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The history of natural numbers began in primitive times. Since ancient times, people have counted objects. For example, in trade, a commodity account was needed, or in construction, a material account. Yes, even in everyday life, too, I had to count things, products, 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 are the numbers that 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 call the set of natural numbers, is denoted by the symbol N.

Table of natural numbers.

natural row.

Natural numbers written in ascending order in a row form natural series or series of natural numbers.

Properties of the natural series:

• The smallest natural number is one.
• In the natural series, the next number is greater than the previous one by one. (1, 2, 3, …) Three dots or three dots are used if it is impossible to complete the sequence of numbers.
• The natural series has no maximum number, it is infinite.

Example #1:
Write the first 5 natural numbers.
Solution:
Natural numbers start with 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 number starts with one.

Example #4:
What is the last number in the natural series? What is the largest natural number?
Answer: The natural number starts from 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 the unit 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 after the numbers: a) 5, b) 67, c) 9998.
Answer: a) 6, b) 68, c) 9999.

Example #7:
How many numbers are 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:
Name the previous number after the number 11.
Answer: 10.

Example #9:
What numbers are used to count objects?
Answer: natural numbers.

The lesson "Notation of Natural Numbers" is the first lesson in the fifth grade mathematics course and is a continuation, and in some moments, a repetition of a similar topic that was studied in the course elementary school. As a result, students often do not perceive the educational material very attentively. Therefore, in order to achieve maximum interest and concentration of attention, it is necessary to introduce new methods of explanation, for example, use the presentation "Notation of Natural Numbers".

The lesson begins with a repetition of a series of digits, 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 natural side by side and an example of the first twenty of its elements is given. Particular attention during the presentation is given to the meaning of the number, depending on its place in the notation of the number. To do this, we considered writing a number by digits. Using effective and non-intrusive animation, students are shown what the same number means depending on where it is: in the units 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 the course of mathematics, 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. The attention of students is also focused on the fact that 0 are not classified as natural numbers.

In the presentation, in detail, using examples, the concepts of single-digit, two-digit, three-digit and four-digit numbers are considered. Records of one million and one billion are considered. Special attention is paid to the correct reading of multi-digit numbers and their division into classes. Using a table for writing a multi-digit number with the allocation of classes and digits, 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 result of mastering new material by students, this presentation development contains a list of questions that fully cover the material presented. This will allow the teacher to respond as quickly as possible to the moments that were not fully understood by the students. as a result of studying this topic.

Since the presentation "Denotation of Natural Numbers" presents the titled topic at an understandable and accessible level, the presentation of the educational material is logical and consistent, it can be successfully used not only during the class-lesson explanation of this topic, but also in self-study or distance learning by schoolchildren.

The simplest number is natural number. They are used in Everyday life for counting items, i.e. to calculate their number and order.

What is a natural number: natural numbers name the numbers that are used for counting items or to indicate the serial number of any item from all homogeneous items.

Integersare 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 largest natural number. When counting the number zero is not used, so zero is a natural number.

natural series of numbers is the sequence of all natural numbers. Write natural numbers:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ...

In natural numbers, each number is one more than the previous one.

How many numbers are in the natural series? The natural series is infinite, there is no largest natural number.

Decimal since 10 units of any category form 1 unit of the highest order. positional so how the value of a digit depends on its place in the number, i.e. from the category where it is recorded.

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 digits of the class is called itsdischarge.

Comparison of natural numbers.

Of the 2 natural numbers, the number that is called earlier in the count is less. 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.

Zero place

There are two approaches to the definition of natural numbers:

• counting (numbering) items ( the first, second, third, fourth, fifth…);
• natural numbers - numbers that arise when quantity designation items ( 0 items, 1 item, 2 items, 3 items, 4 items, 5 items…).

In the first case, the series of natural numbers starts from one, in the second - from zero. There is no common opinion for most mathematicians on the preference of the first or second approach (that is, whether to consider zero as a natural number or not). The vast majority of Russian sources have traditionally adopted the first approach. The second approach, for example, is used in the works Nicolas Bourbaki, where natural numbers are defined as power finite sets. The presence of zero facilitates the formulation and proof of many theorems in the arithmetic of natural numbers, so the first approach introduces the useful notion extended natural series, including zero .

The set of all natural numbers is usually denoted by the symbol . International Standards ISO 31-11(1992) and ISO 80000-2(2009) establish the following designations:

In Russian sources, this standard is not yet observed - in them the symbol N (\displaystyle \mathbb (N) ) denotes natural numbers without zero, and the extended natural series is denoted N 0 , Z + , Z ⩾ 0 (\displaystyle \mathbb (N) _(0),\mathbb (Z) _(+),\mathbb (Z) _(\geqslant 0)) etc.

