Natural numbers and their properties. Studying an exact subject: natural numbers - what are the numbers, examples and properties What are the natural numbers

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 the 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.

Natural numbers and their properties

Natural numbers are used to count objects in life. When writing any natural number, the numbers $0,1,2,3,4,5,6,7,8,9$ are used.

A sequence of natural numbers, each next number in which is $1$ greater than the previous one, forms a natural series, which begins with one (since one is the smallest natural number) and does not have the greatest value, i.e. infinite.

Zero is not considered a natural number.

Properties of the succession relation

All properties of natural numbers and operations on them follow from four properties of succession relations, which were formulated in 1891 by D. Peano:

One is a natural number that does not follow any natural number.

Each natural number is followed by one and only one number

Every natural number other than $1$ follows one and only one natural number

The subset of natural numbers containing the number $1$, and together with each number the number following it, contains all natural numbers.

If the entry of a natural number consists of one digit, it is called single-digit (for example, $2,6.9$, etc.), if the entry consists of two digits, it is called double-digit (for example, $12,18,45$), etc. Similarly. Two-digit, three-digit, four-digit, etc. In mathematics, numbers are called multi-valued.

Property of addition of natural numbers

Commutative property: $a+b=b+a$

The sum does not change when the terms are rearranged

Combinative property: $a+ (b+c) =(a+b) +c$

To add the sum of two numbers to a number, you can first add the first term, and then, to the resulting sum, add the second term

Adding zero does not change the number, and if you add any number to zero, you get the added number.

Properties of Subtraction

Property of subtracting a sum from a number $a-(b+c) =a-b-c$ if $b+c ≤ a$

In order to subtract a sum from a number, you can first subtract the first term from this number, and then the second term from the resulting difference.

The property of subtracting a number from the sum $(a+b) -c=a+(b-c)$ if $c ≤ b$

To subtract a number from a sum, you can subtract it from one term and add another term to the resulting difference.

If you subtract zero from a number, the number will not change

If you subtract it from the number itself, you get zero

Properties of Multiplication

Communicative $a\cdot b=b\cdot a$

The product of two numbers does not change when the factors are rearranged

Conjunctive $a\cdot (b\cdot c)=(a\cdot b)\cdot c$

To multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor

When multiplied by one, the product does not change $m\cdot 1=m$

When multiplied by zero, the product is zero

When there are no parentheses in the product notation, multiplication is performed in order from left to right

Properties of multiplication relative to addition and subtraction

Distributive property of multiplication relative to addition

$(a+b)\cdot c=ac+bc$

In order to multiply a sum by a number, you can multiply each term by this number and add the resulting products

For example, $5(x+y)=5x+5y$

Distributive property of multiplication relative to subtraction

$(a-b)\cdot c=ac-bc$

In order to multiply the difference by a number, multiply the minuend and subtrahend by this number and subtract the second from the first product

For example, $5(x-y)=5x-5y$

Comparison of natural numbers

For any natural numbers $a$ and $b$, only one of three relations can be satisfied: $a=b$, $a

The number that appears earlier in the natural series is considered smaller, and the number that appears later is considered larger. Zero is less than any natural number.

Example 1

Compare the numbers $a$ and $555$, if it is known that there is a certain number $b$, and the following relations hold: $a

Solution: Based on the specified property, because by condition $a

in any subset of natural numbers containing at least one number there is a smallest number

In mathematics, a subset is a part of a set. A set is said to be a subset of another if each element of the subset is also an element of the larger set

Often, to compare numbers, they find their difference and compare it with zero. If the difference is greater than $0$, but the first number is greater than the second, if the difference is less than $0$, then the first number is less than the second.

Rounding natural numbers

When full precision is not needed or is not possible, numbers are rounded, that is, they are replaced by close numbers with zeros at the end.

Natural numbers are rounded to tens, hundreds, thousands, etc.

When rounding a number to tens, it is replaced by the nearest number consisting of whole tens; such a number has the digit $0$ in the units place

When rounding a number to the nearest hundred, it is replaced by the nearest number consisting of whole hundreds; such a number must have the digit $0$ in the tens and ones place. Etc

The numbers to which this is rounded are called the approximate value of the number with an accuracy of the indicated digits. For example, if you round the number $564$ to tens, we find that you can round it down and get $560$, or with an excess and get $570$.

