On the axiomatic way of constructing a theory. Determination of a natural number. Axiomatic theories Axiomatic construction of a system of natural numbers addition

In the axiomatic construction of any mathematical theory, certain rules are observed:

Some concepts of the theory are chosen as major and are accepted without definition;

Each concept of the theory, which is not contained in the list of basic ones, is given a definition, it explains its meaning with the help of basic and previous concepts;

Formulated axioms- proposals that in this theory are accepted without proof; they reveal the properties of basic concepts;

Every proposition of a theory that is not in the list of axioms must be proven; such propositions are called theorems and prove them on the basis of the axioms and theorems that precede the one under consideration.

If the construction of a theory is carried out by an axiomatic method, i.e. according to the above rules, then they say that the theory is built deductively.

In the case of an axiomatic construction of a theory, essentially all statements are deduced by proof from axioms. Therefore, special requirements are imposed on the system of axioms. First of all, it must be consistent and independent.

The system of axioms is called consistent if it is impossible to logically deduce from it two mutually exclusive sentences.

If a system of axioms does not possess this property, it cannot be suitable for substantiating a scientific theory.

A consistent system of axioms is called independent, if none of the axioms of this system is a consequence of other axioms of this system.

In the axiomatic construction of the same theory, different systems of axioms can be used. But they must be equal. In addition, when choosing a particular system of axioms, mathematicians take into account how simple and clear proofs of theorems can be obtained in the future. But if the choice of axioms is conditional, then science itself or a separate theory does not depend on any conditions - they are a reflection of the real world.

Axiomatic construction of the system natural numbers carried out according to the formulated rules. Studying this material, we must see how all the arithmetic of natural numbers can be derived from the basic concepts and axioms. Of course, its presentation in our course will not always be rigorous - we omit some of the proofs due to their great complexity, but we will stipulate each such case.

The exercise

1. What is the essence of the axiomatic way of constructing a theory?

2. Is it true that an axiom is a proposition that does not require proof?

3. What are the basic concepts of the school planimetry course? Remember a few of the axioms from this course. The properties of what concepts are described in them?

4. Define a rectangle by choosing a parallelogram as a generic concept. Name three concepts that should precede the concept of "parallelogram" in a geometry course.

5. What sentences are called theorems? Remember what is logical structure theorem and what it means to prove the theorem.

Basic concepts and axioms. Determination of a natural number

As the basic concept in the axiomatic construction of the arithmetic of natural numbers, we took the relation "follow directly", given on a non-empty set N. Also known are the notions of a set, an element of a set, and other set-theoretic notions, as well as the rules of logic.

The element immediately following the element but, denote but".

The essence of the "follow immediately" relationship is revealed in the following axioms.

Axiom 1. The set N contains an element that does not immediately follow any element of this set. We will call it a unit and denote it by the symbol 1.

Axiom 2. For each element and from N there is only one element a"immediately following but.

Axiom 3. For each element but there is at most one element of N followed immediately by but.

Axiom 4. Any subset M multitudes N coincides with N, if it has the following properties: 1) 1 is contained in M; 2) from the fact that but contained in M, it follows that and but" contained in M.

The formulated axioms are often called Peano's axioms.

Using the relation "follow immediately" and axioms 1-4, we can give the following definition of a natural number.

Definition. Lots of N, for the elements of which the relation "follow immediately" is established, which satisfies axioms 1-4, is called the set of natural numbers, and its elements- natural numbers.

This definition does not say anything about the nature of the elements of the set N. This means that it can be anything. Choosing as

of the set N some specific set, on which a specific relation "immediately follow after" is given, satisfying axioms 1-4, we get model of the given system of axioms. In mathematics, it has been proved that between all such models it is possible to establish a one-to-one correspondence that preserves the "immediately follow" relationship, and all such models will differ only in the nature of the elements, their name and designation. The standard model of the Peano axiom system is the historical development society a number of numbers:

Each number in this series has its own designation and name, which we will consider known.

Considering the natural series of numbers as one of the models of axioms 1-4, it should be noted that they describe the process of the formation of this series, and this happens when the properties of the relation "follow immediately" are revealed in the axioms. So, the natural series begins with the number 1 (axiom 1); each natural number is immediately followed by a single natural number (axiom 2); each natural number immediately follows at most one natural number (axiom 3); starting from the number 1 and passing in order to the immediately following one after another natural numbers, we obtain the entire set of these numbers (axiom 4). Note that Axiom 4 in a formalized form describes the infinity of natural numbers, and the proof of statements about natural numbers is based on it.

In general, any countable set can be a model of the Peano axiom system, for example:! ..

Consider, for example, a sequence of sets in which the set (oo) is the initial element, and each subsequent set is obtained from the previous one by assigning one more circle (Fig. 108, a). Then N there is a set consisting of sets of the type described, and it is a model of the Peano axiom system. Indeed, the set N contains an element (oo) that does not immediately follow any element of this set, i.e.

there is only one set that is obtained from BUT by adding one circle, i.e., Axiom 2 holds. For each set BUT there is at most one set from which the set is formed BUT adding one circle, i.e. Axiom 3 holds. If MÌ N and it is known that the set BUT contained in M, it follows that the set in which there is one more circle than in the set BUT, also contained in M, then M = N(and hence Axiom 4 holds).

Note that none of the axioms can be omitted in the definition of a natural number - for any of them it is possible to construct a set in which the other three axioms are fulfilled, but this axiom is not fulfilled. This position is clearly confirmed by the examples shown in Figures 109 and 110. In Fig. 109, a is shown a set in which axioms 2 and 3 are fulfilled, but axiom 1 is not fulfilled (axiom 4 will not make sense, since there is no element in the set, directly not following any other). Figure 109, b shows a set in which axioms 1, 3, and 4 are fulfilled, but behind the element but two elements immediately follow, and not one, as required in axiom 2. Figure 109, c shows a set in which axioms 1, 2, 4 are satisfied, but the element from immediately follows as an element but, and behind the element b. Figure 110 shows a set in which axioms 1, 2, 3 are fulfilled, but axiom 4 is not satisfied - the set of points lying on the ray, it contains the number immediately following it, but it does not coincide with the entire set of points shown in the figure.