Axioms that make it possible to define the set of natural numbers

Peano axioms for natural numbers

Lots of N (\displaystyle \mathbb (N) ) will be called the set of natural numbers if some element is fixed 1 (unit), function S (\displaystyle S) c domain of definition N (\displaystyle \mathbb (N) ), called the succession function ( S: N (\displaystyle S\colon \mathbb (N) )), and the following conditions are met:

1. element one belongs to this set ( 1 ∈ N (\displaystyle 1\in \mathbb (N) )), that is, is a natural number;
2. the number following the natural number is also a natural number (if , then S (x) ∈ N (\displaystyle S(x)\in \mathbb (N) ) or, in shorter notation, S: N → N (\displaystyle S\colon \mathbb (N) \to \mathbb (N) ));
3. one does not follow any natural number ( ∄ x ∈ N (S (x) = 1) (\displaystyle \nexists x\in \mathbb (N) \ (S(x)=1)));
4. if natural number a (\displaystyle a) immediately follows as a natural number b (\displaystyle b), and for the natural number c (\displaystyle c), then b (\displaystyle b) and c (\displaystyle c) is the same number (if S (b) = a (\displaystyle S(b)=a) and S (c) = a (\displaystyle S(c)=a), then b = c (\displaystyle b=c));
5. (axiom of induction) if any sentence (statement) P (\displaystyle P) proved for a natural number n = 1 (\displaystyle n=1) (induction base) and if from the assumption that it is true for another natural number n (\displaystyle n), it follows that it is true for the following n (\displaystyle n) natural number ( induction hypothesis), then this proposition is true for all natural numbers (let P (n) (\displaystyle P(n))- some single (unary) predicate, whose parameter is a natural number n (\displaystyle n). Then if P (1) (\displaystyle P(1)) and ∀ n (P (n) ⇒ P (S (n))) (\displaystyle \forall n\;(P(n)\Rightarrow P(S(n)))), then ∀ n P (n) (\displaystyle \forall n\;P(n))).

The above axioms reflect our intuitive understanding of the natural series and number line.

The fundamental fact is that these axioms essentially uniquely determine the natural numbers (the categorical nature of the system of Peano's axioms). Namely, it can be proved (see and also a short proof) that if (N , 1 , S) (\displaystyle (\mathbb (N) ,1,S)) and (N ~ , 1 ~ , S ~) (\displaystyle ((\tilde (\mathbb (N) )),(\tilde (1)),(\tilde (S)))) are two models for the system of Peano's axioms, then they must be isomorphic, that is, there is an invertible mapping ( bijection) f: N → N ~ (\displaystyle f\colon \mathbb (N) \to (\tilde (\mathbb (N) ))) such that f (1) = 1 ~ (\displaystyle f(1)=(\tilde (1))) and f (S (x)) = S ~ (f (x)) (\displaystyle f(S(x))=(\tilde (S))(f(x))) for all x ∈ N (\displaystyle x\in \mathbb (N) ).

Therefore, it suffices to fix as N (\displaystyle \mathbb (N) ) any one particular model of the set of natural numbers.

Sometimes, especially in foreign and translated literature, Peano's first and third axioms replace one with zero. In this case, zero is considered a natural number. When defined in terms of classes of equivalent sets, zero is a natural number by definition. It would be unnatural to specifically discard it. In addition, this would significantly complicate the further construction and application of the theory, since in most constructions zero, like the empty set, is not something isolated. Another advantage of considering zero as a natural number is that N (\displaystyle \mathbb (N) ) forms monoid. As already mentioned, in Russian literature, zero is traditionally excluded from the number of natural numbers.

Set-theoretic definition of natural numbers (Frege-Russell definition)

Thus, natural numbers are also introduced, based on the concept of a set, according to two rules:

Numbers given in this way are called ordinal.

Let us describe the first few ordinal numbers and their corresponding natural numbers:

The value of the set of natural numbers

The value of an infinite set is characterized by the concept " cardinality of the set”, which is a generalization of the number of elements of a finite set to infinite sets. In size (i.e. power), the set of natural numbers is greater than any finite set, but less than any interval, for example, the interval (0 , 1) (\displaystyle (0,1)). The set of natural numbers has the same cardinality as the set of rational numbers. A set of the same cardinality as the set of natural numbers is called countable set. Thus, the set of members of any sequences countable. At the same time, there is a sequence in which each natural number occurs an infinite number of times, since the set of natural numbers can be represented as countable an association non-intersecting countable sets (for example, N = ⋃ k = 0 ∞ (⋃ n = 0 ∞ (2 n + 1) 2 k) (\displaystyle \mathbb (N) =\bigcup \limits _(k=0)^(\infty )\left(\ bigcup \limits _(n=0)^(\infty )(2n+1)2^(k)\right))).

Operations on natural numbers

To closed operations(operations that do not output a result from the set of natural numbers) on natural numbers include the following arithmetic operations:

Additionally, two more operations are considered (from a formal point of view, they are not operations on natural numbers, since they are not defined for all pairs of numbers (sometimes they exist, sometimes they don't)):

It should be noted that the operations of addition and multiplication are fundamental. In particular, ring integers determined precisely through binary operations addition and multiplication.

Basic properties

• commutativity additions:

• Commutativity of multiplication:

• Associativity additions:

• Associativity of multiplication:

• distributivity multiplication with respect to addition:

Algebraic structure

Addition turns the set of natural numbers into semigroup with a unit, the role of the unit is played by 0 . Multiplication also transforms the set of natural numbers into a semigroup with unit, while the identity element is 1 . By using closures with respect to addition-subtraction and multiplication-division operations, groups of integers are obtained Z (\displaystyle \mathbb (Z) ) and rational positive numbers Q + ∗ (\displaystyle \mathbb (Q) _(+)^(*)) respectively.