Rule for rounding natural numbers

If to the right of the digit to which the number is rounded there is a digit $5$ or a digit greater than $5$, then $1$ is added to the digit of this digit; otherwise this figure is left unchanged

All digits located to the right of the digit to which the number is rounded are replaced with zeros

Definition

Natural numbers are numbers that are used when counting or to indicate the serial number of an object among similar objects.

For example. Natural numbers will be: $2,37,145,1059,24411$

Natural numbers written in ascending order form a number series. It starts with the smallest natural number 1. The set of all natural numbers is denoted by $N=$1,2,3, \dots n, \ldots$$. It is infinite because there is no greatest natural number. If we add one to any natural number, we get the natural number following the given number.

Example

Exercise. Which of the following numbers are natural numbers?

$$-89 ; 7; \frac(4)(3) ; 34; 2 ; eleven ; 3.2; \sqrt(129) ; \sqrt(5)$$

Answer. $7 ; 34 ; 2 ; 11$

On the set of natural numbers, two basic arithmetic operations are introduced - addition and multiplication. To denote these operations, the symbols are used respectively " + " And " " (or " × " ).

Addition of natural numbers

Each pair of natural numbers $n$ and $m$ is associated with a natural number $s$, called a sum. The sum $s$ consists of as many units as there are in the numbers $n$ and $m$. The number $s$ is said to be obtained by adding the numbers $n$ and $m$, and they write

The numbers $n$ and $m$ are called terms. The operation of adding natural numbers has the following properties:

Commutativity: $n+m=m+n$
Associativity: $(n+m)+k=n+(m+k)$

Read more about adding numbers by following the link.

Example

Exercise. Find the sum of numbers:

$13+9 \quad$ and $ \quad 27+(3+72)$

Solution. $13+9=22$

To calculate the second sum, to simplify the calculations, we first apply to it the associativity property of addition:

$$27+(3+72)=(27+3)+72=30+72=102$$

Answer.$13+9=22 \quad;\quad 27+(3+72)=102$

Multiplication of natural numbers

Each ordered pair of natural numbers $n$ and $m$ is associated with a natural number $r$, called their product. The product $r$ contains as many units as there are in the number $n$, taken as many times as there are units in the number $m$. The number $r$ is said to be obtained by multiplying the numbers $n$ and $m$, and they write

$n \cdot m=r \quad $ or $ \quad n \times m=r$

The numbers $n$ and $m$ are called factors or factors.

The operation of multiplying natural numbers has the following properties:

Commutativity: $n \cdot m=m \cdot n$
Associativity: $(n \cdot m) \cdot k=n \cdot(m \cdot k)$

Read more about multiplying numbers by following the link.

Example

Exercise. Find the product of numbers:

12$\cdot 3 \quad $ and $ \quad 7 \cdot 25 \cdot 4$

Solution. By definition of the multiplication operation:

$$12 \cdot 3=12+12+12=36$$

We apply the associativity property of multiplication to the second product:

$$7 \cdot 25 \cdot 4=7 \cdot(25 \cdot 4)=7 \cdot 100=700$$

Answer.$12 \cdot 3=36 \quad;\quad 7 \cdot 25 \cdot 4=700$

The operation of addition and multiplication of natural numbers is related by the law of distributivity of multiplication relative to addition:

$$(n+m) \cdot k=n \cdot k+m \cdot k$$

The sum and product of any two natural numbers is always a natural number, therefore the set of all natural numbers is closed under the operations of addition and multiplication.

Also, on the set of natural numbers, you can introduce the operations of subtraction and division, as operations inverse to the operations of addition and multiplication, respectively. But these operations will not be uniquely defined for any pair of natural numbers.

The associative property of multiplication of natural numbers allows us to introduce the concept of a natural power of a natural number: the $n$th power of a natural number $m$ is the natural number $k$ obtained by multiplying the number $m$ by itself $n$ times:

To denote the $n$th power of a number $m$, the following notation is usually used: $m^(n)$, in which the number $m$ is called degree basis, and the number $n$ is exponent.

