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Any natural number. Natural numbers and their properties

Mathematics emerged from general philosophy around the sixth century BC. e., and from that moment her victorious march around the world began. Each stage of development introduced something new - elementary counting evolved, transformed into differential and integral calculus, centuries passed, formulas became more and more confusing, and the moment came when “the most complex mathematics began - all numbers disappeared from it.” But what was the basis?

The beginning began

Natural numbers appeared along with the first mathematical operations. One spine, two spines, three spines... They appeared thanks to Indian scientists who developed the first positional

The word “positionality” means that the location of each digit in a number is strictly defined and corresponds to its rank. For example, the numbers 784 and 487 are the same numbers, but the numbers are not equivalent, since the first includes 7 hundreds, while the second only 4. The Indian innovation was picked up by the Arabs, who brought the numbers to the form that we know Now.

In ancient times, numbers were given a mystical meaning; Pythagoras believed that number underlies the creation of the world along with the basic elements - fire, water, earth, air. If we consider everything only from the mathematical side, then what is a natural number? The field of natural numbers is denoted as N and is an infinite series of numbers that are integers and positive: 1, 2, 3, … + ∞. Zero is excluded. Used primarily to count items and indicate order.

What is it in mathematics? Peano's axioms

Field N is the basic one on which elementary mathematics is based. Over time, fields of integer, rational,

What is a natural number was clarified earlier in simple language, below we will consider a mathematical definition based on Peano’s axioms.

The unit is considered natural number.
The number that follows a natural number is a natural number.
There is no natural number before one.
If the number b follows both the number c and the number d, then c=d.
An axiom of induction, which in turn shows what a natural number is: if some statement that depends on a parameter is true for the number 1, then we assume that it also works for the number n from the field of natural numbers N. Then the statement is also true for n =1 from the field of natural numbers N.

Basic operations for the field of natural numbers

Since field N was the first for mathematical calculations, both the domains of definition and the ranges of values of a number of operations below belong to it. They may be closed or not. The main difference is that closed operations are guaranteed to leave the result within the set N, regardless of what numbers are involved. It is enough that they are natural. The outcome of other numerical interactions is no longer so clear and directly depends on what kind of numbers are involved in the expression, since it may contradict the main definition. So, closed operations:

addition - x + y = z, where x, y, z are included in the N field;
multiplication - x * y = z, where x, y, z are included in the N field;
exponentiation - x y, where x, y are included in the N field.

The remaining operations, the result of which may not exist in the context of the definition of “what is a natural number,” are as follows:

Properties of numbers belonging to the field N

All further mathematical reasoning will be based on the following properties, the most trivial, but no less important.

The commutative property of addition is x + y = y + x, where the numbers x, y are included in the field N. Or the well-known “the sum does not change by changing the places of the terms.”
The commutative property of multiplication is x * y = y * x, where the numbers x, y are included in the N field.
The combinational property of addition is (x + y) + z = x + (y + z), where x, y, z are included in the N field.
The matching property of multiplication is (x * y) * z = x * (y * z), where the numbers x, y, z are included in the N field.
distributive property - x (y + z) = x * y + x * z, where the numbers x, y, z are included in the N field.

Pythagorean table

One of the first steps in students’ knowledge of the entire structure of elementary mathematics after they have understood for themselves which numbers are called natural numbers is the Pythagorean table. It can be considered not only from the point of view of science, but also as a most valuable scientific monument.

This multiplication table has undergone a number of changes over time: zero has been removed from it, and numbers from 1 to 10 represent themselves, without taking into account orders (hundreds, thousands...). It is a table in which the row and column headings are numbers, and the contents of the cells where they intersect are equal to their product.

In teaching practice last decades there was a need to memorize the Pythagorean table “in order,” that is, memorization came first. Multiplication by 1 was excluded because the result was a multiplier of 1 or greater. Meanwhile, in the table with the naked eye you can notice a pattern: the product of numbers increases by one step, which is equal to the title of the line. Thus, the second factor shows us how many times we need to take the first one in order to obtain the desired product. This system much more convenient than that practiced in the Middle Ages: even understanding what a natural number is and how trivial it is, people managed to complicate their everyday counting by using a system that was based on powers of two.

Subset as the cradle of mathematics

On at the moment the field of natural numbers N is considered only as one of the subsets complex numbers, but this does not make them less valuable in science. Natural number is the first thing a child learns when studying himself and the world around us. One finger, two fingers... Thanks to him, a person develops logical thinking, as well as the ability to determine cause and deduce effect, paving the way for great discoveries.

Natural numbers can be used for counting (one apple, two apples, etc.)

