IV. Axiomatic construction of a system of natural numbers. Axiomatic theories Municipal educational institution

When axiomatically constructing any mathematical theory, certain rules are observed:

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

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

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

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

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

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

The system of axioms is called consistent, if two mutually exclusive sentences cannot be logically deduced from it.

If a system of axioms does not have 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.

When constructing the same theory axiomatically, different systems of axioms can be used. But they must be equivalent. In addition, when choosing a particular system of axioms, mathematicians take into account how simply and clearly proofs of the 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.

The axiomatic construction of a system of natural numbers is carried out according to the formulated rules. By studying this material, we must see how the entire arithmetic of natural numbers can be derived from basic concepts and axioms. Of course, its presentation in our course will not always be strict - we omit some proofs due to their great complexity, but we will discuss each such case.

Exercise

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

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

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

4. Define a rectangle, 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 the logical structure of the theorem is and what it means to prove the theorem.

Basic concepts and axioms. Definition of natural number

As the basic concept in the axiomatic construction of the arithmetic of natural numbers, the relation “directly follow” is taken, defined on a non-empty set N. The concept of a set, an element of a set and other set-theoretic concepts, as well as the rules of logic, are also considered well-known.

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

The essence of the “directly follow” attitude is revealed in the following axioms.

Axiom 1. In the set N there is an element that does not immediately follow any element of this set. We will call it unity and denote it by the symbol 1.

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

Axiom 3. For each element A There is at most one element in N that is immediately followed by A.

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

The formulated axioms are often called Peano axioms.

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

Definition. A bunch of N, for whose elements the relation “directly follow” is established, satisfying axioms 1-4, is called a set of natural numbers, and its elements- natural numbers.

This definition says nothing about the nature of the elements of the set N. So it can be anything. Choosing as

set N is some specific set on which a specific “directly follow” relation is specified, satisfying axioms 1-4, we get model of a given system of axioms. It has been proven in mathematics that a one-to-one correspondence can be established between all such models, preserving the “directly follow” relation, 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 a series of numbers that emerged in the process of historical development of society:

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 formation of this series, and this happens when the properties of the relation “directly follow” are revealed in the axioms. Thus, the natural series begins with the number 1 (axiom 1); every natural number is immediately followed by a single natural number (axiom 2); every natural number immediately follows at most one natural number (axiom 3); starting from the number 1 and moving in order to the natural numbers immediately following each other, we obtain the entire set of these numbers (axiom 4). Note that axiom 4 formally describes the infinity of the natural series, and the proof of statements about natural numbers is based on it.

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

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

there is a unique set that can be obtained from A by adding one circle, i.e., axiom 2 is satisfied. For each set A there is at most one set from which a set is formed A by adding one circle, i.e. Axiom 3 holds. If MÌ N and it is known that many A contained in M, it follows that a set in which there is one more circle than in the set A, also contained in M, That M = N(and therefore, axiom 4 is satisfied).

Note that in the definition of a natural number, none of the axioms can be omitted - for any of them it is possible to construct a set in which the other three axioms are satisfied, but this axiom is not satisfied. This position is clearly confirmed by the examples given in Figures 109 and 110. Figure 109a shows a set in which axioms 2 and 3 are satisfied, but axiom 1 is not satisfied (axiom 4 will not make sense, since there is no element in the set, directly not following any other). Figure 109b shows a set in which axioms 1, 3 and 4 are satisfied, but behind the element A two elements immediately follow, and not one, as required in axiom 2. Figure 109c shows a set in which axioms 1, 2, 4 are satisfied, but the element With immediately follows as element A, and behind the element b. Figure 110 shows a set in which axioms 1, 2, 3 are satisfied, but axiom 4 is not satisfied - a 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 talk about the “true” nature of the concepts being studied makes these theories too abstract and formal at first glance - it turns out that the same axioms are satisfied by different sets of objects and different relations between them. However, this apparent abstraction is the strength of the axiomatic method: every statement derived logically from these axioms is applicable to any sets 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 by choosing the basic relation “immediately follow” and the axioms that describe its properties. Further construction of the theory involves consideration of the known properties of natural numbers and operations on them. They must be disclosed in definitions and theorems, i.e. are derived purely logically from the relation “directly follow”, and axioms 1-4.