The fact that axiomatic theories do not speak of the "true" nature of the concepts under study makes at first glance these theories too abstract and formal - it turns out that the same axioms are satisfied by different sets of objects and different relationships between them. However, this seeming abstractness is the strength of the axiomatic method: every statement deduced from the given axioms logically can be applied to any set of objects, as long as relations that satisfy the axioms are defined in them.

So, we began the axiomatic construction of a system of natural numbers with the choice of the basic relation "follow immediately" and the axioms in which its properties are described. The further construction of the theory presupposes consideration of the known properties of natural numbers and operations on them. They must be disclosed in definitions and theorems, i.e. are deduced from a purely logical way from the relation “immediately follow”, and axioms 1-4.

The first concept that we will introduce after defining a natural number is the relation "immediately precedes", which is often used when considering the properties of a natural number.

Definition. If a natural number b immediately follows a natural number a, then the number a is called immediately preceding (or preceding) the number b.

The relationship “precedes” has a number of properties. They are formulated in the form of theorems and are proved using Axioms 1 - 4.

Theorem 1... The unit has no preceding natural number.

The truth of this statement follows immediately from Axiom 1.

Theorem 2. Every natural number but, other than 1, has a preceding number b, such that b ¢ = a.

Evidence. Let us denote by M the set of natural numbers, consisting of the number 1 and of all numbers that have a preceding one. If the number but contained in M, then the number but" also in M, since the preceding for but" is the number but. This means that the set M contains 1, and from the fact that the number but belongs to the set M, it follows that the number but" belongs M. Then, by Axiom 4, the set M coincides with the set of all natural numbers. This means that all natural numbers, except 1, have a preceding number.

Note that by virtue of Axiom 3, numbers other than 1 have a single preceding number.

The axiomatic construction of the theory of natural numbers is not considered either in primary or secondary school. However, those properties of the relationship "immediately follow", which are reflected in the Peano's axioms, are the subject of study in the initial course of mathematics. Already in the first grade, when considering the numbers of the first ten, it becomes clear how each number can be obtained. In this case, the concepts "should" and "precede" are used. Each new number acts as a continuation of the studied segment of the natural series of numbers. Students are convinced that each number is followed by the next, and moreover, only one, that the natural series of numbers is infinite. And of course, knowledge of the axiomatic theory will help the teacher to methodically correctly organize the assimilation of the features of the natural series of numbers by children.

Exercises

1. Can Axiom 3 be formulated as follows: “For each element but of N there is only one element immediately followed by a "?

2. Highlight the condition and conclusion in Axiom 4, write them down using the symbols Î, =>.

3. Continue the definition of a natural number: “An element of the set Î, Þ is called a natural number.

Addition

According to the rules for constructing an axiomatic theory, the definition of addition of natural numbers must be introduced using only the relation “immediately follow” and the concepts of “natural number” and “preceding number”.

We preface the definition of addition with the following reasoning. If to any natural number but add 1, then we get the number but", immediately following a, i.e. but + 1 = but", and, therefore, we get the rule for adding 1 to any natural number. But how to add to the number but natural number b, other than 1? Let's use the following fact: if it is known that 2 + 3 = 5, then the sum 2 + 4 is equal to the number 6, which immediately follows the number 5. This happens because in the sum 2 + 4 the second term is the number immediately following the number 3 . Thus, the sum but+ b " can be found if the amount is known but+ b. These facts are the basis for the definition of the addition of natural numbers in the axiomatic theory. In addition, it uses the concept of an algebraic operation.

Definition. The addition of natural numbers is an algebraic operation that has the following properties:

1) ("but Î N ) a + 1 = a ",

2) (" but, b Î) a + b "= (a + b)".

Number but+ b called the sum of numbers but and b, and the numbers themselves but and b -terms.

As you know, the sum of any two natural numbers is also a natural number, and for any natural numbers but and b sum but+ b- the only one. In other words, the sum of natural numbers exists and is unique. A feature of the definition is that it is not known in advance whether there exists an algebraic operation with the indicated properties, and if it exists, is it unique? Therefore, with the axiomatic construction of the theory of natural numbers, the following statement is proved:

Theorem 3. The addition of natural numbers exists and is unique.

This theorem consists of two statements (two theorems):

1) addition of natural numbers exists;

2) addition of natural numbers is unique.

As a rule, existence and uniqueness are linked together, but they are often independent of each other. The existence of an object does not imply its uniqueness. (For example, if you say that you have a pencil, that does not mean that there is only one.) The uniqueness statement means that there cannot be two objects with given properties. Uniqueness is often proved by contradiction: it is assumed that there are two objects that satisfy a given condition, and then they build a chain of deductive inferences leading to a contradiction.

To verify the truth of Theorem 3, we first prove that if in the set N there is an operation with properties 1 and 2, then this operation is unique; then we prove that an addition operation with properties 1 and 2 exists.

Proof of the uniqueness of addition. Suppose that in the set N there are two addition operations with properties 1 and 2. We denote one of them by the sign +, and the other by the sign M. For these operations we have:

1) a + 1 = but"; 1) butÅ = a "\

2) a + b "= (a + b)" 2) butÅ b "= (aÅ b) ".

Let us prove that

("a, bÎ N )a + b = aÅ b. (1)

Let the number but is chosen arbitrarily, and b M b, for which equality (1) is true.

It is easy to verify that 1 Î M. Indeed, from the fact that but+ 1 = but"=butЕ 1 it follows that a + 1 = aÅ 1.

Let us now prove that if bÎ M, then b "Î М, those. if a a + b = aÅ b, then but+ b "= aÅ b ". As a + b - aÅ b, then by axiom 2 (a + b) "= (aÅ b) ", and then a + b "- (a + b)" = (aÅ b) "= aÅ b ". Since the set M contains 1 and together with each number b contains also the number b ¢ then by Axiom 4, the set M coincides with N, and hence equality (1) b. Since the number but was chosen arbitrarily, then equality (1) is true for any natural but and b, those. operations + and Е on the set N may differ from each other only in designations.

Proof of the existence of addition. Let us show that an algebraic operation with properties 1 and 2 indicated in the definition of addition exists.

Let be M - many of those and only those numbers but, for which it is possible to determine a + b so that conditions 1 and 2 are satisfied. Let us show that 1 Î M. To do this, for any b put

1 + b = b ¢.(2)

1) 1 + 1 = 1 ¢ - according to rule (2), i.e. equality holds a + 1 = but" at but= 1.

