Integers– natural numbers are numbers that are used to count objects. Plenty of everyone natural numbers 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 last number You can always add one and 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
When writing any number using digits, the place in which the digit appears in the number is critical. 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 edge 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.
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.
Natural numbers can be used for counting (one apple, two apples, etc.)Integers(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 natural number arithmetic, so the first approach introduces the 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 axiomsWe 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 of values 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 a 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:
- Commutativity of multiplication:
- Addition associativity:
- Multiplication associativity:
- Distributivity of multiplication relative to addition:
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
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 of time
Natural numbers appeared along with the first ones mathematical operations. One spine, two spines, three spines... 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 integer, rational, complex numbers.
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 is a natural number was clarified earlier in simple language, below we will consider a mathematical definition based on Peano’s 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
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 are closed and 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 – 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
One of the first steps in students’ knowledge of the entire structure elementary mathematics after they have figured out for themselves which numbers are called natural numbers, the Pythagorean table appears. 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 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. 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.
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 a controversial issue here, 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 neither single word about which one 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 math papers (perhaps not in last resort due to the lack of a uniform definition). In my experience, I often see the terms in mathematical articles non-negative integers And positive integers(which are interpreted unambiguously) rather than integers. 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 integers, 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. How 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
In 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 the 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 very big piece information, 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 naturally in the brain. 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 are a fundamental concept, the original abstraction. They cannot be defined. 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: “A natural series is every specific system Peano, that is, Peano’s model of 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 most. And it gives no reason to doubt the definition of a natural number. After all, after reading the article, I didn’t 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. 95.181.136.132 10:03, 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)
- 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)
- 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 most. And it gives no reason to doubt the definition of a natural number. After all, after reading the article, I didn’t 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. 95.181.136.132 10:03, November 7, 2014 (UTC)
- “Nothing comes into existence naturally in the brain.” Recent studies 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 numbers.
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 mean numbers used to count something or someone sequentially. The smallest natural number is considered to be one. In most cases, zero is not a natural number. Negative numbers 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, which Above 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 familiar to humans and intuitive, because they surround us since childhood. In the article below we will give a basic understanding of the meaning of natural numbers and describe the basic skills of writing and reading them. The entire theoretical part will be accompanied by examples.
Yandex.RTB R-A-339285-1
General understanding of natural numbers
At a certain stage in the development of mankind, the task of counting certain objects and designating their quantity arose, which, in turn, required finding a tool to solve this problem. Natural numbers became such a tool. The main purpose of natural numbers is also clear - to give an idea of the number of objects or serial number a specific object, if we are talking about a set.
It is logical that for a person to use natural numbers, it is necessary to have a way to perceive and reproduce them. So, a natural number can be voiced or depicted, which is natural ways transfer of information.
Let's look at the basic skills of voicing (reading) and representing (writing) natural numbers.
Decimal notation of a natural number
Let us remember how the following characters are represented (we will indicate them separated by commas): 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 . We call these signs numbers.
Now let's take it as a rule that when depicting (recording) any natural number, only the indicated numbers are used without the participation of any other symbols. Let the digits when writing a natural number have the same height, are written one after another in a line and there is always a digit other than zero on the left.
Let us indicate examples of the correct recording of natural numbers: 703, 881, 13, 333, 1,023, 7, 500,001. The spacing between numbers is not always the same; this will be discussed in more detail below when studying the classes of numbers. The given examples show that when writing a natural number, all the digits from the above series do not have to be present. Some or all of them may be repeated.
Definition 1
Records of the form: 065, 0, 003, 0791 are not records of natural numbers, because On the left is the number 0.
The correct recording of a natural number, made taking into account all the described requirements, is called decimal notation of a natural number.
Quantitative meaning of natural numbers
As already mentioned, natural numbers initially carry a quantitative meaning, among other things. Natural numbers, as a numbering tool, are discussed in the topic on comparing natural numbers.
Let's proceed to natural numbers, the entries of which coincide with the entries of digits, i.e.: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 .
Let's imagine a certain object, for example, like this: Ψ. We can write down what we see 1 item. The natural number 1 is read as "one" or "one". The term "unit" also has another meaning: something that can be considered as a single whole. If there is a set, then any element of it can be designated as one. For example, out of a set of mice, any mouse is one; any flower from a set of flowers is one.