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

Definition. If a natural number b immediately follows a natural number a, then the number a is said to immediately precede (or precede) the number b.

The relation “precedes” has a number of properties. They are formulated as theorems and proven 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 A, different from 1, has a preceding number b, such that b ¢ = a.

Proof. Let us denote by M the set of natural numbers consisting of the number 1 and all numbers that have a predecessor. If the number A contained in M, that's the number A" also available in M, since it precedes for A" is the number A. This means that many M contains 1, and from the fact that the number A belongs to the set M, it follows that the number A" 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 schools. However, those properties of the relation “directly follow”, which are reflected in 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. The concepts “follows” and “precedes” 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 thing, that the natural series of numbers is infinite. And of course, knowledge of axiomatic theory will help the teacher methodically and competently organize children’s assimilation of the features of the natural series of numbers.

Exercises

1.Can axiom 3 be formulated as follows: “For each element A from N there is a single element that is immediately followed by a"?

2. Select the condition and conclusion in axiom 4, write them using the symbols О, =>.

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

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 “natural number” and “preceding number”.

Let us preface the definition of addition with the following considerations. If to any natural number A add 1, we get the number A", immediately following a, i.e. A + 1 = A", and, therefore, we get the rule for adding 1 to any natural number. But how to add to a number A natural number b, different from 1? Let's use the following fact: if we know 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 amount A+ b" can be found if the amount is known A+ b. These facts form the basis for the definition of addition of natural numbers in axiomatic theory. In addition, it uses the concept of algebraic operation.

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

1) ("A Î N ) a + 1=a",

2) (" A, b Î) a + b" = (a + b)".

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

As is known, the sum of any two natural numbers is also a natural number, and for any natural numbers A And b sum A+ b- the only one. In other words, the sum of natural numbers exists and is unique. The peculiarity of the definition is that it is not known in advance whether there is an algebraic operation that has the specified properties, and if it exists, is it unique? Therefore, when constructing the axiomatic theory of natural numbers, the following statements are proven:

Theorem 3. Addition of natural numbers exists and it 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 most 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, this does not mean that there is only one.) A uniqueness statement means that there cannot be two objects with given properties. Uniqueness is often proven by contradiction: one assumes that there are two objects that satisfy a given condition, and then builds a chain of deductive inferences that leads 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 will prove that the operation of addition with properties 1 and 2 exists.

Proof of the uniqueness of addition. Let us assume that in the set N There are two addition operations that have properties 1 and 2. We denote one of them by the + sign, and the other by the Å sign. For these operations we have:

1) a + 1 = A"; 1) AÅ =a"\

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

Let's prove that

("a, bÎ N )a + b=aÅ b. (1)

Let the number A chosen at random, and b M b, for which equality (1) is true.

It is easy to verify that 1 О M. Indeed, from the fact that A+ 1 = A"=AÅ 1 it follows that a + 1 =aÅ 1.

Let us now prove that if bÎ M, That b" О М, those. If a + b = aÅ b, That A+ b" = aÅ b". Because a + b - aÅ b, then according to axiom 2 (a + b)" = (aÅ b)", and then a + b" - (a + b)" = (aÅ b)" = aÅ b". Since many M contains 1 and together with each number b also contains a number b¢ then by axiom 4, the set M coincides with N, which means equality (1) b. Since the number A was chosen arbitrarily, then equality (1) is true for any natural A And b, those. operations + and Å on a 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 specified in the definition of addition exists.