2)1 + b "= (b ")¢ b= (1 + b) "- according to rule (2), i.e., the equality a + b "= (a + b) " at a = 1.

So, 1 belongs to the set M.

Let's pretend that but belongs M. Based on this assumption, we will show that but" contained in M, those. that you can define addition but" and any number b so that conditions 1 and 2 are satisfied. To do this, we put:

but"+ b =(a + b) ".(3)

Since, by assumption, the number a + b is defined, then by Axiom 2, the number (but+ b) ". Let us verify that conditions 1 and 2 are satisfied in this case:

1)a "+ 1 = (a + 1)" = (but")". In this way, but"+ 1 = (a")".

2)a "+ b" = (a + b ¢) "= ((a + b) ")"= (a "+ b)". In this way, a "+ b" = = (a "+ b)".

So, we have shown that the set M contains 1 and together with each number but contains number but". By Axiom 4, we conclude that the set M there are many natural numbers. Thus, there is a rule that allows for any natural numbers but and b uniquely find such a natural number a + b, that properties 1 and 2, formulated in the definition of addition, hold.

Let us show how the well-known table of addition of single-digit numbers can be derived from the definition of addition and Theorem 3.

Let's agree on the following notation: 1 "= 2; 2" = 3; 3 ¢ = 4; 4 "= 5, etc.

We draw up a table in the following sequence: first, we add one to any single-digit natural number, then two, then three, etc.

1 + 1 = 1 ¢ based on property 1 of the definition of addition. But 1 ¢ we agreed to denote 2, therefore, 1 + 1 = 2.

Similarly, 2 + 1 = 2 "= 3; 3 + 1 = 3" = 4, etc.

Let us now consider the cases associated with the addition of the number 2 to any single-digit natural number.

1 + 2 = 1 + 1 ¢ - used the accepted designation. But 1 + 1 ¢ = = (1 + 1) "according to property 2 from the definition of addition, 1 + 1 is 2, as stated above. Thus,

1 +2 = 1 + 1" = (1 +1)" = 2" = 3.

Similarly 2 + 2 = 2 + 1 "= (2 + 1)" = 3 "= 4; 3 + 2 = 3 + 1 ¢= (3 + 1) "= = 4" = 5, etc.

If we continue this process, we get the entire table of addition of single-digit numbers.

The next step in the axiomatic construction of a system of natural numbers is to prove the properties of addition, with the property of associativity being considered first, then commutativity, etc.

Theorem 4.(" a, b, cÎ N ) (a + b)+ from= but+ (b+ from).

Evidence. Let the natural numbers but and b are chosen arbitrarily, and from takes on various natural meanings. Let us denote by M the set of all those and only those natural numbers c for which the equality (a + b) + c = a + (b + c) right.

First, let us prove that 1 Î M, those. make sure equality is fair (but+ b)+ 1 = but+ (b+ 1) Indeed, by the definition of addition, we have (a + b)+ 1 = (but+ b) "= but+ b "= but+ (b+ 1).

Let us now prove that if c Î M, then c "Î M, those. from equality (but+ b)+ c = a+ (b + c) equality follows (but+ b)+ from"= but+ (b + c "). (but+ b)+ from"= ((but + b)+ from)". Then, based on the equality (but+ b) + c= a + (b + c) you can write: ((but+ b)+ c) "= (a+ (b+ from))". From where, by the definition of addition, we get: ( a + (b+ c)) "= a + (b + c)" = a + (b + c ") .

M contains 1, and from the fact that from contained in M, it follows that and from" contained in M. Therefore, according to Axiom 4, M= N, those. equality ( but + b)+ from= a + (b + c) true for any natural number from, and since the numbers but and b were chosen arbitrarily, then it is true and for any natural numbers but and b, Q.E.D.

Theorem 5.("a, bÎ N) a+ b= b+ but.

Evidence. It consists of two parts: first, they prove that (" aÎ N) but+1 = 1+a and then that (" a, bÎ N ) a + b = b+ but.

1 . Let us prove that (" butÎN) a+ 1 = 1 + a. Let be M - set of all those and only those numbers but, for which the equality but+ 1 = 1 + but true.

Since 1 + 1 = 1 + 1 is a true equality, then 1 belongs to the set M.

Let us now prove that if butÎ M, then but"Î M, i.e., from the equality a + 1 = 1 + but equality follows a "+ 1 = 1 + but". Really, a "+ 1 = (a + 1) + 1 by the first addition property. Further, the expression (a + 1) + 1 can be converted to the expression (1 + a) + 1, using the equality but+ 1 = 1 + but. Then, based on the associative law, we have: (1 + but)+ 1 = 1 + (but+ 1). And finally, by the definition of addition, we get: 1 + (a + 1) = 1 + a ".

Thus, we have shown that the set M contains 1 and together with each number but contains also the number but". Therefore, according to the axiom A, M = I, those. equality but+ 1 = 1 + but true for any natural but.

2 . Let us prove that (" a, bÎ N ) but+ b = b+ but. Let be but - an arbitrarily chosen natural number, and b takes on various natural meanings. Let us denote by M the set of all those and only those natural numbers b, for which the equality a + b = b+ but true.

Since at b = 1 we obtain the equality but+ 1 = 1 + but, the truth of which is proved in item 1, then 1 is contained in M.

Let us now prove that if b belongs M, then and b " also belongs M, those. from equality but+ b = b+ but equality follows but+ b "= b "+ but. Indeed, by the definition of addition, we have: but+ b "= (but+ b) ". As but+ b= b+ but, then (but+ b) "= (b+ but)". Hence, by the definition of addition: (b+ but)"= b+ but"= b+ (a+ 1). Based on the fact that a + 1 = 1 + but, we get: b+ (a + 1) = b+ (1 + but). Applying the associative property and the definition of addition, we perform the transformations: b + (1 + a) = (b + 1) + a = b "+ a.

So, we have proved that 1 is contained in the set M and together with each number b lots of M contains also the number b ¢, immediately following b ¢. By Axiom 4, we obtain M= AND, those. equality a+ b= b+ but true for any natural number b, as well as for any natural but, as his choice was arbitrary.

Theorem 6. ("a, bÎ N) a + b¹ b.