Natural numbers(from lat. naturalis- natural; natural numbers) - numbers that arise naturally when counting (for example, 1, 2, 3, 4, 5...). The sequence of all natural numbers arranged in ascending order is called natural next to.

There are two approaches to defining natural numbers:

counting (numbering) items ( first, second, third, fourth, fifth"…);
natural numbers are 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 consensus among most mathematicians on whether the first or second approach is preferable (that is, whether zero should be considered a natural number or not). The overwhelming majority of Russian sources traditionally adopt the first approach. The second approach, for example, is used in the works of Nicolas Bourbaki, where the natural numbers are defined as cardinalities of finite sets.

Negative and non-integer (rational, real, ...) numbers are not considered natural numbers.

The set of all natural numbers It is customary to denote the symbol N (\displaystyle \mathbb (N)) (from lat. naturalis- natural). The set of natural numbers is infinite, since for any natural number n (\displaystyle n) there is a natural number greater than n (\displaystyle n) .

The presence of zero makes it easier to formulate and prove many theorems in the arithmetic of natural numbers, so the first approach introduces useful concept extended natural range, including zero. The extended series is denoted N 0 (\displaystyle \mathbb (N) _(0)) or Z 0 (\displaystyle \mathbb (Z) _(0)) .

Axioms that allow us to determine the set of natural numbers

Peano's axioms for natural numbers

Main article: Peano's axioms

We will call a set N (\displaystyle \mathbb (N) ) a set of natural numbers if some element is fixed 1 (unit) belonging to N (\displaystyle \mathbb (N) ) (1 ∈ N (\displaystyle 1\in \mathbb (N) )), and a function S (\displaystyle S) with domain N (\displaystyle \mathbb (N) ) and the range N (\displaystyle \mathbb (N) ) (called the succession function; S: N → N (\displaystyle S\colon \mathbb (N) \to \mathbb (N) )) so that the following conditions are met:

one is a natural number (1 ∈ N (\displaystyle 1\in \mathbb (N) ));
the number following the natural number is also a natural number (if x ∈ N (\displaystyle x\in \mathbb (N) ) , then S (x) ∈ N (\displaystyle S(x)\in \mathbb (N) )) ;
one does not follow any natural number (∄ x ∈ N (S (x) = 1) (\displaystyle \nexists x\in \mathbb (N) \ (S(x)=1)));
if a natural number a (\displaystyle a) immediately follows both a natural number b (\displaystyle b) and a natural number c (\displaystyle c) , then b = c (\displaystyle b=c) (if S (b ) = a (\displaystyle S(b)=a) and S (c) = a (\displaystyle S(c)=a) , then b = c (\displaystyle b=c));
(axiom of induction) if any sentence (statement) P (\displaystyle P) has been proven for the 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 next natural number (\displaystyle n) ( inductive hypothesis), then this sentence is true for all natural numbers (let P (n) (\displaystyle P(n)) be some one-place (unary) predicate whose parameter is the 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 listed axioms reflect our intuitive understanding of the natural series and the number line.

The fundamental fact is that these axioms essentially uniquely define the natural numbers (the categorical nature of the Peano axiom system). Namely, it can be proven (see 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 Peano axiom system, then they are necessarily 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 is enough to fix as N (\displaystyle \mathbb (N) ) any one specific model of the set of natural numbers.

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

According to set theory, the only object for constructing any mathematical systems is a set.

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

S (n) = n ∪ ( n ) (\displaystyle S(n)=n\cup \left\(n\right\)) .

Numbers defined in this way are called ordinal.

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

0 = ∅ (\displaystyle 0=\varnothing ) ;
1 = ( 0 ) = ( ∅ ) (\displaystyle 1=\left\(0\right\)=\left\(\varnothing \right\)) ;
2 = ( 0 , 1 ) = ( ∅ , ( ∅ ) ) (\displaystyle 2=\left\(0,1\right\)=(\big \()\varnothing ,\;\left\(\varnothing \ right\)(\big \))) ;
3 = ( 0 , 1 , 2 ) = ( ∅ , ( ∅ ) , ( ∅ , ( ∅ ) ) ) (\displaystyle 3=\left\(0,1,2\right\)=(\Big \() \varnothing ,\;\left\(\varnothing \right\),\;(\big \()\varnothing ,\;\left\(\varnothing \right\)(\big \))(\Big \) )).

Zero as a natural number

Sometimes, especially in foreign and translated literature, one is replaced by zero in the first and third Peano axioms. In this case, zero is considered a natural number. When defined through classes of equal sets, zero is a natural number by definition. It would be unnatural to deliberately reject 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 separate. Another advantage of treating zero as a natural number is that it makes N (\displaystyle \mathbb (N) ) a monoid.