Now imagine: Ψ Ψ . We see one object and another object, i.e. in the recording it will be 2 items. The natural number 2 is read as “two”.
Further, by analogy: Ψ Ψ Ψ – 3 items (“three”), Ψ Ψ Ψ Ψ – 4 (“four”), Ψ Ψ Ψ Ψ Ψ – 5 (“five”), Ψ Ψ Ψ Ψ Ψ Ψ – 6 (“six”), Ψ Ψ Ψ Ψ Ψ Ψ Ψ – 7 (“seven”), Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ – 8 (“eight”), Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ – 9 (“ nine").
From the indicated position, the function of a natural number is to indicate quantities items.
Definition 1
If the record of a number coincides with the record of the number 0, then such a number is called "zero". Zero is not a natural number, but it is considered along with other natural numbers. Zero denotes absence, i.e. zero items means none.
Single digit natural numbers
It is an obvious fact that when writing each of the natural numbers discussed above (1, 2, 3, 4, 5, 6, 7, 8, 9), we use one sign - one digit.
Definition 2
Single digit natural number– a natural number, which is written using one sign – one digit.
There are nine single-digit natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9.
Two-digit and three-digit natural numbers
Definition 3Two-digit natural numbers- natural numbers, when writing which two signs are used - two digits. In this case, the numbers used can be either the same or different.
For example, the natural numbers 71, 64, 11 are two-digit.
Let's consider what meaning is contained in two-digit numbers. We will rely on the quantitative meaning of single-digit natural numbers that is already known to us.
Let's introduce such a concept as “ten”.
Let's imagine a set of objects that consists of nine and one more. In this case, we can talk about 1 ten (“one dozen”) objects. If you imagine one ten and one more, then we are talking about 2 tens (“two tens”). Adding one more to two tens, we get three tens. And so on: continuing to add one ten at a time, we will get four tens, five tens, six tens, seven tens, eight tens and, finally, nine tens.
Let's look at a two-digit number as a set of single-digit numbers, one of which is written on the right, the other on the left. The number on the left will indicate the number of tens in a natural number, and the number on the right will indicate the number of units. In the case where the number 0 is located on the right, then we are talking about the absence of units. The above is the quantitative meaning of two-digit natural numbers. There are 90 of them in total.
Definition 4
Three-digit natural numbers– natural numbers, when writing which three signs are used – three digits. The numbers can be different or repeated in any combination.
For example, 413, 222, 818, 750 are three-digit natural numbers.
To understand the quantitative meaning of three-digit natural numbers, we introduce the concept "a hundred".
Definition 5
One hundred (1 hundred) is a set consisting of ten tens. A hundred and another hundred make 2 hundreds. Add one more hundred and get 3 hundreds. By gradually adding one hundred at a time, we get: four hundred, five hundred, six hundred, seven hundred, eight hundred, nine hundred.
Let's consider the notation of a three-digit number itself: the single-digit natural numbers included in it are written one after another from left to right. The rightmost single digit number indicates the number of units; the next single-digit number to the left is by the number of tens; the leftmost single digit number is in the number of hundreds. If the entry contains the number 0, it indicates the absence of units and/or tens.
Thus, the three-digit natural number 402 means: 2 units, 0 tens (there are no tens that are not combined into hundreds) and 4 hundreds.
By analogy, the definition of four-digit, five-digit, and so on natural numbers is given.
Multi-digit natural numbers
From all of the above, it is now possible to move on to the definition of multi-valued natural numbers.
Definition 6
Multi-digit natural numbers– natural numbers, when writing which two or more characters are used. Multi-digit natural numbers are two-digit, three-digit, and so on numbers.
One thousand is a set that includes ten hundred; one million consists of a thousand thousand; one billion – one thousand million; one trillion – one thousand billion. Even larger sets also have names, but their use is rare.
Similar to the principle above, we can consider any multi-digit natural number as a set of single-digit natural numbers, each of which, being in a certain place, indicates the presence and number of units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions , hundreds of millions, billions and so on (from right to left, respectively).
For example, the multi-digit number 4,912,305 contains: 5 units, 0 tens, three hundreds, 2 thousand, 1 ten thousand, 9 hundred thousand and 4 million.
To summarize, we looked at the skill of grouping units into various sets (tens, hundreds, etc.) and saw that the numbers in the notation of a multi-digit natural number indicate the number of units in each of such sets.
