, c)9998.
Answer: a)6, b)68, c)9999.
Example #7:
How many numbers are there in the natural series between the numbers: a) 1 and 5, b) 14 and 19.
Solution:
a) 1, 2, 3, 4, 5 – three numbers are between the numbers 1 and 5.
b) 14, 15, 16, 17, 18, 19 – four numbers are between the numbers 14 and 19.
Example #8:
Say the previous number after 11.
Answer: 10.
Example #9:
What numbers are used when counting objects?
Answer: natural numbers.