Evidence. Let be but - a natural number chosen arbitrarily, and b takes on various natural meanings. Let us denote by M the set of those and only those natural numbers b, for which Theorem 6 is true.

Let us prove that 1 Î M. Indeed, since but+ 1 = but"(by the definition of addition), and 1 does not follow any number (axiom 1), then but+ 1 ¹ 1.

Let us now prove that if bÎ M, then b "Î M, those. from what a + bÎ b it follows that a + b "¹ b ". Indeed, by the definition of addition, a + b "= (a + b)", but since a + bÎ b, then (a + b) "¹ b " and therefore a + b ¢=b ¢.

By Axiom 4, the sets M and N coincide, therefore, for any natural numbers a + bÎ b, Q.E.D.

The approach to addition, considered in the axiomatic construction of a system of natural numbers, is the basis of the initial teaching of mathematics. Obtaining numbers by adding 1 is closely related to the principle of constructing a natural series, and the second property of addition is used in calculations, for example, in such cases: 6 + 3 = (6+ 2) + 1 = 8 + 1 = 9.

All proven properties are studied in the beginner's mathematics course and are used to transform expressions.

Exercises

1. Is it true that each natural number is obtained from the previous one by adding one?

2. Using the definition of addition, find the meaning of the expressions:

a) 2 + 3; b) 3 + 3; c) 4 + 3.

3. What transformations of expressions can be performed using the associativity property of addition?

4. Convert the expression by applying the associative addition property:

a) (12 + 3) +17; b) 24 + (6 + 19); c) 27 + 13 + 18.

5. Prove that (" a, bÎ N) a + b¹ but.

6. Find out how the various primary school mathematics textbooks are formulated:

a) the commutative property of addition;

b) the associative property of addition.

7 In one of the textbooks for elementary school, the rule of adding a number to the sum is considered on a specific example (4 + 3) + 2 and the following ways of finding the result are proposed:

a) (4 + 3) + 2 = 7 + 2 = 9;

b) (4 + 3) + 2 = (4 + 2) + 3 = 6 + 3 = 9;

c) (4 + 3) + 2 = 4 + (2 + 3) = 4 + 5 = 9.

Justify the performed transformations. Can it be argued that the rule for adding a number to a sum is a consequence of the associative property of addition?

8 .It is known that a + b= 17. What is equal to:

but) a + (b + 3); b) (but+ 6) + b; c) (13+ b)+a?

9 .Describe possible ways to calculate the value of an expression of the form a + b + c. Give reasons for these methods and illustrate them with specific examples.

Multiplication

According to the rules for constructing an axiomatic theory, it is possible to define the multiplication of natural numbers using the relation “follow immediately after” and the concepts introduced earlier.

We preface the definition of multiplication with the following arguments. If any natural number but multiply by 1, you get but, those. the equality holds a × 1 = but and we get the rule for multiplying any natural number by 1. But how to multiply the number but for a natural number b, other than 1? Let's use the following fact: if it is known that 7 × 5 = 35, then to find the product 7 × 6, it is enough to add 7 to 35, since 7 × 6 = 7 × (5 + 1) = 7 × 5 +7. Thus, the product a × b " can be found if the product is known: a × b "= a × b+ but.

The noted facts form the basis for the definition of multiplication of natural numbers. In addition, it uses the concept of an algebraic operation.

Definition. Multiplication of natural numbers is an algebraic operation that has the following properties:

1) ("a Î N) a × 1= a;

2) ("a, Î N) a × b "= a × b+ but.

Number a × b called product numbers but and b, and the numbers themselves but and b-multipliers.

A feature of this definition, as well as the definition of addition of natural numbers, is that it is not known in advance whether there exists an algebraic operation with the indicated properties, and if it exists, then whether it is unique. In this regard, it becomes necessary to prove this fact.

Theorem 7. Multiplication of natural numbers exists, and it is unique.

The proof of this theorem is similar to the proof of Theorem 3.

Using the definition of multiplication, Theorem 7, and the addition table, You can derive the multiplication table for single numbers. We do it in the following sequence: first we consider multiplication by 1, then by 2, etc.

It is easy to see that multiplication by 1 is performed by property 1 in the definition of multiplication: 1 × 1 = 1; 2 × 1 = 2; 3 × 1 = 3, etc.

Let us now consider the cases of multiplication by 2: 1 × 2 = 1 × 1 "= 1 × 1 + 1 = 1 + 1 = 2 - the transition from the product 1 × 2 to the product 1 × 1 ¢ is carried out according to the previously adopted notation; the transition from expression 1 × 1 to the expression 1 × 1 + 1 - based on the second property of multiplication; the product 1 × 1 is replaced by the number 1 in accordance with the result already obtained in the table; and, finally, the value of the expression 1 + 1 is found in accordance with the addition table.

2 × 2 = 2 × 1 "= 2 × 1 +2 = 2 + 2 = 4;

3 × 2 = 3 × 1 ¢ = 3 × 1 + 3 = 3 + 3 = 6.

If we continue this process, we get the entire multiplication table for single numbers.

As you know, multiplication of natural numbers is commutative, associative and distributive with respect to addition. In the case of an axiomatic construction of the theory, it is convenient to prove these properties, starting with distributivity.

But due to the fact that the commutativity property will be proved later, it is necessary to consider distributivity on the right and left with respect to addition.

Theorem 8. ("a, b, cÎ N) (but+ b) × c = a × c+ b × c.

Evidence. Let natural numbers a and b are chosen arbitrarily, and from takes on various natural meanings. Let us denote by M the set of all those and only those natural numbers c for which the equality (a + b) × c = a × c+ b × c.

Let us prove that 1 Î M, those. that equality ( a + b) × 1 = but× 1 + b × 1 true. According to property 1 from the definition of multiplication we have: (a + b) × 1= a + b = a × 1+ b× 1.

Let us now prove that if fromÎ M, then from"Î M, those. that from the equality ( a + b) c = a × c+ b × c equality follows (but+ b) × c "= a × c"+ b × c ". By the definition of multiplication, we have: ( a + b) × c "= (a + b) × c+ (a + b). As (a + b) × c = a × c + b × c, then ( a + b) × c+ (a + b)= (a × c + b × c) + (a+ b). Using the associative and commutative properties of addition, we perform the transformations: ( a× from+ b × c)+ (but+ b) =(a× from + b × c+ but)+ b =(a × c + a + b × c)+ b= = ((a × c+ a) + b × c)+ b = (a × c+ a) + (b × c+ b). And finally, by the definition of multiplication, we get: (a × c+ a) + (b × c+ b) = a × c "+ b × c ".