Let M - the set of those and only those numbers A, for which it is possible to determine a + b so that conditions 1 and 2 are satisfied. Let us show that 1 О M. To do this, for any b let's put

1+b=b¢.(2)

1)1 + 1 = 1¢ - according to rule (2), i.e. equality holds a + 1 = A" at A= 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 A belongs M. Based on this assumption, we will show that A" contained in M, those. that addition can be defined A" and any number b so that conditions 1 and 2 are satisfied. To do this, we set:

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

Since by assumption the number a + b is defined, then by axiom 2, the number is also determined in a unique way (A+ b)". Let's check that conditions 1 and 2 are met:

1)a" + 1 = (a + 1)" = (A")". Thus, A"+ 1 = (a")".

2)a" + b" = (a+ b¢)"= ((a + b)")"= (a" + b)". Thus, a" + b" = = (a" + b)".

So, we showed that the set M contains 1 and together with each number A contains a number A". According to axiom 4, we conclude that the set M there are many natural numbers. Thus, there is a rule that allows for any natural numbers A And b uniquely find such a natural number a + b, that properties 1 and 2 formulated in the definition of addition are satisfied.

Let us show how from the definition of addition and Theorem 3 one can derive the well-known table for adding single-digit numbers.

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

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

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

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

Let us now consider cases involving the addition of the number 2 to any single-valued natural number.

1+2 = 1 + 1¢ - we used the accepted notation. 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 adding single-digit numbers.

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

Theorem 4.(" a,b,cО N )(a + b)+ With= A+ (b+ With).

Proof. Let the natural numbers A And b chosen randomly, and With 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.

Let us first prove that 1 О M, those. let's make sure the equality is fair (A+ b)+ 1 = A+ (b+ 1) Indeed, by the definition of addition, we have (a + b)+ 1 = (A+ b)"= A+ b"= A+ (b+ 1).

Let us now prove that if c О М, then c" О M, those. from equality (A+ b)+ c = a+ (b + c) equality follows (A+ b)+ With"= A+ (b + c"). (A+ b)+ With"= ((A + b)+ With)". Then, based on the equality (A+ b) + c= a + (b + c) can be written: ((A+ b)+ c)" = (a+ (b+ With))". 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 With contained in M, it follows that With" contained in M. Therefore, according to axiom 4, M= N, those. equality ( A + b)+ With= a + (b + c) true for any natural number With, and since the numbers A And b were chosen arbitrarily, then it is true for any natural numbers A And b, Q.E.D.

Theorem 5.(" a, bÎ N) a+ b= b+ A.

Proof. It consists of two parts: first they prove that (" aО N) A+1 = 1+a and then what(" a, bО N ) a + b=b+ A.

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

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

Let us now prove that if AÎ M, That A"Î M, i.e. from equality a + 1 = 1 + A equality follows a" + 1 = 1 + A". Really, a" + 1 = (a + 1) + 1 by the first property of addition. Next, the expression (a + 1) + 1 can be converted into the expression (1 + a) + 1, using the equality A+ 1 = 1 + A. Then, based on the associative law, we have: (1 + A)+ 1 = 1 + (A+ 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 A also contains a number A". Therefore, according to the axiom A, M = I, those. equality A+ 1 = 1 + A true for any natural A.

2 . Let us prove that (" a, bÎ N ) A+ b = b+ A. Let A - 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 equality a + b =b+ A true.

Since when b = 1 we get the equality A+ 1 = 1 + A, the truth of which is proven in paragraph 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 A+ b =b+ A equality follows A+ b"= b"+ A. Indeed, by the definition of addition, we have: A+ b"= (A+ b)". Because A+ b= b+ A, That (A+ b)" =(b+ A)". Hence, by definition of addition: (b+ A)"= b+ A"= b+ (a+ 1). Based on the fact that a + 1 = 1 + A, we get: b+ (a + 1) = b+ (1 + A). Using the associative property and the definition of addition, we perform the transformations: b + (1 + a) = (b+1) + a = b" + a.

So, we have proven that 1 is contained in the set M and along with each number b a bunch of M also contains a number b¢, immediately following b¢. By axiom 4 we get that M= AND, those. equality a+ b= b+ A true for any natural number b, as well as for any natural A, because his choice was arbitrary.

