Presentation "notation of natural numbers". Integers
Place zero
There are two approaches to determining 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 is, for example, taken in the works of Nicolas Bourbaki, where the natural numbers are defined as cardinalities of finite sets. 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 set of all natural numbers is usually denoted by the symbol. International standards ISO 31-11 (1992) and ISO 80000-2 (2009) establish the following designations:
In Russian sources this standard is not yet observed - in them the symbol N (\displaystyle \mathbb (N) ) denotes the natural numbers without zero, and the extended natural series is denoted N 0 , Z + , Z ⩾ 0 (\displaystyle \mathbb (N) _(0),\mathbb (Z) _(+),\mathbb (Z) _(\geqslant 0)) etc.
Axioms that allow us to determine the set of natural numbers
Peano's axioms for natural numbers
A bunch of N (\displaystyle \mathbb (N) ) will be called a set of natural numbers if some element is fixed 1 (unit), function S (\displaystyle S) with domain of definition N (\displaystyle \mathbb (N) ), called the follow function ( S: N (\displaystyle S\colon \mathbb (N) )), and the following conditions are met:
- element one belongs to this set ( 1 ∈ N (\displaystyle 1\in \mathbb (N) )), that is, is a natural number;
- the number following the natural number is also a natural number (if , then S (x) ∈ N (\displaystyle S(x)\in \mathbb (N) ) or, in shorter notation, S: N → N (\displaystyle S\colon \mathbb (N) \to \mathbb (N) ));
- 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 as a natural number b (\displaystyle b), and for a natural number c (\displaystyle c), That b (\displaystyle b) And c (\displaystyle c) is the same number (if S (b) = a (\displaystyle S(b)=a) And S (c) = a (\displaystyle S(c)=a), That b = c (\displaystyle b=c));
- (axiom of induction) if any sentence (statement) P (\displaystyle P) proven for natural numbers n = 1 (\displaystyle n=1) (induction base) and if from the assumption that it is true for another natural number n (\displaystyle n), it follows that it is true for the following n (\displaystyle n) natural number ( inductive hypothesis), then this sentence is true for all natural numbers (let P(n) (\displaystyle P(n))- some one-place (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)))), That ∀ 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, as well as a brief 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 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.
Sometimes, especially in foreign and translated literature, in the first and third Peano axioms one is replaced by zero. In this case, zero is considered a natural number. When defined through classes of equipower 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 N (\displaystyle \mathbb (N) ) forms a monoid. As already mentioned, in Russian literature traditionally zero is excluded from the list of natural numbers.
Set-theoretic definition of natural numbers (Frege-Russell definition)
Thus, natural numbers are also introduced based on the concept of a set, according to two rules:
Numbers defined in this way are called ordinal.
Let us describe the first few ordinal numbers and the corresponding natural numbers:
Magnitude of the set of natural numbers
The size of an infinite set is characterized by the concept “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 is the same in cardinality as the set 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:
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)):
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 closures with respect to the operations of addition-subtraction and multiplication-division, groups of integers are obtained Z (\displaystyle \mathbb (Z) ) and rational positive numbers Q + ∗ (\displaystyle \mathbb (Q)_(+)^(*)) respectively.
The history of natural numbers began in primitive times. Since ancient times, people have counted objects. For example, in trade you needed an account of goods or in construction an account of materials. Yes, even in everyday life I also had to count things, food, livestock. At first, numbers were used only for counting in life, in practice, but later, with the development of mathematics, they became part of science.
Integers- these are the numbers we use when counting objects.
For example: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ….
Zero is not a natural number.
All natural numbers, or let's say the set of natural numbers, are denoted by the symbol N.
Table of natural numbers.
Natural series.
Natural numbers written in a row in ascending order form natural series or a series of natural numbers.
Properties of the natural series:
- The smallest natural number is one.
- In a natural series, the next number is greater than the previous one by one. (1, 2, 3, ...) Three dots or ellipses are placed if it is impossible to complete the sequence of numbers.