So, we have shown that the set M contains 1, and from the fact that it contains c, it follows that and from" contained in M. By Axiom 4, we obtain M= N. This means that the equality ( a + b) × c = a × c + b × c true for any natural numbers from, as well as for any natural a and b, since they were chosen arbitrarily.

Theorem 9. (" a, b, cÎ N) a × (b + c) = a × b + a × c.

This property is left distributive with respect to addition. It is proved in the same way as it is done for right distributivity.

Theorem 10.(" a, b, cÎ N) (a × b) × c = a × (b × c).

This is the associativity property of multiplication. Its proof is based on the definition of multiplication and Theorems 4-9.

Theorem 11. ("a, b,Î N) a × b.

The proof of this theorem is similar in form to the proof of the commutative property of addition.

The approach to multiplication, considered in the axiomatic theory, is the basis for learning multiplication in primary school... Multiplication by 1 is usually defined, and the second multiplication property is used when compiling a multiplication table for single-digit numbers and calculations.

In the initial course, we study all the properties of multiplication that we have considered: commutativity, associativity, and distributivity.

Exercises

1 ... Using the definition of multiplication, find the values of the expressions:

a) 3 × 3; 6) 3 × 4; c) 4 × 3.

2. Write down the distributive property of left multiplication with respect to addition and prove it. What transformations of expressions are possible based on it? Why did it become necessary to consider the distributivity of left and right multiplication with respect to addition?

3. Prove the associativity property of multiplication of natural numbers. What transformations of expressions are possible based on it? Is this property being studied in elementary school?

4. Prove the commutative property of multiplication. Give examples of its use in an elementary math course.

5. What properties of multiplication can be used when finding the value of an expression:

a) 5 × (10 + 4); 6) 125 × 15 × 6; c) (8 × 379) × 125?

6. It is known that 37 - 3 = 111. Using this equality, calculate:

a) 37 × 18; b) 185 × 12.

Justify all performed transformations.

7 ... Determine the meaning of an expression without performing written calculations. Justify the answer:

a) 8962 × 8 + 8962 × 2; b) 63402 × 3 + 63402 × 97; c) 849+ 849 × 9.

8 ... What properties of multiplication will primary school students use when completing the following tasks:

Is it possible, without calculating, to say which expressions will have the same values:

a) 3 × 7 + 3 × 5; b) 7 × (5 + 3); c) (7 + 5) × 3?

Are the equalities true:

a) 18 × 5 × 2 = 18 × (5 × 2); c) 5 × 6 + 5 × 7 = (6 + 7) × 5;

b) (3 × 10) × 17 = 3 × 10 × 17; d) 8 × (7 + 9) = 8 × 7 + 9 × 8?

Is it possible, without performing calculations, to compare the values of the expressions:

a) 70 × 32 + 9 × 32 ... 79 × 30 + 79 × 2;

b) 87 × 70 + 87 × 8 ... 80 × 78 + 7 × 78?

Department of Education of the Administration of the Kirovsky District of Volgograd

Municipal educational institution

gymnasium number 9

Section mathematics

On this topic:Integers

Pupils of 6 b grade

Shanina Lisa

Leader:

Mathematic teacher

Date of writing the work:

Manager's signature:

Volgograd 2013

Introduction page 3

§one. Basic concepts and definitions page 4

§2. Axiomatics of natural number p. 5

§3. "ABOUT SOME SECRETS THAT STORE NUMBERS" p.8

§four. Great mathematicians p. 10

Conclusion p. 12

References p. 13

Introduction

What are natural numbers? Everything! Oh how good. Who can explain? Um, um, "positive integers", no, it won't work. We'll have to explain what "integers" are, which is more difficult. Still have versions? The number of apples? We don't seem to understand why we need to explain.

Natural numbers are some mathematical objects, in order to make some statements about them, to introduce operations on them (addition, multiplication), we need some kind of formal definition. Otherwise, the addition operation will remain the same informal, at the level "there were two heaps of apples, put them into one". And it will become impossible to prove theorems that use addition, this is sad.

Yes, yes, it is absolutely correct to remember that points and lines are indefinable concepts. But we have axioms that define properties that can be used in proofs. For example, "through any two points on the plane, you can draw a straight line and, moreover, only one." And so on. Here's something I would like.

In this paper, we will consider natural numbers, Peano's axioms and the mysteries of numbers.

Relevance and novelty of work is that the area of Peano's axioms is not disclosed in school textbooks and their role is not shown.

The purpose of this work is study of the question of the natural number and the secrets of numbers.

The main hypothesis of work is Peano's axiom and the mystery of numbers.

§one. Basic concepts and definitions

Number - it is an expression of a certain amount.

Natural number an element of an unbounded sequence.

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

There are two approaches to defining natural numbers - numbers used when:

enumeration (numbering) of objects (first, second, third, ...);

designation of the number of items (no items, one item, two items, ...).

Axiom – these are the basic starting points (self-evident principles) of one theory or another, from which, by deduction, that is, by purely logical means, the rest of the content of this theory is extracted.

A number that has only two divisors (this number itself and one) is called - simple number.

Composite Number is a number that has more than two divisors.

§2. Axiomatics of Natural Number

Natural numbers are obtained when counting objects and when measuring quantities. But if, when measuring, numbers appear other than natural numbers, then counting leads only to natural numbers. To keep score, you need a sequence of numbers that starts with one and which allows you to move from one number to another and as many times as necessary. In other words, you need a natural line segment. Therefore, solving the problem of substantiating the system of natural numbers, first of all it was necessary to answer the question of what is a number as an element of a natural number. The answer to this was given in the works of two mathematicians - German Grassmann and Italian Peano. They proposed an axiomatics in which the natural number was justified as an element of an unboundedly continuing sequence.

The axiomatic construction of a system of natural numbers is carried out according to the formulated rules .