In Russian literature, zero is usually excluded from the number of natural numbers (0 ∉ N (\displaystyle 0\notin \mathbb (N) )), and the set of natural numbers with zero is denoted as N 0 (\displaystyle \mathbb (N) _(0) ) . If zero is included in the definition of natural numbers, then the set of natural numbers is written as N (\displaystyle \mathbb (N) ) , and without zero - as N ∗ (\displaystyle \mathbb (N) ^(*)) .

In the international mathematical literature, taking into account the above and to avoid ambiguities, the set ( 1 , 2 , … ) (\displaystyle \(1,2,\dots \)) is usually called the set of positive integers and denoted Z + (\displaystyle \ mathbb(Z)_(+)) . The set ( 0 , 1 , … ) (\displaystyle \(0,1,\dots \)) is often called the set of non-negative integers and is denoted by Z ⩾ 0 (\displaystyle \mathbb (Z) _(\geqslant 0)) .

The position of the set of natural numbers (N (\displaystyle \mathbb (N))) among the sets of integers (Z (\displaystyle \mathbb (Z))), rational numbers(Q (\displaystyle \mathbb (Q) )), real numbers(R (\displaystyle \mathbb (R))) and irrational numbers(R ∖ Q (\displaystyle \mathbb (R) \setminus \mathbb (Q) ))

Magnitude of the set of natural numbers

The size of an infinite set is characterized by the concept of “cardinality of a set,” which is a generalization of the number of elements of a finite set to infinite sets. In magnitude (that is, cardinality), the set of natural numbers is larger than any finite set, but smaller 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 a countable set. Thus, the set of terms of any sequence is countable. At the same time, there is a sequence in which each natural number appears an infinite number of times, since the set of natural numbers can be represented as a countable union of disjoint 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

Closed operations (operations that do not derive a result from the set of natural numbers) on natural numbers include the following arithmetic operations:

addition: term + term = sum;
multiplication: factor × factor = product;
exponentiation: a b (\displaystyle a^(b)) , where a (\displaystyle a) is the base of the degree, b (\displaystyle b) is the exponent. If a (\displaystyle a) and b (\displaystyle b) are natural numbers, then the result will be a natural number.

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 everyone pairs of numbers (sometimes exist, sometimes not)):

subtraction: minuend - subtrahend = difference. In this case, the minuend must be greater than the subtrahend (or equal to it, if we consider zero to be a natural number);
division with remainder: dividend / divisor = (quotient, remainder). The quotient p (\displaystyle p) and the remainder r (\displaystyle r) from dividing a (\displaystyle a) by b (\displaystyle b) are defined as follows: a = p ⋅ b + r (\displaystyle a=p\cdot b+ r) , and 0 ⩽ r b (\displaystyle 0\leqslant r can be represented as a = p ⋅ 0 + a (\displaystyle a=p\cdot 0+a) , that is, any number could be considered partial, and the remainder a (\displaystyle a) .

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

Basic properties

Commutativity of addition:

a + b = b + a (\displaystyle a+b=b+a) .

Commutativity of multiplication:

a ⋅ b = b ⋅ a (\displaystyle a\cdot b=b\cdot a) .

Addition associativity:

(a + b) + c = a + (b + c) (\displaystyle (a+b)+c=a+(b+c)) .

Multiplication associativity:

(a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) (\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)) .

Distributivity of multiplication relative to addition:

( a ⋅ (b + c) = a ⋅ b + a ⋅ c (b + c) ⋅ a = b ⋅ a + c ⋅ a (\displaystyle (\begin(cases)a\cdot (b+c)=a \cdot b+a\cdot c\\(b+c)\cdot a=b\cdot a+c\cdot a\end(cases))) .

Algebraic structure

Addition turns the set of natural numbers into a semigroup with unit, the role of unit is played by 0 . Multiplication also turns the set of natural numbers into a semigroup with identity, with the identity element being 1 . Using closure with respect to the operations of addition-subtraction and multiplication-division, we obtain groups of integers Z (\displaystyle \mathbb (Z) ) and rationals positive numbers Q + ∗ (\displaystyle \mathbb (Q) _(+)^(*)) respectively.

Set-theoretic definitions

Let us use the definition of natural numbers as equivalence classes of finite sets. If we denote the equivalence class of a set A, generated by bijections, using square brackets: [ A], the basic arithmetic operations are defined as follows:

[ A ] + [ B ] = [ A ⊔ B ] (\displaystyle [A]+[B]=) ;
[ A ] ⋅ [ B ] = [ A × B ] (\displaystyle [A]\cdot [B]=) ;
[ A ] [ B ] = [ A B ] (\displaystyle ([A])^([B])=) ,

A ⊔ B (\displaystyle A\sqcup B) - disjoint union of sets;
A × B (\displaystyle A\times B) - direct product;
A B (\displaystyle A^(B)) - a set of mappings from B V A.