Example

Exercise. Find the value of the expression $2^(5)$

Solution. By definition of the natural power of a natural number, this expression can be written as follows

$$2^(5)=2 \cdot 2 \cdot 2 \cdot 2 \cdot 2=32$$

In mathematics, there are several different sets of numbers: real, complex, integer, rational, irrational, ... In our Everyday life We most often use natural numbers, since we encounter them when counting and when searching, designating the number of objects.

In contact with

What numbers are called natural numbers?

From ten digits you can write absolutely any existing sum of classes and ranks. Natural values are considered to be those which are used:

When counting any objects (first, second, third, ... fifth, ... tenth).
When indicating the number of items (one, two, three...)

N values are always integer and positive. There is no largest N because the set of integer values is unlimited.

Attention! Natural numbers are obtained when counting objects or when indicating their quantity.

Absolutely any number can be decomposed and presented in the form of digit terms, for example: 8.346.809=8 million+346 thousand+809 units.

Set N

The set N is in the set real, integer and positive. On the diagram of sets, they would be located in each other, since the set of natural ones is part of them.

The set of natural numbers is denoted by the letter N. This set has a beginning, but no end.

There is also an extended set N, where zero is included.

Smallest natural number

In most math schools, the smallest value of N is considered a unit, since the absence of objects is considered emptiness.

But in foreign mathematical schools, for example in French, it is considered natural. The presence of zero in the series makes the proof easier some theorems.

A series of values N that includes zero is called extended and is denoted by the symbol N0 (zero index).

Series of natural numbers

N series is a sequence of all N sets of digits. This sequence has no end.

The peculiarity of the natural series is that the next number will differ by one from the previous one, that is, it will increase. But the meanings cannot be negative.

Attention! For ease of counting, there are classes and categories:

Units (1, 2, 3),
Tens (10, 20, 30),
Hundreds (100, 200, 300),
Thousands (1000, 2000, 3000),
Tens of thousands (30,000),
Hundreds of thousands (800.000),
Millions (4000000), etc.

All N

All N are in the set of real, integer, non-negative values. They are theirs integral part.

These values go to infinity, they can belong to the classes of millions, billions, quintillions, etc.

For example:

Five apples, three kittens,
Ten rubles, thirty pencils,
One hundred kilograms, three hundred books,
A million stars, three million people, etc.

Sequence in N

In different mathematical schools you can find two intervals to which the sequence N belongs:

from zero to plus infinity, including ends, and from one to plus infinity, including ends, that is, everything positive integer answers.

N sets of digits can be either even or odd. Let's consider the concept of oddity.

Odd (any odd number ends in the numbers 1, 3, 5, 7, 9.) with two have a remainder. For example, 7:2=3.5, 11:2=5.5, 23:2=11.5.

What does even N mean?

Any even sums of classes end in numbers: 0, 2, 4, 6, 8. When even N is divided by 2, there will be no remainder, that is, the result is the whole answer. For example, 50:2=25, 100:2=50, 3456:2=1728.

Important! A number series of N cannot consist only of even or odd values, since they must alternate: even is always followed by odd, followed by even again, etc.

Properties N

Like all other sets, N has its own special properties. Let's consider the properties of the N series (not extended).

The value that is the smallest and that does not follow any other is one.
N represent a sequence, that is, one natural value follows another(except for one - it is the first).
When we perform computational operations on N sums of digits and classes (add, multiply), then the answer it always turns out natural meaning.
Permutation and combination can be used in calculations.
Each subsequent value cannot be less than the previous one. Also in the N series the following law will apply: if the number A is less than B, then in the number series there will always be a C for which the equality holds: A+C=B.
If we take two natural expressions, for example A and B, then one of the expressions will be true for them: A = B, A is greater than B, A is less than B.
If A is less than B, and B is less than C, then it follows that that A is less than C.
If A is less than B, then it follows that: if we add the same expression (C) to them, then A + C is less than B + C. It is also true that if these values are multiplied by C, then AC is less than AB.
If B is greater than A, but less than C, then it is true: B-A is less than C-A.