- The natural series does not have a greatest number, it is infinite.
Example #1:
Write the first 5 natural numbers.
Solution:
Natural numbers start from one.
1, 2, 3, 4, 5
Example #2:
Is zero a natural number?
Answer: no.
Example #3:
What is the first number in natural series?
Answer: The natural series starts from one.
Example #4:
What is the last number in the natural series? What is the largest natural number?
Answer: The natural series begins with one. Each next number is greater than the previous one by one, so the last number does not exist. There is no largest number.
Example #5:
The unit in the natural series has previous number?
Answer: no, because one is the first number in the natural series.
Example #6:
Name the next number in the natural series: a)5, b)67, c)9998.
Answer: a)6, b)68, c)9999.
Example #7:
How many numbers are there in the natural series between the numbers: a) 1 and 5, b) 14 and 19.
Solution:
a) 1, 2, 3, 4, 5 – three numbers are between the numbers 1 and 5.
b) 14, 15, 16, 17, 18, 19 – four numbers are between the numbers 14 and 19.
Example #8:
Say the previous number after 11.
Answer: 10.
Example #9:
What numbers are used when counting objects?
Answer: natural numbers.
The simplest number is natural number. They are used in Everyday life for counting objects, i.e. to calculate their number and order.
What is a natural number: natural numbers name the numbers that are used to counting items or to indicate the serial number of any item from all homogeneous items.
Integers- these are numbers starting from one. They are formed naturally when counting.For example, 1,2,3,4,5... -first natural numbers.
Smallest natural number- one. There is no greatest natural number. When counting the number Zero is not used, so zero is a natural number.
Natural number series is the sequence of all natural numbers. Writing 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 one.
How many numbers are there in the natural series? The natural series is infinite; the largest natural number does not exist.
Decimal since 10 units of any digit form 1 unit of the highest digit. Positionally so how the meaning of a digit depends on its place in the number, i.e. from the category where it is written.
Classes of natural numbers.
Any natural number can be written using 10 Arabic numerals:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
To read natural numbers, they are divided, starting from the right, into groups of 3 digits each. 3 first the numbers on the right are the class of units, the next 3 are the class of thousands, then the classes of millions, billions andetc. Each of the class digits is called itsdischarge.
Comparison of natural numbers.
Of 2 natural numbers, the smaller is the number that is called earlier when counting. For example, number 7 less 11 (written like this:7 < 11 ). When one number is greater than the second, it is written like this:386 > 99 .
Table of digits and classes of numbers.
|
1st class unit |
1st digit of the unit 2nd digit tens 3rd place hundreds |
|
2nd class thousand |
1st digit of unit of thousands 2nd digit tens of thousands 3rd category hundreds of thousands |
|
3rd class millions |
1st digit of unit of millions 2nd category tens of millions 3rd category hundreds of millions |
|
4th class billions |
1st digit of unit of billions 2nd category tens of billions 3rd category hundreds of billions |
|
Numbers from 5th grade and above are considered large numbers. Units of the 5th class are trillions, 6th class - quadrillions, 7th class - quintillions, 8th class - sextillions, 9th class - eptillions. Basic properties of natural numbers.
Operations on natural numbers. 4. Division of natural numbers is the inverse operation of multiplication. If b ∙ c = a, That
Formulas for division: a: 1 = a a: a = 1, a ≠ 0 0: a = 0, a ≠ 0 (A∙ b) : c = (a:c) ∙ b (A∙ b) : c = (b:c) ∙ a Numerical expressions and numerical equalities. A notation where numbers are connected by action signs is numerical expression. For example, 10∙3+4; (60-2∙5):10. Records where 2 numeric expressions are combined with an equal sign are numerical equalities. Equality has left and right sides. The order of performing arithmetic operations. Adding and subtracting numbers are operations of the first degree, while multiplication and division are operations of the second degree. When a numerical expression consists of actions of only one degree, they are performed sequentially from left to right. When expressions consist of actions of only the first and second degrees, then the actions are performed first second degree, and then - actions of the first degree. When there are parentheses in an expression, the actions in the parentheses are performed first. For example, 36:(10-4)+3∙5= 36:6+15 = 6+15 = 21. |
Numbers intended for counting objects and answering the question “how many?” ("How many
balls?", "How many apples?", "How many soldiers?"), are called natural.
