Numbers intended for counting items and answering the question "How much?" ("how much
balls? "," How many apples? "," How many soldiers? ") are called natural.
If you write them down in order, from a smaller number to more, then the natural range of numbers will be:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 99, 100, 101, …, 999, 1000, 1001 …
Natural series of numbers begins with numbers 1.
Each next natural number is 1 more than the previous one.
Natural series of numbers are infinite.
The numbers are even and odd. Even numbers are divided into two, and odd numbers are not divided into two.
A number of odd numbers:
1, 3, 5, 7, 9, 11, 13, …, 99, 101, …, 999, 1001, 1003 …
A number of even numbers:
2, 4, 6, 8, 10, 12, 14, …, 98, 100, …, 998, 1000, 1002 …
In a natural row, odd and even numbers alternate:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …, 99, 100, …, 999, 1000 …
How to compare natural numbers
When comparing two natural numbers, more that stands in a natural row:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 …
So, seven more than three, and five more units.
In mathematics to record the word "less", the sign "<», а для записи слова «больше» - знак « > ».
The sharp corner of the "more" icons and "less" is always directed towards smaller of two numbers.
Record 7\u003e 3 read as "seven more than three".
Record 3.< 7 читается как «три меньше семи».
Recording 5\u003e 1 read as "five more than one".
Record 1.< 5 читается как «один меньше пяти».
The word "equal" in mathematics is replaced by the "\u003d" sign:
When the numbers are large, it is difficult to immediately say which one worth the right in a natural row.
When comparing two natural numbers with a different number of numbers, more of them, in which the numbers are greater.
For example, 233 000< 1 000 000, потому что в первом числе шесть цифр, а во втором - семь.
Multivissal natural numbers with the same number of digits are compared to be bonded, starting with the older discharge.
First, the units of the oldest discharge are compared, then the next following it, the next and so on. For example, comparing numbers 5401 and 5430:
5401 \u003d 5 thousand 4 hundred 0 dozen 1 unit;
5430 \u003d 5 thousand 4 hundred 3 dozen 0 units.
Compare units of thousands. In the category of units of thousands of Numbers 5401 - 5 units, in the discharge of units of thousands of 5430 - 5 units. By comparing the unit of thousands, it is also impossible to say which of the numbers is more.
Compare hundreds. In the discharge of hundreds of numbers 5401 - 4 units, in the discharge of hundreds of Numbers 5430 - also 4 units. It is necessary to continue comparison.
Compare tens. In the discharge of tens of numbers 5401 - 0 units, in the discharge of tens of numbers 5430 - 3 units.
Comparing, we get 0< 3, поэтому 5401 < 5430.
Numbers can be placed in descending order and in ascending order.
If in the record of several natural numbers every next number is less than the previous one, then they say that numbers are written in descending order.
We write numbers 5, 22, 13, 800 in descending order.
Laying more. The number 5 is unambiguous, 13 and 22 - double-digit, 800 - three-digit number and, therefore, the largest. We write in the first place 800.
Of the two-digit numbers 13 and 22 greater than 22. We write in number 800 number 22, and then 13.
The smallest number is an unambiguous number 5. We write it last.
800, 22, 13, 5 - recording data numbers in the order of their decrease.
If in the record of several natural numbers every next number more than the previous one, they say that the numbers are recorded in ascending order.
And how to record numbers 15, 2, 31, 278, 298 in ascending order?
Among the numbers 15, 2, 31, 278, 298 pairing smaller.
This is an unambiguous number 2. We write it first.
Of the two-digit numbers 15 and 31, choose less - 15, we write it in second place, and after it - 31.
Of the three-digit numbers 278 - the smaller, write it in number 31, and the last write the number 298.
2, 15, 21, 278, 298 - recording data numbers in ascending order
The lesson "The designation of natural numbers" is the first lesson in the course of the mathematics of the fifth grade and is a continuation, and in some points, and the repetition of a similar topic, which studied in the course of primary school. As a result, the disciples often do not carefully perceive the educational material. Therefore, to achieve maximum interest and concentration of attention, it is necessary to introduce new explanation methods, for example, to use the presentation of the "Natural Numbers" presentation.