Five axioms can be viewed as an axiomatic definition of basic concepts:

1 is a natural number;

The next natural number is a natural number;

1 does not follow any natural number;

If a natural number but follows natural number b and beyond the natural number from then b and from are identical;

If any proposition is proved for 1 and if from the assumption that it is true for a natural number n, it follows that it is true for the following n a natural number, then this proposition is true for all natural numbers.

Unit Is the first number in the natural series , and also one of the digits in the decimal notation system.

It is believed that the designation of the unit of any category by the same sign (rather close to the modern one) first appeared in Ancient Babylon in about 2 thousand years BC. e.

The ancient Greeks, who considered only natural numbers as numbers, considered each of them as a collection of units. The very same unit is assigned special place: it was not counted as a number.

I. Newton wrote: "... by number we mean not so much a collection of units as an abstract ratio of one quantity to another quantity, conventionally taken by us as a unit." Thus, the unit has already taken its rightful place among other numbers.

Arithmetic operations on numbers have a variety of properties. They can be described in words, for example: "The sum does not change from the change of the places of the terms." Can be written in letters: a + b = b + a. Can be expressed in special terms.

We apply the basic laws of arithmetic, often out of habit, without realizing it:

1) displacement law (commutativity), - the property of addition and multiplication of numbers, expressed by identities:

a + b = b + a a * b = b * a;

2) the combination law (associativity), - the property of addition and multiplication of numbers, expressed by identities:

(a + b) + c = a + (b + c) (a * b) * c = a * (b * c);

3) distribution law (distributivity), - a property that connects addition and multiplication of numbers and expressed by identities:

a * (b + c) = a * b + a * c (b + c) * a = b * a + c * a.

After proving the displaceable, combinative and distributive (with respect to addition) laws of the action of multiplication, the further construction of the theory of arithmetic operations on natural numbers does not present any fundamental difficulties.

At present, in our minds or on a piece of paper, we do only the simplest calculations, more and more often entrusting more complex computational work to calculators and computers. However, the operation of all computers - simple and complex - is based on the simplest operation - the addition of natural numbers. It turns out that the most complex calculations can be reduced to addition, only this operation must be done many millions of times.

§3. . "ABOUT SOME SECRETS THAT KEEP NUMBERS"

Mersenne numbers.

The search for prime numbers has been going on for several centuries.

A number that has only two divisors (this number itself and one) is called a prime number

A composite number is one that has more than two divisors. For example: the French monk Maren Mersenne (1 year) wrote down the formula for the number "for simplicity", which received the name of the Mersenne number.

These are numbers of the form M p = 2P -1, where p = prime number.

I checked: is this formula workable for all prime numbers

By now, numbers greater than 2 have been tested for simplicity for all p up to 50,000. E ”as a result, more than 30 Mersenne primes have been found.

3.1 Perfect numbers.

Among the composite numbers, there is such a group of numbers that are called perfect if the number was equal to the sum of all its divisors (excluding the number itself). For example:

496=1+2+4+8+16+31+62+124+248

3.2. Friendly numbers

The scientist Pythagoras traveled a lot in the countries of the East: he was in Egypt and Babylon. There Pythagoras also got acquainted with Eastern mathematics. Pythagoras believed that the secret of the world is hidden in the numerical laws, numbers have their own special life meaning. Among the composite numbers, there are pairs of numbers, each of which is equal to the sum of the divisors of the other.

For example: 220 and 284

220=1+2+4+5+10+11+20+22+44+55+110=284

234=1+2+4+71+142=220

I found a couple more friendly numbers with my calculator.

For example: 1184 and 1210

1184=1+2+4+8+16+32+37+74+148+296+592=1210

1210 = 1 + 2 + 5 + 10 + 1.1 + 22 + 55 + 110 + 121 + 242 + 605 = 1184 etc. etc.

Friendly numbers- two natural numbers, for which the sum of all divisors of the first number (except for itself) is equal to the second number and the sum of all divisors of the second number (except for itself) is equal to the first number .. Usually, when talking about friendly numbers, they mean pairs of two different numbers.

Friendly numbers

Friendly numbers are a pair of numbers, each of which equals the sum of its divisors (for example, 220 and 284).

§four. Great mathematicians

Hermann Gunther Grassmann ( German Hermann Günther Grassmann, 1809-1877) - physicist, mathematician and philologist.

After Grassmann was educated in Stetin, he entered the University of Berlin, at the Faculty of Theology. Having successfully passed both exams in theology, for a long time he did not abandon the thought of devoting himself to the work of a preacher, and he retained his aspiration for theology until the end of his life. At the same time, he became interested in mathematics. In 1840 he passed an additional examination to acquire the right to teach mathematics, physics, mineralogy and chemistry. .

Differential "href =" / text / category / differentcial / "rel =" bookmark "> differential equations, definition and scope of the concept of a curve, etc.) and the formal-logical foundation of mathematics. Its axiomatics of natural numbers has become common use. his example is a continuous (Jordan) curve that completely fills a certain square.

Sir Isaac Newton (English Sir Isaac Newton, December 25, 1642 - March 20, 1727 according to the Julian calendar, in force in England until 1752; or January 4, 1643 - March 31, 1727 according to the Gregorian calendar) - English physicist, mathematician and astronomer, one of the founders of classical physics ... The author of the fundamental work "Mathematical Principles of Natural Philosophy", in which he outlined the law of universal gravitation and three laws of mechanics, which became the basis classical mechanics... Developed differential and integral calculus, color theory and many other mathematical and physical theories.

Marin Mersenne (obsolete transliteration of Marin Mersenne; French Marin Mersenne; September 8, 1588 - September 1, 1648) - French mathematician, physicist, philosopher and theologian. During the first half of the 17th century, he was essentially the coordinator of the scientific life of Europe, conducting active correspondence with almost all prominent scientists of that time. He also has serious personal scientific achievements in the field of acoustics, mathematics and music theory.

Conclusion

We meet with numbers at every step and are so used to it that we hardly realize how important they are in our life. Numbers are part of human thinking.

After completing this work, I learned the axioms of natural numbers, great mathematicians, some secrets about numbers. There are ten digits in total, and the numbers that can be represented with their help are infinite.

Mathematics is unthinkable without numbers. Different ways of representing numbers help scientists create mathematical models, theories, explaining unsolved natural phenomena.

Bibliography

1. Kordema schoolchildren in mathematics: (Material for class. And extracurricular activities). - M .: Education, 1981 .-- 112 p.