If you write them in order, from smallest number to largest, you get a natural series of numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 99, 100, 101, …, 999, 1000, 1001 …
The natural series of numbers begins with the number 1.
Each next natural number is 1 greater than the previous one.
The natural series of numbers is infinite.
Numbers can be even or odd. Even numbers are divisible by two, and Not even numbers are not divisible by two.
Series of odd numbers:
1, 3, 5, 7, 9, 11, 13, …, 99, 101, …, 999, 1001, 1003 …
Series of even numbers:
2, 4, 6, 8, 10, 12, 14, …, 98, 100, …, 998, 1000, 1002 …
In the natural series, 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, the one to the right in the natural series is greater:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 …
So, seven is more than three, and five is more than one.
In mathematics, the word “less” is written using the sign “<», а для записи слова «больше» - знак « > ».
The sharp corner of the greater than and less than symbols always points towards the smaller of the two numbers.
The entry 7 > 3 is read as “seven over three.”
Entry 3< 7 читается как «три меньше семи».
The entry 5 > 1 is read as “five over one.”
Entry 1< 5 читается как «один меньше пяти».
The word “equal” in mathematics is replaced with the sign “=”:
When the numbers are large, it is difficult to immediately say which one is to the right in the natural series.
When comparing two natural numbers with different numbers of digits, the one with the most digits is greater.
For example, 233,000< 1 000 000, потому что в первом числе шесть цифр, а во втором - семь.
Multi-digit natural numbers with the same number of digits are compared bitwise, starting with the most significant digit.
First, the units of the most significant digit are compared, then the next one, the next one, and so on. For example, let's compare the numbers 5401 and 5430:
5401 = 5 thousand 4 hundreds 0 tens 1 unit;
5430 = 5 thousand 4 hundreds 3 tens 0 units.
Comparing units of thousands. In the place of units of thousands of the number 5401 there are 5 units, in the place of units of thousands of the number 5430 there are 5 units. By comparing units of thousands, it is still impossible to say which number is larger.
Comparing hundreds. In the hundreds place of the number 5401 there are 4 units, in the hundreds place the number 5430 is also 4 units. We must continue the comparison.
Comparing tens. In the tens place of the number 5401 there are 0 units, in the tens place of the number 5430 there are 3 units.
Comparing, we get 0< 3, поэтому 5401 < 5430.
Numbers can be arranged in descending or ascending order.
If in a record of several natural numbers each next number is less than the previous one, then the numbers are said to be written in descending order.
Let's write down the numbers 5, 22, 13, 800 in descending order.
Let's find larger number. The number 5 is a single digit number, 13 and 22 are two digit numbers, 800 is a three digit number and therefore the largest. We write 800 in the first place.
Of the two-digit numbers 13 and 22, the greater is 22. After the number 800 we write the number 22, and then 13.
The smallest number is the single-digit number 5. We write it last.
800, 22, 13, 5 - recording these numbers in descending order.
If in a record of several natural numbers each next number is greater than the previous one, then the numbers are said to be written in ascending order.
How to write the numbers 15, 2, 31, 278, 298 in ascending order?
Among the numbers 15, 2, 31, 278, 298 we will find the smaller one.
This is a single-digit number 2. Let's write it in first place.
From the two-digit numbers 15 and 31, choose the smaller one - 15, write it in second place, and after it - 31.
Of the three-digit numbers, 278 is the smallest, we write it after the number 31, and the last we write the number 298.
2, 15, 21, 278, 298 - writing these numbers in ascending order