Theorem 6.("a,bÎ N) a + b¹ b.

Proof. Let A - a natural number chosen at random, 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 О M. Indeed, since A+ 1 = A"(by definition of addition), and 1 does not follow any number (axiom 1), then A+ 1 ¹ 1.

Let us now prove that if bÎ M, That 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 because a +bÎ b, That (a + b)"¹ b" and, therefore, a +b¢=b¢.

According to the axiom there are 4 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 the system of natural numbers, is the basis of initial mathematics education. Obtaining numbers by adding 1 is closely related to the principle of constructing the natural series, and the second property of addition is used in calculations, for example, in the following cases: 6 + 3 = (6+ 2)+ 1=8 + 1= 9.

All proven properties are studied in the initial 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. Transform an expression using the associative property of addition:

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

5. Prove that (" a, bÎ N) a + b¹ A.

6. Find out how mathematics is formulated in various elementary school textbooks:

a) the commutative property of addition;

b) associative property of addition.

7 .One of the elementary school textbooks discusses the rule for adding a number to a sum using a specific example (4 + 3) + 2 and suggests the following ways to find the result:

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 transformations performed. Is it possible to say 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:

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

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

Multiplication

According to the rules for constructing an axiomatic theory, the multiplication of natural numbers can be determined using the “directly follow” relation and the concepts introduced earlier.

Let us preface the definition of multiplication with the following considerations. If any natural number A multiply by 1, you get A, those. there is equality a× 1 = A and we get the rule for multiplying any natural number by 1. But how to multiply a number A to a natural number b, different from 1? Let's use the following fact: if we know 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 work a×b" can be found if the work is known: a×b" = a×b+ A.

The noted facts form the basis for the definition of multiplication of natural numbers. In addition, it uses the concept of 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"= а×b+ A.

Number а×b called work numbers A And b, and the numbers themselves A And b-multipliers.

The peculiarity 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 that has the indicated properties, and if it exists, then whether it is unique. In this regard, there is a need 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 a multiplication table for single-digit numbers. We do this 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 accepted 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 according to the result already obtained in the table, and finally the value of the expression 1+1 is found according to the addition table. Similarly:

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-digit numbers.

As is known, multiplication of natural numbers is commutative, associative and distributive with respect to addition. When building a theory axiomatically, it is convenient to prove these properties, starting with distributivity.

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

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

Proof. Let natural numbers a and b chosen randomly, and With 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×с.

Let us prove that 1 О M, those. that equality ( a + b)× 1 = A×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 WithÎ M, That With"Î M, those. which from the equality ( a + b)c = a×c+ b×с equality follows (A+ b)×c" = a×c"+ b×с". By definition of multiplication, we have: ( a + b)×c"= (a + b)×s+ (a + b). Because (a + b)×c=a×c + b×c, That ( a + b)×c+ (a+b)= (a×c + b×c) + (a+ b). Using the associative and commutative property of addition, we perform the transformations: ( a× With+ b×с)+ (A+ b) =(a× With + b×с+ A)+ b =(a×c + a + b×c)+ b= = ((a×c+ a) + b×с)+ b = (a×c+ a) + (b×с+ b). And finally, by the definition of multiplication, we get: (a×c+ a) + (b×с+ b) =a×c"+ b×с".

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

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

This is the property of left distributivity with respect to addition. It is proved in a similar way to how it was done for right distributivity.

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

This is the associative property of multiplication. His 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 axiomatic theory, is the basis for teaching multiplication in elementary school. Multiplication by 1 is generally defined, and the second property of multiplication is used in single-digit multiplication tables 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 meanings of the expressions:

a) 3×3; 6) 3x4; c) 4×3.

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

3. Prove the associative property of multiplication of natural numbers. What expression transformations are possible based on it? Is this property taught in elementary school?

4. Prove the commutative property of multiplication. Give examples of its use in an elementary mathematics 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 transformations performed.