It can be shown that the resulting operations on classes are introduced correctly, that is, they do not depend on the choice of class elements, and coincide with inductive definitions.

What is a natural number? History, scope, properties

The beginning began

Natural numbers appeared along with the first mathematical operations. One root, two roots, three roots... They appeared thanks to Indian scientists who developed the first positional number system.
The word “positionality” means that the location of each digit in a number is strictly defined and corresponds to its rank. For example, the numbers 784 and 487 are the same numbers, but the numbers are not equivalent, since the first includes 7 hundreds, while the second only 4. The Indian innovation was picked up by the Arabs, who brought the numbers to the form that we know Now.

In ancient times, numbers were given a mystical meaning; the greatest mathematician Pythagoras believed that number underlies the creation of the world along with the basic elements - fire, water, earth, air. If we consider everything only from the mathematical side, then what is a natural number? The field of natural numbers is denoted as N and is an infinite series of numbers that are integers and positive: 1, 2, 3, … + ∞. Zero is excluded. Used primarily to count items and indicate order.

What is a natural number in mathematics? Peano's axioms

Field N is the basic one on which elementary mathematics is based. Over time, fields of integers, rationals, and complex numbers were identified.

The work of the Italian mathematician Giuseppe Peano made possible the further structuring of arithmetic, achieved its formality and prepared the way for further conclusions that went beyond the field area N. What a natural number is was clarified earlier in simple language; below we will consider the mathematical definition based on the Peano axioms.

One is considered a natural number.
The number that follows a natural number is a natural number.
There is no natural number before one.
If the number b follows both the number c and the number d, then c=d.
An axiom of induction, which in turn shows what a natural number is: if some statement that depends on a parameter is true for the number 1, then we assume that it also works for the number n from the field of natural numbers N. Then the statement is also true for n =1 from the field of natural numbers N.

Basic operations for the field of natural numbers

addition – x + y = z, where x, y, z are included in the N field;
multiplication – x * y = z, where x, y, z are included in the N field;
exponentiation – xy, where x, y are included in the N field.

The remaining operations, the result of which may not exist in the context of the definition of “what is a natural number,” are as follows:

Properties of numbers belonging to the field N

All further mathematical reasoning will be based on the following properties, the most trivial, but no less important.

The commutative property of addition is x + y = y + x, where the numbers x, y are included in the field N. Or the well-known “the sum does not change by changing the places of the terms.”
The commutative property of multiplication is x * y = y * x, where the numbers x, y are included in the N field.
The combinational property of addition is (x + y) + z = x + (y + z), where x, y, z are included in the field N.
The matching property of multiplication is (x * y) * z = x * (y * z), where the numbers x, y, z are included in the N field.
distributive property – x (y + z) = x * y + x * z, where the numbers x, y, z are included in the N field.

Pythagorean table

In the practice of teaching in recent decades, there has been a need to memorize the Pythagorean table “in order,” that is, memorization came first. Multiplication by 1 was excluded because the result was a multiplier of 1 or greater. Meanwhile, in the table with the naked eye you can notice a pattern: the product of numbers increases by one step, which is equal to the title of the line. Thus, the second factor shows us how many times we need to take the first one in order to obtain the desired product. This system is much more convenient than the one that was practiced in the Middle Ages: even understanding what a natural number is and how trivial it is, people managed to complicate their everyday counting by using a system that was based on powers of two.

Subset as the cradle of mathematics

At the moment, the field of natural numbers N is considered only as one of the subsets of complex numbers, but this does not make them any less valuable in science. Natural number is the first thing a child learns when studying himself and the world around him. One finger, two fingers... Thanks to it, a person develops logical thinking, as well as the ability to determine the cause and deduce the effect, paving the way for great discoveries.

Discussion:Natural number

Controversy around zero

Somehow I can’t imagine zero as a natural number... It seems the ancients didn’t know zero at all. And TSB does not consider zero a natural number. So at least this is a controversial statement. Can we say something more neutral about zero? Or are there compelling arguments? --.:Ajvol:. 18:18, 9 Sep 2004 (UTC)

Rolled back last change. --Maxal 20:24, 9 Sep 2004 (UTC)

The French Academy at one time issued a special decree according to which 0 was included in the set of natural numbers. Now this is a standard, in my opinion there is no need to introduce the concept of “Russian natural number”, but to adhere to this standard. Naturally, it should be mentioned that once upon a time this was not the case (not only in Russia but everywhere). Tosha 23:16, 9 Sep 2004 (UTC)