Reading natural numbers, classes
In the theory above, we indicated the names of natural numbers. In Table 1 we indicate how to correctly use the names of single-digit natural numbers in speech and in letter writing:
| Number | Masculine | Feminine | Neuter gender |
|
1 |
One |
One |
One |
| Number | Nominative case | Genitive | Dative | Accusative | Instrumental case | Prepositional |
| 1 2 3 4 5 6 7 8 9 |
One Two Three Four Five Six Seven Eight Nine |
One Two Three Four Five Six Semi Eight Nine |
Alone Two Three Four Five Six Semi Eight Nine |
One Two Three Four Five Six Seven Eight Nine |
One Two Three Four Five Six Family Eight Nine |
About one thing About two About three About four Again About six About seven About eight About nine |
To correctly read and write two-digit numbers, you need to memorize the data in Table 2:
| Number |
Masculine, feminine and neuter gender |
| 10 11 12 13 14 15 16 17 18 19 20 30 40 50 60 70 80 90 |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Fourty Fifty Sixty Seventy Eighty Ninety |
| Number | Nominative case | Genitive | Dative | Accusative | Instrumental case | Prepositional |
| 10 11 12 13 14 15 16 17 18 19 20 30 40 50 60 70 80 90 |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Fourty Fifty Sixty Seventy Eighty Ninety |
Ten |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Magpie Fifty Sixty Seventy Eighty Ninety |
Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Fourty Fifty Sixty Seventy Eighty Ninety |
Ten Eleven twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Thirty Magpie Fifty sixty Seventy Eighty nineteen |
About ten About eleven About twelve About thirteen About fourteen About fifteen About sixteen About seventeen About eighteen About nineteen About twenty About thirty Oh magpie About fifty About sixty About seventy About eighty Oh ninety |
To read other two-digit natural numbers, we will use the data from both tables; we will consider this with an example. Let's say we need to read the two-digit natural number 21. This number contains 1 unit and 2 tens, i.e. 20 and 1. Turning to the tables, we read the indicated number as “twenty-one”, while the conjunction “and” between the words does not need to be pronounced. Let's say we need to use the indicated number 21 in a certain sentence, indicating the number of objects in the genitive case: “there are no 21 apples.” sound in in this case the pronunciation will be as follows: “there are not twenty-one apples.”
Let us give another example for clarity: the number 76, which is read as “seventy-six” and, for example, “seventy-six tons.”
| Number | Nominative | Genitive | Dative | Accusative | Instrumental case | Prepositional |
| 100 200 300 400 500 600 700 800 900 |
One hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
hundred Two hundred Three hundred Four hundred Five hundred Six hundred Semistam Eight hundred Nine hundred |
One hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
hundred Two hundred Three hundred Four hundred Five hundred Six hundred Seven hundred Eight hundred Nine hundred |
Oh hundred About two hundred About three hundred About four hundred About five hundred About six hundred About the seven hundred About eight hundred About nine hundred |
To fully read a three-digit number, we also use the data from all of the indicated tables. For example, given the natural number 305. This number corresponds to 5 units, 0 tens and 3 hundreds: 300 and 5. Taking the table as a basis, we read: “three hundred and five” or in declension by case, for example, like this: “three hundred and five meters.”
Let's read one more number: 543. According to the rules of the tables, the indicated number will sound like this: “five hundred forty-three” or in declension according to cases, for example, like this: “there are no five hundred forty-three rubles.”
Let's move on to general principle reading multi-digit natural numbers: to read a multi-digit number, you need to divide it from right to left into groups of three digits, and the leftmost group can have 1, 2 or 3 digits. Such groups are called classes.
The rightmost class is the class of units; then the next class, to the left - the class of thousands; further – the class of millions; then comes the class of billions, followed by the class of trillions. The following classes also have a name, but the natural numbers consisting of large quantity characters (16, 17 or more) are rarely used in reading; it is quite difficult to perceive them by ear.