7 . Determine the value of an expression without performing written calculations. Justify your 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 to compare the values of the expressions without performing calculations:

a) 70×32+ 9×32... 79×30 + 79×2;

b) 87×70 + 87×8 ... 80×78 +7×78?

Polysemy

Polysemy, or polysemy of words, arises due to the fact that language represents a system that is limited in comparison with the infinite variety of real reality, so that in the words of Academician Vinogradov, “Language is forced to distribute countless meanings under one or another rubric of basic concepts.” (Vinogradov “Russian language” 1947). It is necessary to distinguish between the different uses of words in one lexical-semantic variant and the actual difference of the word. So, for example, the word (das)Ol can designate a number of different oils, except cow's (for which there is a word Butter). However, it does not follow from this that, denoting different oils, the word Ol will have a different meaning each time: in all cases its meaning will be the same, namely oil (everything except cow's). Just like for example the meaning of the word Tisch table regardless of what type of table the word denotes in this particular case. The situation is different when the word Ol means oil. Here, what comes to the fore is no longer the similarity of oil in terms of oiliness with various types of oil, but the special quality of oil - flammability. And at the same time, words denoting various types of fuel will be associated with the word Ol: Kohl, Holz, etc. This gives us the opportunity to distinguish two meanings from the word Ol (or, in other words, two lexical-semantic options): 1) oil (not an animal) 2) oil.
Typically, new meanings arise by transferring one of the existing words to a new object 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, transfer is based on the similarity of things in color, shape, nature of movement, and so on. With all metaphorical changes, some sign of the original concept remains

Homonymy

The polysemy of a word is such a big and multifaceted problem that a wide variety of problems in lexicology are somehow related to it. In particular, the problem of homonymy comes into contact with this problem in some aspects.
Homonyms are words that sound the same but have different meanings. In some cases, homonyms arise from polysemy that has undergone a process of destruction. But homonyms can also arise as a result of random sound coincidences. The key that opens the door, and the key - a spring or a scythe - a hairstyle and a scythe - an agricultural tool - these words have different meanings and different origins, but coincidentally coincide in their sound.
Homonyms are distinguished by lexical (they relate to one part of speech, for example, a key is for opening a lock and a key is a spring. source), morphological (they relate to different parts of speech, for example, three is a numeral, three is a verb in the 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 the English language.
Homophones and homographs must be distinguished from homonyms. Homophones are different words that, although different in their spelling, are the same in pronunciation, for example: onion - meadow, Seite - page and Saite - string.
Homographs are such different words that have the same spelling, although they are pronounced differently (both in terms of the sound composition and the place of stress in the word), for example, Castle - castle.

Synonymy

Synonyms are words that are close in meaning, but sound differently, expressing shades of one concept.
There are three types of synonyms:
1. Conceptual or ideographic. They differ from each other in lexical meaning. This difference is manifested in the varying degrees of the designated attribute (frost - cold, strong, powerful, mighty), in the nature of its designation (padded jacket - quilted jacket - padded jacket), in the volume of the expressed concept (banner - flag, daring - bold), in the degree of coherence of the lexical meanings (brown - hazel, black - raven).
2. Synonyms are stylistic or functional. They differ from each other in the sphere of use, for example, eyes - eyes, face - face, forehead - forehead. Synonyms emotionally - evaluative. These synonyms openly express the speaker’s attitude towards the designated person, object or phenomenon. For example, a child can be solemnly called a child, affectionately a little boy and a little boy, contemptuously a boy and a sucker, and also intensified and contemptuously a puppy, a sucker, a brat.
3. Antonyms - combinations of words that are opposite in their lexical meaning, for example: top - bottom, white - black, talk - silent, loud - quiet.