The French Academy is not a decree for us. There is also no established opinion on this matter in the English-language mathematical literature. See for example, --Maxal 23:58, 9 Sep 2004 (UTC)

Somewhere over there it says: “If you are writing an article about a controversial issue, then try to present all points of view, providing links to different opinions.” Bes island 23:15, 25 Dec 2004 (UTC)

I don't see it here controversial issue, but I see: 1) disrespect for other participants by significantly changing/deleting their text (it is customary to discuss them before making significant changes); 2) replacing strict definitions (indicating the cardinality of sets) with vague ones (is there a big difference between “numbering” and “denoting quantity”?). Therefore, I’m rolling back again, but I’m leaving a final comment. --Maxal 23:38, 25 Dec 2004 (UTC)

Disrespect is exactly how I regard your kickbacks. So let's not talk about that. My edit doesn't change the essence article, it just clearly formulates two definitions. The previous version of the article formulated the definition of “without zero” as the main one, and “with zero” as a kind of dissidence. This absolutely does not meet the requirements of Wikipedia (see quote above), and, incidentally, it does not completely scientific style presentation in the previous version. I added the wording “cardinality of a set” as an explanation to “denotation of quantity” and “enumeration” to “numbering”. And if you don’t see the difference between “numbering” and “denoting quantities,” then, let me ask, why then do you edit mathematical articles? Bes island 23:58, 25 Dec 2004 (UTC)

As for “does not change the essence” - the previous version emphasized that the difference in definitions is only in the attribution of zero to natural numbers. In your version, the definitions are presented as radically different. As for the “basic” definition, then it should be so, because this article in Russian Wikipedia, which means you basically need to stick to what you said generally accepted in Russian mathematics schools. I ignore the attacks. --Maxal 00:15, 26 Dec 2004 (UTC)

In fact, the only obvious difference is zero. In fact, this is precisely the cardinal difference, coming from different understandings of the nature of natural numbers: in one version - as quantities; in the other - as numbers. This absolutely different concepts, no matter how hard you try to hide the fact that you don’t understand this.

Regarding the fact that in Russian Wikipedia it is required to cite the Russian point of view as the dominant one. Look carefully here. Look at the English article about Christmas. It doesn’t say that Christmas should be celebrated on December 25, because that’s how it’s celebrated in England and the USA. Both points of view are given there (and they differ no more and no less than the difference between natural numbers “with zero” and “without zero”), and not a single word about which of them is supposedly truer.

In my version of the article, both points of view are designated as independent and equally entitled to exist. The Russian standard is indicated by the words you referred to above.

Perhaps, from a philosophical point of view, the concepts of natural numbers are indeed absolutely different, but the article offers essentially mathematical definitions, where all the difference is 0 ∈ N (\displaystyle 0\in \mathbb (N) ) or 0 ∉ N (\displaystyle 0\not \in \mathbb (N) ) . The dominant point of view or not is a delicate matter. I appreciate the phrase observed in most of the Western world on December 25 from an English article about Christmas as an expression of the dominant point of view, despite the fact that no other dates are given in the first paragraph. By the way, in the previous version of the article on natural numbers there were also no direct instructions on how necessary to determine natural numbers, simply the definition without zero was presented as more common (in Russia). In any case, it is good that a compromise has been found. --Maxal 00:53, 26 Dec 2004 (UTC)

The expression “In Russian literature, zero is usually excluded from the number of natural numbers” is somewhat unpleasantly surprising; gentlemen, zero is not considered a natural number, unless otherwise stated, throughout the world. The same French, as far as I read them, specifically stipulate the inclusion of zero. Of course N 0 (\displaystyle \mathbb (N) _(0)) is used more often, but if, for example, I like women, I won’t change men into women. Druid. 2014-02-23

Unpopularity of natural numbers

It seems to me that natural numbers are an unpopular subject in mathematics papers (perhaps not least due to the lack of a common definition). In my experience, I often see the terms in mathematical articles non-negative integers And positive integers(which are interpreted unambiguously) rather than natural numbers. Interested parties are asked to express their (dis)agreement with this observation. If this observation finds support, then it makes sense to indicate it in the article. --Maxal 01:12, 26 Dec 2004 (UTC)

Without a doubt, you are right in the summary part of your statement. This is all precisely because of differences in definition. In some cases I myself prefer to indicate “positive integers” or “non-negative integers” instead of “natural” in order to avoid discrepancies regarding the inclusion of zero. And, in general, I agree with the operative part. Bes island 01:19, 26 Dec 2004 (UTC) In the articles - yes, perhaps it is so. However, in longer texts, as well as where the concept is used often, they usually use natural numbers, however, first explaining “what” natural numbers we are talking about - with or without zero. LoKi 19:31, July 30, 2005 (UTC)