To make the recording easier to read, classes are separated from each other by a small indentation. For example, 31,013,736, 134,678, 23,476,009,434, 2,533,467,001,222.
| Class trillion |
Class billions |
Class millions |
Class of thousands | Unit class |
| 134 | 678 | |||
| 31 | 013 | 736 | ||
| 23 | 476 | 009 | 434 | |
| 2 | 533 | 467 | 001 | 222 |
To read a multi-digit number, we call the numbers that make it up one by one (from left to right by class, adding the name of the class). The name of the class of units is not pronounced, and those classes that make up three digits 0 are also not pronounced. If one class contains one or two digits on the left, then they are not used in any way when reading. For example, 054 will be read as “fifty-four” or 001 as “one”.
Example 1
Let's look at the reading of the number 2,533,467,001,222 in detail:
We read the number 2 as a component of the class of trillions - “two”;
By adding the name of the class, we get: “two trillion”;
We read the next number, adding the name of the corresponding class: “five hundred thirty-three billion”;
We continue by analogy, reading the next class to the right: “four hundred sixty-seven million”;
In the next class we see two digits 0 located on the left. According to the above reading rules, digits 0 are discarded and do not participate in reading the record. Then we get: “one thousand”;
Reading last class units without adding its name - “two hundred twenty-two”.
Thus, the number 2 533 467 001 222 will sound like this: two trillion five hundred thirty-three billion four hundred sixty-seven million one thousand two hundred twenty-two. Using this principle, we will read the other given numbers:
31,013,736 – thirty-one million thirteen thousand seven hundred thirty-six;
134 678 – one hundred thirty-four thousand six hundred seventy-eight;
23 476 009 434 – twenty-three billion four hundred seventy-six million nine thousand four hundred thirty-four.
Thus, the basis for correctly reading multi-digit numbers is the skill of dividing a multi-digit number into classes, knowledge of the corresponding names and understanding of the principle of reading two- and three-digit numbers.
As is already clear from all of the above, its value depends on the position at which the digit appears in the notation of a number. That is, for example, the number 3 in the natural number 314 indicates the number of hundreds, namely 3 hundreds. The number 2 is the number of tens (1 ten), and the number 4 is the number of units (4 units). In this case, we will say that the number 4 is in the ones place and is the value of the ones place in the given number. The number 1 is in the tens place and serves as the value of the tens place. The number 3 is located in the hundreds place and is the value of the hundreds place.
Definition 7
Discharge- this is the position of a digit in the notation of a natural number, as well as the value of this digit, which is determined by its position in a given number.
The categories have their own names, we have already used them above. From right to left there are digits: units, tens, hundreds, thousands, tens of thousands, etc.
For ease of remembering, you can use the following table (we indicate 15 digits):
Let’s clarify this detail: the number of digits in a given multi-digit number is the same as the number of characters in the number’s notation. Eg, this table contains the names of all digits for a number with 15 digits. Subsequent discharges also have names, but are used extremely rarely and are very inconvenient to hear.
With the help of such a table, it is possible to develop the skill of determining the digit by writing a given natural number into the table so that the rightmost digit is written in the units digit and then in each digit one by one. For example, let’s write the multi-digit natural number 56,402,513,674 like this:
Pay attention to the number 0, located in the tens of millions digit - it means the absence of units of this digit.
Let us also introduce the concepts of the lowest and highest digits of a multi-digit number.
Definition 8
Lowest (junior) rank of any multi-digit natural number – the units digit.
Highest (senior) category of any multi-digit natural number – the digit corresponding to the leftmost digit in the notation of a given number.
So, for example, in the number 41,781: the lowest digit is the ones digit; The highest rank is the rank of tens of thousands.
Logically it follows that it is possible to talk about the seniority of the digits relative to each other. Each subsequent digit, when moving from left to right, is lower (younger) than the previous one. And vice versa: when moving from right to left, each next digit is higher (older) than the previous one. For example, the thousands place is older than the hundreds place, but younger than the millions place.
Let us clarify that when solving some practical examples It is not the natural number itself that is used, but the sum bit terms a given number.
Briefly about the decimal number system
Definition 9Notation– a method of writing numbers using signs.
Positional number systems– those in which the meaning of a digit in a number depends on its position in the number record.
According to this definition, we can say that, while studying natural numbers and the way they are written above, we used the positional number system. The number 10 plays a special place here. We count in tens: ten units make a ten, ten tens will unite into a hundred, etc. The number 10 serves as the base of this number system, and the system itself is also called decimal.
In addition to it, there are other number systems. For example, computer science uses the binary system. When we keep track of time, we use the sexagesimal number system.
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