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 them to be contrasted. So the concepts black and white denote opposite color concepts.
2. Antonyms of complementary and conversion opposites: war - peace, husband - wife, married - single, possible - impossible, closed - open.
3. Antonyms of the dichotomous division of concepts. They are often the same root words: folk - anti-national, legal - illegal, humane - inhumane.
Of interest is the so-called intraword antonymy, when the meanings of words that have the same material shell are contrasted. For example, in Russian the verb to lend someone money means “to lend,” and to borrow money from someone already means to borrow money from someone. Intraword opposition of meanings is called enantiosemy.

6. Axiomatic construction of a system of natural numbers. Axiomatic method of constructing a mathematical theory. Requirements for the axiom system: consistency, independence, completeness. Peano's axiomatics. The concept of a natural number from an axiomatic position. Models of the Peano axiom system. Addition and multiplication of natural numbers from axiomatic positions. Orderliness 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. Method of mathematical induction. Introduction of zero and construction of a set of non-negative integers. Theorem on division with remainder.

Basic concepts and definitions

Number - it is an expression of a certain quantity.

Natural number element of an indefinitely continuing sequence.

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

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

listing (numbering) items (first, second, third, ...);

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

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

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

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

§2. Axiomatics of natural numbers

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

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

The five axioms can be considered 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 A follows a natural number b and beyond the natural number With, That b And With are identical;

If any proposition is proven 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 natural number, then this sentence is true for all natural numbers.

Unit– this is the first number of the natural series , as well as one of the digits in the decimal number system.

It is believed that the designation of a unit of any category with the same sign (quite close to the modern one) appeared for the first time in Ancient Babylon approximately 2 thousand years BC. e.

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

I. Newton wrote: “... by number we understand not so much a collection of units as an abstract relation of one quantity to another quantity, conventionally accepted by us as a unit.” Thus, one 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 by changing the places of the terms.” You can write it 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) commutative law (commutativity), - the property of addition and multiplication of numbers, expressed by identities:

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

2) combinational 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) distributive law (distributivity), - a property that connects the addition and multiplication of numbers and is expressed by identities:

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

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

Currently, in our heads or on a piece of paper, we do only the simplest calculations, increasingly 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, but this operation must be done many millions of times.

Axiomatic methods in mathematics

One of the main reasons for the development of mathematical logic is the widespread 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 relationships 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. Next, enter axioms those. the main provisions of the theory under consideration, accepted without evidence. All further content of the theory is logically derived from the axioms. For the first time, the axiomatic construction of a mathematical theory was undertaken by Euclid in the construction of geometry.

GOUVPO

Tula State Pedagogical University

Named after L.N. Tolstoy

NUMERICAL SYSTEMS

Tula 2008

Numerical systems

The manual is intended for students of mathematical specialties of a pedagogical university and was developed in accordance with the state standard for the course “Numerical Systems”. Theoretical material is presented. Solutions to typical tasks are analyzed. Exercises for solving in practical classes are provided.

Compiled by -

Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Algebra and Geometry, TSPU named after. L. N. Tolstoy Yu. A. Ignatov

Reviewer -

Candidate of Physical and Mathematical Sciences, Professor of the Department of Mathematical Analysis, TSPU named after. L. N. Tolstoy I. V. Denisov

Educational edition

Numerical systems

Compiled by

IGNATOV Yuri Alexandrovich

NUMERICAL SYSTEMS

This course covers the foundations of mathematics. It provides a strict axiomatic construction of the basic numerical systems: natural, integer, rational, real, complex, as well as quaternions. It is based on the theory of formal axiomatic systems, discussed in the course of mathematical logic.

In each paragraph, the theorems are numbered first. If it is necessary to refer to a theorem from another paragraph, stepwise numbering is used: the paragraph number is placed before the theorem number. For example, Theorem 1.2.3 is Theorem 3 from paragraph 1.2.

Integers

Axiomatic theory of natural numbers

The axiomatic theory is defined by the following elements:

A set of constants;

A set of functional symbols to indicate operations;

A set of predicate symbols to represent relations;

A list of axioms connecting the above elements.