The lesson begins with the repetition of a number of numbers, as well as the concept of a natural number and its decimal record. It is explained that the sequence of all natural numbers is called naturally near and is given an example of the first twenty items. Special attention during the presentation is given to the value of the figures depending on her places in the record of the number. To do this, the record of the number on discharges is considered. Using spectacular and non-intrusive animation, students are demonstrated, which means the same figure depending on where it is: in the discharge of units, in the discharge of tens, etc.
It is not rare to see that, along with the fact that the number zero is often used both in daily life, and in the course of mathematics, schoolchildren have difficulty when they need to explain what is the number. To increase the effectiveness of understanding, the concept of zero is given an example of an account in a football match. Also emphasized the attention of students to the fact that 0 not refer to natural numbers.
In the presentation in detail, using examples, the concepts of unambiguous, two-digit, three-digit and four-digit numbers are considered. Considered entries of one million and billion. Separate attention is paid to the correct reading of multivalued numbers and their breakdown to classes. Using a table to record a multi-valued number with the selection of classes and discharges, it is demonstrated that the left class, in contrast to all other, can have less than three digits.
In order to check the result of the assimilation of new material students, this presentation development contains a list of questions that fully covers the stated material. This will allow the teacher to react as quickly as possible to the moments that remained not fully understood schoolchildren. as a result of studying this topic.
Since the presentation of the "Natural Numbers" presentation sets the subject at an understandable and affordable level, the exposition of educational material is logical and consistently, it can be successfully used not only during a cool-urgent explanation of this topic, but also with independent or remote schoolchildren.
Purpose of zero
There are two approaches to the definition of natural numbers:
- count (numbering) subjects ( first, second, the third, fourth, fifth…);
- natural numbers - numbers arising from number designation subjects ( 0 items, 1 subject, 2 subjects, 3 subjects, 4 subjects, 5 items…).
In the first case, a number of natural numbers begins with a unit, in the second - from scratch. There is no one for the majority of mathematicians on the preference of the first or second approach (that is, if the zero is natural or not). In the overwhelming majority of Russian sources, the first approach is traditionally accepted. The second approach, for example, is applied in the writings of Nicolas Bombaki, where natural numbers are defined as the power of finite sets. The presence of zero facilitates the wording and proof of many theorems of arithmetic natural numbers, so the first approach introduces a useful concept extended Natural Rowincluding zero.
The set of all natural numbers is made to designate the symbol. ISO International Standards 31-11 (1992) and ISO 80000-2 (2009) establish the following notation:
In Russian sources, this standard is not yet respected - they are symbol N (\\ DisplayStyle \\ MathBB (N)) denotes natural numbers without zero, and the extended natural row is indicated N 0, z +, z ⩾ 0 (\\ displaystyle \\ mathbb (n) _ (0), \\ mathbb (z) _ (+), \\ mathbb (z) _ (\\ geqslant 0)) etc.