Numbers

Is it worth listing the names of numbers (one, two, three, etc.) in the last part of this article? Wouldn't it make more sense to put this in the Number article? Still, this article, in my opinion, should be more mathematical in nature. What do you think? --LoKi 19:32, July 30, 2005 (UTC)

In general, it’s strange how you can get an ordinary natural number from *empty* sets? In general, no matter how much you combine emptiness with emptiness, nothing will come out except emptiness! Is this not an alternative definition at all? Posted at 21:46, July 17, 2009 (Moscow)

Categoricality of the Peano axiom system

I added a remark about the categorical nature of the Peano axiom system, which in my opinion is fundamental. Please format the link to the book correctly [[Participant: A_Devyatkov 06:58, June 11, 2010 (UTC)]]

Peano's axioms

Almost all foreign literature and on Wikipedia, Peano's axioms begin with "0 is a natural number." Indeed, in the original source it is written “1 is a natural number.” However, in 1897 Peano makes a change and changes 1 to 0. This is written in the "Formulaire de mathematiques", Tome II - No. 2. page 81. This is a link to electronic version on the desired page:

http://archive.org/stream/formulairedemat02peangoog#page/n84/mode/2up (French).

Explanations for these changes are given in "Rivista di matematica", Volume 6-7, 1899, page 76. Also a link to the electronic version on the desired page:

http://archive.org/stream/rivistadimatema01peangoog#page/n69/mode/2up (Italian).

0=0

What are the “axioms of digital turntables”?

I would like to roll back the article to the latest patrolled version. Firstly, someone renamed Peano's axioms to Piano's axioms, which is why the link stopped working. Secondly, a certain Tvorogov added a very large piece of information to the article, which, in my opinion, is completely inappropriate in this article. It is written in a non-encyclopedic manner; in addition, the results of Tvorogov himself are given and a link to his own book. I insist that the section about “axioms of digital turntables” should be removed from this article. P.s. Why was the section about the number zero removed? mesyarik 14:58, March 12, 2014 (UTC)

The topic is not covered, a clear definition of natural numbers is necessary

Please don't write heresy like " Natural numbers (natural numbers) are numbers that arise naturally when counting.“Nothing arises in the brain naturally. Exactly what you put there will be there.

How can a five-year-old explain which number is a natural number? After all, there are people who need to be explained as if they were five years old. How does a natural number differ from an ordinary number? Examples needed! 1, 2, 3 is natural, and 12 is natural, and -12? and three-quarters, or for example 4.25 natural? 95.181.136.132 15:09, November 6, 2014 (UTC)

Natural numbers - fundamental concept, the original abstraction. They cannot be determined. You can go as deep into philosophy as you like, but in the end you either have to admit (accept on faith?) some rigid metaphysical position, or admit that absolute definition no, natural numbers are part of an artificial formal system, a model that man (or God) came up with. I found an interesting treatise on this topic. How do you like this option, for example: “Any specific Peano system is called a natural series, that is, a model of Peano’s axiomatic theory.” Feel better? RomanSuzi 17:52, November 6, 2014 (UTC)
- It seems that with your models and axiomatic theories you are only complicating everything. This definition will be understood in best case scenario two out of a thousand people. Therefore, I think that the first paragraph is missing a sentence " In simple words: natural numbers are positive integers starting from one inclusive." This definition sounds normal to the majority. And it gives no reason to doubt the definition of a natural number. After all, after reading the article, I did not fully understand what natural numbers are and the number 807423 is a natural or natural numbers are those that make up this number, i.e. 8 0 7 4 2 3. Often complications only spoil everything. Information about natural numbers should be on this page and not in numerous links to other pages. November 7, 2014 (UTC)
  - Here it is necessary to distinguish between two tasks: (1) clearly (even if not strictly) explain to the reader who is far from mathematics what a natural number is, so that he understands more or less correctly; (2) give such a strict definition of a natural number, from which its basic properties follow. You correctly advocate the first option in the preamble, but it is precisely this that is given in the article: a natural number is a mathematical formalization of counting: one, two, three, etc. Your example (807423) can certainly be obtained when counting, which means this also a natural number. I don’t understand why you confuse a number and the way it is written in numbers; this is a separate topic, not directly related to the definition of a number. Your version of explanation: “ natural numbers are positive integers starting from one inclusive"is no good, because it is impossible to define less general concept(natural number) through a more general (number), not yet defined. It's hard for me to imagine a reader who knows what a positive integer is, but has no idea what a natural number is. LGB 12:06, November 7, 2014 (UTC)
    - Natural numbers cannot be defined in terms of integers. RomanSuzi 17:01, November 7, 2014 (UTC)