2., Shor of arithmetic problems of increased difficulty. - M .: Education, 1968 .-- 238 p.

3. Perelman arithmetic. - M .: AO Century, 1994 .-- 164 p.

4. Malygin of historicism in teaching mathematics in secondary school. - M .: State educational and pedagogical publishing house of the Ministry of education of the RSFSR, 1963. - 223 p.

5., Shevkin. - M .: UC of pre-university education of Moscow State University, 1996. - 303 p.

6. Mathematical encyclopedic Dictionary... / Ch. ed. ; Ed. number:,. - M .: Sov. encyclopedia, 1988 .-- 847 p.

7. Savin's Dictionary of a Young Mathematician. - M .: Pedagogy, 1985 .-- 352s.

Polysemy

Polysemy, or polysemy of words, arises due to the fact that language is a system that is limited in comparison with the infinite variety of reality, so that, in the words of Academician Vinogradov, "Language is forced to distribute an infinite variety of meanings under one or another heading of basic concepts." (Vinogradov "Russian language" 1947). It is necessary to distinguish between different uses of words in one lexico-semantic version and the actual difference of the word. So, for example, the word (das) Ol can denote a number of different oils, except for cow (for which the word Butter exists). However, it does not follow from this that, denoting different oils, the word Ol will each time have a different meaning: in all cases its meaning will be the same, namely butter (everything except cow). As well as, for example, the meaning of the word Tisch table no matter what kind of table the word means in this particular case. The situation is different when the word Ol denotes oil. Here it is no longer the similarity of oil in terms of lubricity with various types of oil that is highlighted, but the special quality of oil - combustibility. And at the same time, the words denoting different kinds fuel: Kohl, Holz, etc. This gives us the opportunity to distinguish two meanings of the word Ol (or, in other words, two lexico-semantic variants): 1) oil (not animal) 2) oil.
Usually, new meanings arise by transferring one of the existing words to new item or phenomenon. This is how figurative meanings are formed. They are based on either the similarity of objects, or the connection of one object with another. Several types of name transfer are known. The most important of them are metaphor or metonymy.
In metaphor, the transfer is based on the similarity of things in color, shape, nature of movement, and so on. With all the metaphorical changes, some sign of the original concept remains

Homonymy

The ambiguity of the word is such a large and multifaceted problem that the most diverse problems of lexicology in one way or another turn out to be connected with it. In particular, the problem of homonymy comes into contact with this problem, with some of its sides.
Homonyms are words that sound the same, but different in meaning. Homonyms in a number of cases arise from polysemy, which has undergone a process of destruction. But homonyms can also arise as a result of random sound coincidences. The key with which the door is opened, and the key - a spring or a braid - a hairstyle and a braid - an agricultural tool - these words have different meanings and different origins, but coincidentally coincided in their sound.
Homonyms distinguish between lexical (refer to one part of speech, for example a key - to open a lock and a key - a spring), morphological (refer to different parts of speech, for example, three - a numeral, three - a verb in an imperative mood), lexico-grammatical, which are created as a result of conversion when a given word passes into another part of speech. for example in English. look- look and look-look. There are especially many lexical and grammatical homonyms in English language.
Homophones and homographs must be distinguished from homonyms. Homophones are called different words, which, differing in their attribution, coincide in pronunciation, for example: bow - meadow, Seite - page and Saite - string.
Homographs are such different words that coincide in spelling, although they are pronounced differently (both in terms of the sound composition and the place of stress in the word), for example, Castle is a castle.

Synonymy

Synonyms are similar in meaning, but different sounding words that express the shades of one concept.
There are three types of synonyms:
1. Conceptual, or ideographic. They differ from each other in lexical meaning. This difference manifests itself in varying degrees of the signified feature (frost - cold, strong, powerful, mighty), in the nature of its designation (quilted jacket - quilted jacket - quilted jacket), in both the expressed concept (banner - flag, daring - bold), in the degree of lexical coherence values (brown - brown, black - black).
2. Style synonyms, or functional. They differ from each other in the sphere of use, for example, eyes - eyes, face - face, forehead - brow. Synonyms emotionally - evaluative. These synonyms openly express the speaker's attitude to the designated person, object or phenomenon. For example, a child can be solemnly called a child, an affectionate little boy and a boy, a scornful boy and a milk sucker, and also emphatically - contemptuously a puppy, a sucker, a jerk.
3. Antonyms - combinations of words that are opposite in their lexical meaning, for example: top - bottom, white - black, speak - be silent, loudly - softly.

Antonymy

There are three types of antonyms:
1. Antonyms of gradual and coordinated opposition, for example, white - black, quiet - loud, close - distant, good - evil, and so on. These antonyms have something in common in their meaning, which allows their opposition. So the concepts black and white denote opposite color concepts.
2. Antonyms of the complementary and converting opposition: war is peace, husband is wife, married is single, you can - cannot, close - open.
3. Antonyms of dichotomous division of concepts. They are often the same root words: popular - anti-popular, legal - illegal, humane - inhuman.
Interest is, etc. intraword antonymy, when the meanings of words that have the same material shell are opposed. For example, in Russian the verb to lend money to someone means "to lend", and to borrow money from someone already means to borrow from someone. The intraword opposition of meanings is called enantiosemia.

6. Axiomatic construction of a system of natural numbers. An axiomatic method for constructing a mathematical theory. Requirements for the system of axioms: consistency, independence, completeness. Peano's axiomatics. The concept of a natural number from axiomatic positions. Models of the Peano axiom system. Addition and multiplication of natural numbers from axiomatic positions. The ordering of the set of natural numbers. Properties of the set of natural numbers. Subtraction and division of a set of natural numbers from axiomatic positions. The method of mathematical induction. Introducing zero and constructing a set of non-negative integers. Division theorem with remainder.

Basic concepts and definitions

Number - it is an expression of a certain amount.

Natural number an element of an unbounded sequence.

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

There are two approaches to defining natural numbers - numbers used when:

enumeration (numbering) of objects (first, second, third, ...);

designation of the number of items (no items, one item, two items, ...).

Axiom - these are the basic starting points (self-evident principles) of one theory or another, from which, by deduction, that is, by purely logical means, the rest of the content of this theory is extracted.

A number that has only two divisors (this number itself and one) is called - a prime number.