Axioms allowing to determine the set of natural numbers
Peano Axioms for Natural Numbers
Lots of N (\\ DisplayStyle \\ MathBB (N)) Let's call a set of natural numbers if some element is fixed 1 (Unit), function S (\\ DisplayStyle S) c area of \u200b\u200bdefinition N (\\ DisplayStyle \\ MathBB (N))called the following function ( S: N (\\ DisplayStyle S \\ Colon \\ MathBB (N))), and the following conditions are met:
- element unit belongs to this set ( 1 ∈ N (\\ DisplayStyle 1 \\ In \\ MathBB (N))), that is, is a natural number;
- the number next to natural is also natural (if, then S (x) ∈ N (\\ displaystyle s (x) \\ in \\ mathbb (n)) or in a shorter record S: N → N (\\ DisplayStyle S \\ Colon \\ MathBB (N) \\ To \\ MathBB (N)));
- the unit should not be at any natural number ( ∄ x ∈ N (S (x) \u003d 1) (\\ displaystyle \\ Nexists X \\ in \\ mathbb (n) \\ (s (x) \u003d 1)));
- if a natural number A (\\ DisplayStyle a) directly follows as in a natural number B (\\ DisplayStyle B)and in natural numbers C (\\ DisplayStyle C)T. B (\\ DisplayStyle B) and C (\\ DisplayStyle C) - this is the same number (if S (b) \u003d a (\\ displaystyle s (b) \u003d a) and S (C) \u003d A (\\ DisplayStyle S (C) \u003d a)T. B \u003d C (\\ DisplayStyle B \u003d C));
- (induction axiom) if any sentence (statement) P (\\ DisplayStyle P) Proved for a natural number n \u003d 1 (\\ displaystyle n \u003d 1) (base induction) and if from the assumption that it is true for another natural number N (\\ DisplayStyle N), implies that it is true for the following N (\\ DisplayStyle N) Natural Number ( induction assumption), then this offer is true for all natural numbers (let P (N) (\\ DisplayStyle P (N)) - some single (unary) predicate, whose parameter is a natural number N (\\ DisplayStyle N). Then, if P (1) (\\ DisplayStyle P (1)) and ∀ n (p (n) ⇒ P (s (n)) (\\ DisplayStyle \\ Forall N \\; (p (n) \\ rightarrow p (s (n)))))T. ∀ N p (n) (\\ DisplayStyle \\ Forall N \\; P (n))).
The listed axioms reflect our intuitive view of a natural range and numeric line.
The principal fact is that these axioms are essentially definitely determined by natural numbers (categorithics of the system of Peano Axiom). Namely, you can prove (see, as well as 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)))) - Two models for the Axiom Peaano system, then they need is isomorphic, that is, there is a reversible display (bijection) f: n → n ~ (\\ displaystyle f \\ colon \\ mathbb (n) \\ to (\\ tilde (\\ mathbb (n)))) such that f (1) \u003d 1 ~ (\\ displaystyle f (1) \u003d (\\ tilde (1))) and f (s (x)) \u003d s ~ (f (x)) (\\ displaystyle f (s (x)) \u003d (\\ tilde (s)) (F (x))) for all x ∈ N (\\ displaystyle x \\ in \\ mathbb (n)).
Therefore, it is enough to fix as N (\\ DisplayStyle \\ MathBB (N)) No one specific model of multiple natural numbers.
Sometimes, especially in foreign and translation literature, in the first and third axis peano replace the unit to zero. In this case, zero is considered a natural number. When determined through the classes of equilibrium sets, zero is a natural number by definition. Specially discarding it would be unnatural. In addition, it would significantly complicate the further construction and application of the theory, since in most designs zero, as well as the empty set is not something separate. Another advantage to consider zero natural number is that at the same time N (\\ DisplayStyle \\ MathBB (N)) Forms monoid. As mentioned, in Russian literature, zero is traditionally excluded from among natural numbers.
Theoretical and multiple definition of natural numbers (FREGE - Russell)
Thus, natural numbers are introduced, based on the concept of a set, in two rules:
The numbers specified in this way are called the internal.
We describe the first first ordinal numbers and the corresponding natural numbers:
The magnitude of the set of natural numbers
The magnitude of the infinite set is characterized by the concept of "power of the set", which is a generalization of the number of elements of the final set on endless sets. In magnitude (i.e., power) Many natural numbers are greater than any finite set, but less than any interval, for example, interval (0, 1) (\\ DisplayStyle (0,1)). Many natural numbers in power are the same as many rational numbers. Many of the same power as many natural numbers is called a countable set. So, many members of any sequence are counting. At the same time, there is a sequence into which each natural number includes an infinite number of times, since the set of natural numbers can be represented as a countable union of non-cycle counting sets (for example, N \u003d ⋃ k \u003d 0 ∞ (⋃ n \u003d 0 ∞ (2 n + 1) 2 k) (\\ displaystyle \\ mathbb (n) \u003d \\ bigcup \\ limits _ (k \u003d 0) ^ (\\ infty) \\ left (\\ Natural Operations).