“Nothing comes into existence naturally in the brain.” Latest Research show (I can’t find any links right now) that the human brain is prepared to use language. Thus, naturally, we already have in our genes the readiness to master a language. Well, for natural numbers this is what is needed. The concept of “1” can be shown with your hand, and then, by induction, you can add sticks, getting 2, 3, and so on. Or: I, II, III, IIII, ..., IIIIIIIIIIIIIIIIIIIIIIIIII. But maybe you have specific suggestions for improving the article, based on authoritative sources? RomanSuzi 17:57, November 6, 2014 (UTC)

What is a natural number in mathematics?

Vladimir z

Natural numbers are used to number objects and to count their quantity. For numbering, positive integers are used, starting from 1.

And to count the number, they also include 0, indicating the absence of objects.

Whether the concept of natural numbers contains the number 0 depends on the axiomatics. If the presentation of any mathematical theory requires the presence of 0 in the set of natural numbers, then this is stipulated and considered an immutable truth (axiom) within the framework of this theory. The definition of the number 0, both positive and negative, comes very close to this. If we take the definition of natural numbers as the set of all NON-NEGATIVE integers, then the question arises, what is the number 0 - positive or negative?

IN practical application, as a rule, the first definition that does not include the number 0 is used.

Pencil

Natural numbers are positive integers. Natural numbers are used to count (number) objects or to indicate the number of objects or to indicate the serial number of an object in a list. Some authors artificially include zero in the concept of “natural numbers”. Others use the formulation "natural numbers and zero." This is unprincipled. The set of natural numbers is infinite, because with any large natural number you can perform the operation of addition with another natural number and get an even larger number.

Negative and non-integer numbers are not included in the set of natural numbers.

Sayan Mountains

Natural numbers are numbers that are used for counting. They can only be positive and whole. What does this mean in the example? Since these numbers are used for counting, let's try to calculate something. What can you count? For example, people. We can count people like this: 1 person, 2 people, 3 people, etc. The numbers 1, 2, 3 and others used for counting will be natural numbers. We never say -1 (minus one) person or 1.5 (one and a half) person (excuse the pun:), so -1 and 1.5 (like all negative and fractional numbers) are not natural.

Lorelei

Natural numbers are those numbers that are used when counting objects.

The smallest natural number is one. The question often arises whether zero is a natural number. No, it is not in most Russian sources, but in other countries the number zero is recognized as a natural number...

Moreljuba

Natural numbers in mathematics refer to numbers used to count something or someone sequentially. The smallest natural number is considered to be one. Zero in most cases does not belong to the category of natural numbers. Negative numbers are also not included here.

Greetings Slavs

Natural numbers, also known as natural numbers, are those numbers that arise in the usual way when counting them and that are greater than zero. The sequence of each natural number, arranged in ascending order, is called a natural series.

Elena Nikityuk

The term natural number is used in mathematics. A positive integer is called a natural number. The smallest natural number is considered to be “0”. To calculate anything, these same natural numbers are used, for example 1,2,3... and so on.

Natural numbers are the numbers with which we count, that is, one, two, three, four, five and others are natural numbers.

These are necessarily positive numbers greater than zero.

Fractional numbers also do not belong to the set of natural numbers.

-Orchid-

Natural numbers are needed to count something. They are a series of only positive numbers, starting with one. It is important to know that these numbers are exclusively integers. You can calculate anything with natural numbers.

Marlena

Natural numbers are integers that we usually use when counting objects. Zero as such is not included in the realm of natural numbers, since we usually do not use it in calculations.

Inara-pd

Natural numbers are the numbers we use when counting - one, two, three and so on.

Natural numbers arose from the practical needs of man.

Natural numbers are written using ten digits.

Zero is not a natural number.

What is a natural number?

Naumenko

Natural numbers are numbers. used when numbering and counting natural (flower, tree, animal, bird, etc.) objects.

Integers are called NATURAL numbers, THEIR OPPOSITES AND ZERO,

Explain. what are naturals through integers is incorrect!! !

Numbers can be even - divisible by 2 by a whole and odd - not divisible by 2 by a whole.

Prime numbers are numbers. having only 2 divisors - one and itself...
The first of your equations has no solutions. for the second x=6 6 is a natural number.

Natural numbers (natural numbers) are numbers that arise naturally when counting (both in the sense of enumeration and in the sense of calculus).

The set of all natural numbers is usually denoted by \mathbb(N). The set of natural numbers is infinite, since for any natural number there is a larger natural number.