Attention! All of the above inequalities are also valid in the opposite direction.

What are the components of multiplication called?

In many simple and even complex problems, finding the answer depends on the skills of schoolchildren.

In order to multiply quickly and correctly and be able to solve inverse problems, you need to know the components of multiplication.

15. 10=150. In this expression there are 15 and 10 are multipliers, and 150 is a product.

Multiplication has properties that are necessary when solving problems, equations and inequalities:

Rearranging the factors will not change the final product.
To find an unknown factor, you need to divide the product by a known factor (true for all factors).

For example: 15 . X=150. Let's divide the product by a known factor. 150:15=10. Let's do a check. 15 . 10=150. According to this principle, they even decide complex linear equations(to simplify them).

Important! A product can consist of more than just two factors. For example: 840=2 . 5. 7. 3. 4

What are natural numbers in mathematics?

Places and classes of natural numbers

Conclusion

Let's summarize. N is used when counting or indicating the number of items. The series of natural sets of numbers is infinite, but it includes only integer and positive sums of digits and classes. Multiplication is also necessary in order to to count objects, as well as for solving problems, equations and various inequalities.

Numbers are an abstract concept. They are a quantitative characteristic of objects and can be real, rational, negative, integer and fractional, as well as natural.

The natural series is usually used when counting, in which quantity notations naturally arise. Acquaintance with counting begins in early childhood. What kid avoided funny rhymes that used elements of natural counting? "One, two, three, four, five... The bunny went out for a walk!" or "1, 2, 3, 4, 5, 6, 7, 8, 9, 10, the king decided to hang me..."

For any natural number, you can find another one greater than it. This set is usually denoted by the letter N and should be considered infinite in the direction of increase. But this set has a beginning - it is one. Although there are French natural numbers, the set of which also includes zero. But the main distinguishing features of both sets is the fact that they do not include either fractional or negative numbers.

The need to count a variety of objects arose in prehistoric times. Then the concept of “natural numbers” was supposedly formed. Its formation occurred throughout the entire process of changing a person’s worldview and the development of science and technology.

However, they could not yet think abstractly. It was difficult for them to understand what the commonality of the concepts of “three hunters” or “three trees” was. Therefore, when indicating the number of people, one definition was used, and when indicating the same number of objects of a different kind, a completely different definition was used.

And it was extremely short. It contained only the numbers 1 and 2, and the count ended with the concepts of “many”, “herd”, “crowd”, “heap”.

Later, a more progressive and broader account was formed. An interesting fact is that there were only two numbers - 1 and 2, and the next numbers were obtained by adding.

An example of this was the information that has reached us about the numerical series of the Australian tribe. They had 1 for the word “Enza”, and 2 for the word “petcheval”. The number 3 therefore sounded like “petcheval-Enza”, and 4 sounded like “petcheval-petcheval”.

Most peoples recognized fingers as the standard of counting. Further development of the abstract concept of “natural numbers” followed the path of using notches on a stick. And then it became necessary to designate a dozen with another sign. The ancient people found our way out - they began to use another stick, on which notches were made to indicate tens.

The ability to reproduce numbers expanded enormously with the advent of writing. At first, numbers were depicted as lines on clay tablets or papyrus, but gradually other writing icons began to be used. This is how Roman numerals appeared.

Much later, they appeared that opened up the possibility of writing numbers with a relatively small set of characters. Today it is not difficult to write down such huge numbers as the distance between planets and the number of stars. You just have to learn to use degrees.

Euclid in the 3rd century BC in the book “Elements” establishes the infinity of the numerical set, and Archimedes in “Psamita” reveals the principles for constructing the names of arbitrarily large numbers. Almost until the middle of the 19th century, people did not face the need for a clear formulation of the concept of “natural numbers”. The definition was required with the advent of the axiomatic mathematical method.

And in the 70s of the 19th century he formulated a clear definition of natural numbers, based on the concept of set. And today we already know that natural numbers are all integers, starting from 1 to infinity. Young children, taking their first step in becoming acquainted with the queen of all sciences - mathematics - begin to study these very numbers.