Closed operations (operations that do not withdraw the result from a plurality of natural numbers) over natural numbers include the following arithmetic operations:
Further consider two more operations (from a formal point of view that are not operations over natural numbers, since they are not defined for
All par numbers (sometimes exist, sometimes no)): It should be noted that the operations of addition and multiplication are fundamental. In particular, the ring of integers is determined precisely through binary operations of addition and multiplication.
Basic properties
Commitivity of addition:
- a + b \u003d b + a (\\ displaystyle a + b \u003d b + a)
- A ⋅ B \u003d B ⋅ A (\\ DisplayStyle A \\ Cdot B \u003d B \\ Cdot a)
- (A + B) + C \u003d A + (B + C) (\\ DisplayStyle (A + B) + C \u003d A + (B + C))
- (A ⋅ B) ⋅ C \u003d A ⋅ (B ⋅ C) (\\ DisplayStyle (A \\ CDOT B) \\ CDOT C \u003d A \\ CDOT (B \\ CDOT C))
- (A ⋅ (B + C) \u003d A ⋅ B + A ⋅ C (B + C) ⋅ A \u003d B ⋅ A + C ⋅ A (\\ DisplayStyle (\\ Begin (Cases) A \u200b\u200b\\ Cdot (B + C) \u003d A \\ Cdot B + A \\ Cdot C \\\\ (B + C) \\ Cdot A \u003d B \\ Cdot A + C \\ CDOT A \\ ED (Cases))
Addition turns the set of natural numbers in a semigroup with one, the role of unit performs
. Multiplication also converts many natural numbers into a semigroup with a unit, while the unit element is 0 . Using closures regarding the operations of addition-subtraction and multiplying-division, groups of integers are obtained. 1 Z (\\ displaystyle \\ mathbb (z)) and rational positive numbers Q + * (\\ displaystyle \\ mathbb (q) _ (+) ^ (*)) respectively. Integers
- Numbers that apply for account items Any natural number can be written with ten . Numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Such a record calleddecimal the sequence of all natural numbers is called
Natural near most .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ...
small natural number - unit (1). In a natural row, each next number is 1 more than the previous one. Natural seriesinfinite there is no greatest number in it.the value of the digit depends on its place in the number of numbers. For example, a number 4 means: 4 units, if it is in last place in the number of numbers
(in the discharge of units); ten,4 if it is standing in the penultimate place (in the discharge of dozens);4 hundredsif it stands in third place from the end (in discharge hundreds).
Digital0 means absence of units of this dischargein the decimal record of the number. It also serves to designate the number " zero" This number means "not one". Account 0: 3 football matches says that the first team did not hit any goal into the opponent's gate.
Zero do not refer to natural numbers. And really the subject of the subjects never start from scratch.
If the recording of a natural number consists of one mark – one digit, then it is called unambiguous.Those. unambiguous natural number - the natural number, the record of which consists of one mark – one digit. For example, numbers 1, 6, 8 are unambiguous.
Double-digit natural number - a natural number, the record of which consists of two characters - two digits.
For example, numbers 12, 47, 24, 99 are double-digit.
Also, according to the number of signs, this number gives names and other numbers:
numbers 326, 532, 893 - three-digit;
numbers 1126, 4268, 9999 - four-digitetc.
Double-digit, three-digit, four-digit, five-digit, etc. Numbers are called multigalous numbers .
To read multi-valued numbers, they are broken, starting on the right, on the groups of three digits in each (the leftmost group may consist of one or two digits). These groups call classes.
Million - This is a thousand thousand (1000 thousand), it is recorded 1 million or 1,000,000.
Billion - This is 1000 million. It is written 1 billion or 1,000,000,000.
The three first digits on the right make up the class of units, the three are the following - class thousand, then classes of millions, billions, etc. go. (Fig. 1).
Fig. 1. Class of millions, class thousand and class of units (left to right)
The number15389000286 is recorded in the discharge grid (Fig. 2).
Fig. 2. Discharge mesh: number of 15 billion 389 million 286
This number has 286 units in the class of units, zero units in the class of thousands, 389 units in the class of millions of and15 units in the class billion.