Composite Number is a number that has more than two divisors.

§2. Axiomatics of Natural Number

The axiomatic construction of a system of natural numbers is carried out according to the formulated rules.

Five axioms can be viewed as an axiomatic definition of basic concepts:

1 is a natural number;

The next natural number is a natural number;

1 does not follow any natural number;

If a natural number but follows natural number b and beyond the natural number from then b and from are identical;

Unit Is the first number in the natural series , and also one of the digits in the decimal notation system.

It is believed that the designation of the unit of any category by the same sign (rather close to the modern one) first appeared in Ancient Babylon in about 2 thousand years BC. e.

The ancient Greeks, who considered only natural numbers as numbers, considered each of them as a collection of units. The unit itself has a special place: it was not considered a number.

We apply the basic laws of arithmetic, often out of habit, without realizing it:

1) displacement law (commutativity), - the property of addition and multiplication of numbers, expressed by identities:

a + b = b + a a * b = b * a;

2) the combination law (associativity), - the property of addition and multiplication of numbers, expressed by identities:

(a + b) + c = a + (b + c) (a * b) * c = a * (b * c);

3) distribution law (distributivity), - a property that connects addition and multiplication of numbers and expressed by identities:

a * (b + c) = a * b + a * c (b + c) * a = b * a + c * a.

Axiomatic methods in mathematics

One of the main reasons for the development of mathematical logic is the widespread use of axiomatic method in the construction of various mathematical theories, first of all, geometry, and then arithmetic, group theory, etc. Axiomatic method can be defined as a theory that is built on a pre-selected system of undefined concepts and the relationship between them.

In the axiomatic construction of a mathematical theory, a certain system of undefined concepts and relations between them is preliminarily selected. These concepts and relationships are called basic. Further introduced axioms those. the main provisions of the theory under consideration, taken without proof. All further content of the theory is deduced logically from the axioms. For the first time, the axiomatic construction of a mathematical theory was undertaken by Euclid in the construction of geometry.

Axiomatic method in mathematics.

Basic concepts and relations of the axiomatic theory of the natural series. Determination of a natural number.

Addition of natural numbers.

Multiplication of natural numbers.

Properties of the set of natural numbers

Subtraction and division of natural numbers.

Axiomatic method in mathematics

In the axiomatic construction of any mathematical theory, certain rules:

1. Some concepts of the theory are chosen as major and are accepted without definition.

2. Formulated axioms, which in this theory are accepted without proof, they reveal the properties of basic concepts.

3. Each concept of the theory that is not included in the list of basic ones is given definition, it explains its meaning with the help of the main and preceding this concept.

4. Every proposition of a theory that is not included in the list of axioms must be proven. Such proposals are called theorems and prove them on the basis of the axioms and theorems that precede the considered one.

The axiom system should be:

a) consistent: we must be sure that, drawing all kinds of conclusions from a given system of axioms, we will never come to a contradiction;

b) independent: no axiom should be a consequence of other axioms of this system.

in) complete, if within its framework it is always possible to prove either the given statement or its denial.

The first experiment in the axiomatic construction of a theory can be considered the presentation of geometry by Euclid in his "Elements" (3rd century BC). A significant contribution to the development of the axiomatic method of constructing geometry and algebra was made by N.I. Lobachevsky and E. Galois. At the end of the 19th century. the Italian mathematician Peano developed a system of axioms for arithmetic.

Basic concepts and relations of the axiomatic theory of natural numbers. Determination of a natural number.

As a basic (undefined) concept in some set N selected attitude , and also used set-theoretic concepts, as well as the rules of logic.

The element immediately following the element but, denote but".

The "immediately follow" relationship satisfies the following axioms:

Peano's axioms:

Axiom 1... In the set N there is an element directly not next beyond any element of this set. Let's call it unit and denote by the symbol 1 .

Axiom 2... For each item but of N there is only one element but" immediately following but .

Axiom 3... For each item but of N there is at most one element immediately followed by but .

Axiom 4. Any subset M multitudes N coincides with N if it has the following properties: 1) 1 contained in M ; 2) from the fact that but contained in M , it follows that and but" contained in M.

Definition 1... Lots of N for the elements of which the relation "Directly follow»Satisfying Axioms 1-4 is called set of natural numbers and its elements are natural numbers.

This definition does not say anything about the nature of the elements of the set N . This means that it can be anything. Choosing as a set N some specific set, on which a specific relation "immediately follow" is set, satisfying axioms 1-4, we get model of this system axioms.

The standard model of the Peano axiom system is a series of numbers that arose in the process of the historical development of society: 1,2,3,4, ... The natural series begins with the number 1 (axiom 1); each natural number is immediately followed by a single natural number (axiom 2); each natural number immediately follows at most one natural number (axiom 3); starting from the number 1 and passing in order to the immediately following one after another natural numbers, we obtain the entire set of these numbers (axiom 4).

So, we began the axiomatic construction of a system of natural numbers with the choice of the basic directly follow relationship and the axioms that describe its properties. The further construction of the theory presupposes consideration of the known properties of natural numbers and operations on them. They must be disclosed in definitions and theorems, i.e. are deduced from a purely logical way from the relation “immediately follow”, and axioms 1-4.

The first concept that we will introduce after defining a natural number is attitude "Immediately precedes" , which is often used when considering the properties of the natural range.

Definition 2. If a natural number b directly follows natural number but, that number but called immediately preceding(or preceding) number b .

The relationship "precedes" possesses a number of properties.

Theorem 1. The unit has no preceding natural number.

Theorem 2. Every natural number but other than 1 has a single preceding number b, such that b "= but.

Addition of natural numbers

According to the rules for constructing an axiomatic theory, the definition of addition of natural numbers must be introduced using only the relation "Directly follow", and concepts "Natural number" and "Preceding number".

We preface the definition of addition with the following reasoning. If to any natural number but add 1, then we get the number but", immediately following but, i.e. but+ 1= a " and, therefore, we get the rule for adding 1 to any natural number. But how to add to the number but natural number b, other than 1? Let's use the following fact: if it is known that 2 + 3 = 5, then the sum 2 + 4 = 6, which immediately follows the number 5. This happens because in the sum 2 + 4 the second term is the number immediately following the number 3. So 2 + 4 = 2 + 3 " =(2+3)". Generally form we have, .