Anna Semenchenko

numbers that arise naturally when counting (both in the sense of enumeration and in the sense of calculus).
There are two approaches to defining natural numbers - numbers used in:
listing (numbering) items (first, second, third, ...);
designation of the number of items (no items, one item, two items, ...). Adopted in the works of Bourbaki, where natural numbers are defined as cardinalities of finite sets.
Negative and non-integer (rational, real, ...) numbers are not natural numbers.
The set of all natural numbers is usually denoted by a sign. The set of natural numbers is infinite, since for any natural number there is a larger natural number.

Natural numbers are one of the oldest mathematical concepts.

In the distant past, people did not know numbers and when they needed to count objects (animals, fish, etc.), they did it differently than we do now.

The number of objects was compared with parts of the body, for example, with fingers on a hand, and they said: “I have as many nuts as there are fingers on my hand.”

Over time, people realized that five nuts, five goats and five hares have common property- their number is five.

Remember!

Natural numbers- these are numbers, starting from 1, obtained by counting objects.

1, 2, 3, 4, 5…

Smallest natural number — 1 .

Largest natural number does not exist.

When counting, the number zero is not used. Therefore, zero is not considered a natural number.

People learned to write numbers much later than to count. First of all, they began to depict one with one stick, then with two sticks - the number 2, with three - the number 3.

| — 1, || — 2, ||| — 3, ||||| — 5 …

Then they appeared special signs for denoting numbers - the predecessors of modern numbers. The numerals we use to write numbers originated in India approximately 1,500 years ago. The Arabs brought them to Europe, which is why they are called Arabic numerals.

There are ten numbers in total: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Using these numbers you can write any natural number.

Remember!

Natural series is the sequence of all 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 1.

The natural series is infinite; there is no greatest natural number in it.

The counting system we use is called decimal positional.

Decimal because 10 units of each digit form 1 unit of the most significant digit. Positional because the meaning of a digit depends on its place in the number record, that is, on the digit in which it is written.

Important!

The classes following the billion are named according to the Latin names of numbers. Each subsequent unit contains a thousand previous ones.

1,000 billion = 1,000,000,000,000 = 1 trillion (“three” is Latin for “three”)
1,000 trillion = 1,000,000,000,000,000 = 1 quadrillion (“quadra” is Latin for “four”)
1,000 quadrillion = 1,000,000,000,000,000,000 = 1 quintillion (“quinta” is Latin for “five”)

However, physicists have found a number that exceeds the number of all atoms ( tiny particles matter) throughout the Universe.

This number received a special name - googol. Googol is a number with 100 zeros.

Natural numbers– natural numbers are numbers that are used to count objects. The set of all natural numbers is sometimes called the natural series: 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

In writing any number using digits, the place in which the digit appears in the number has crucial. 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 units 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 side 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.

Where does learning mathematics begin? Yes, that's right, from studying natural numbers and operations with them.Natural numbers (fromlat. naturalis- natural; natural numbers) -numbers that occur naturally when counting (for example, 1, 2, 3, 4, 5, 6, 7, 8, 9...). The sequence of all natural numbers arranged in ascending order is called a natural series.

There are two approaches to defining natural numbers:

counting (numbering) items ( first, second, third, fourth, fifth"…);
natural numbers are 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 begins with one, in the second - with zero. There is no consensus among most mathematicians on whether the first or second approach is preferable (that is, whether zero should be considered a natural number or not). The overwhelming majority of Russian sources traditionally adopt the first approach. The second approach, for example, is used in the worksNicolas Bourbaki , where the natural numbers are defined aspower finite sets .

Negative and integer (rational , real ,...) numbers are not considered natural numbers.

The set of all natural numbers usually denoted by the symbol N (fromlat. naturalis- natural). The set of natural numbers is infinite, since for any natural number n there is a natural number greater than n.

The presence of zero makes it easier to formulate and prove many theorems in natural number arithmetic, so the first approach introduces the useful concept extended natural range , including zero. The extended series is designated N 0 or Z 0 .

TOclosed operations (operations that do not derive a result from the set of natural numbers) on natural numbers include the following arithmetic operations:

addition: term + term = sum;
multiplication: factor × factor = product;
exponentiation: a b , where a is the base of the degree, b is the exponent. If a and b are natural numbers, then the result will be a natural number.

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 allpairs of numbers (sometimes exist, sometimes not)):

subtraction: minuend - subtrahend = difference. In this case, the minuend must be greater than the subtrahend (or equal to it, if we consider zero a natural number)
division with remainder: dividend / divisor = (quotient, remainder). The quotient p and the remainder r from dividing a by b are defined as follows: a=p*r+b, with 0<=r

It should be noted that the operations of addition and multiplication are fundamental. In particular,