For a formal axiomatic theory, rules of inference are also indicated with the help of which theorems are proved. In this case, all statements are written in the form of formulas, the meaning of which does not matter, and transformations are made on these formulas according to given rules. In a substantive axiomatic theory, the rules of inference are not specified. Proofs are carried out on the basis of ordinary logical constructions that take into account the meaning of the statements being proved.

This course builds meaningful theories of basic numerical systems.

The most important requirement for an axiomatic theory is its consistency. The proof of consistency is carried out by constructing a model of a theory in another theory. Then the consistency of the theory under consideration is reduced to the consistency of the theory in which the model is constructed.

For a system of integers, the model is built within the framework of a system of natural numbers, for rational numbers - within a system of integers, etc. The result is a chain of axiomatic theories, in which each theory is based on the previous one. But for the first theory in this chain, namely the theory of natural numbers, there is nowhere to build a model. Therefore, for a system of natural numbers it is necessary to construct a theory for which the existence of a model is beyond doubt, although it is impossible to strictly prove it.

The theory should be very simple. For this purpose, we consider the system of natural numbers only as a tool for counting objects. The operations of addition, multiplication, and order relations must be determined after the theory in the indicated form has been constructed.

For the needs of counting, the system of natural numbers must be a sequence in which the first element (unit) is defined and for each element the next one is defined. In accordance with this, we obtain the following theory.

Constant: 1 (unit).

Function symbol: "¢". Denotes the unary "follow" operation, that is A¢ – the number following A. In this case the number A called previous For A¢.

There are no special predicate characters. The usual equality relation and set-theoretic relations are used. Axioms for them will not be indicated.

The set on which the theory is based is denoted N.

Axioms:

(N1) (" a) a¢ ¹ 1 (one does not follow any number).

(N2) (" a)("b) (a¢ = b¢ ® a = b) (each number has at most one predecessor).

(N3) M Í N, 1О M, ("a)(aÎ M ® a¢Î M) Þ M = N(axiom of mathematical induction).

The above axiomatics was proposed (with minor changes) by the Italian mathematician Peano at the end of the 19th century.

It is not difficult to derive some theorems from the axioms.

Theorem 1. (Method of mathematical induction). Let R(n) – a predicate defined on a set N. Let it be true R(1) and (" n)(P(n)® P(n¢)). Then R(n) is an identically true predicate on N.

Proof. Let M– set of natural numbers n, for which R(n) is true. Then 1О M according to the conditions of the theorem. Next, if nÎ M, That P(n) true by definition M, P(n¢) is true according to the conditions of the theorem, and n¢Î M a-priory M. All premises of the induction axiom are satisfied, therefore, M = N. According to definition M, it means that R(n) is true for all numbers from N. The theorem has been proven.

Theorem 2. Any number A No. 1 has a antecedent, and only one.

Proof. Let M– the set of natural numbers containing 1 and all numbers that have a predecessor. Then 1О M. If aÎ M, That a¢Î M, because a¢ has a antecedent (the condition is not even used here aÎ M). So, by the axiom of induction M = N. The theorem has been proven.

Theorem 3. Any number is different from the next one.

Exercise. Having determined the natural numbers 1¢ = 2, 2¢ = 3, 3¢ = 4, 4¢ = 5, 5¢ = 6, prove that 2 ¹ 6.

Addition of natural numbers

The following recursive definition is given for the addition of natural numbers.

Definition. Addition of natural numbers is a binary operation that applies to natural numbers A And b matches the number a+b, having the following properties:

(S1) A + 1 = A¢ for anyone A;

(S2) a+b¢ = ( a+b)¢ for any A And b.

It is required to prove that this definition is correct, that is, an operation that satisfies the given properties exists. This task seems very simple: it is enough to carry out induction on b, counting A fixed. In this case, it is necessary to select a set M values b, for which the operation a+b is defined and satisfies conditions (S1) and (S2). When performing an inductive transition, we must assume that for b operation is performed, and prove that it is performed for b¢. But in property (S2), which must be satisfied for b, there is already a link to a+b¢. This means that this property automatically assumes the existence of an operation for a+b¢, and therefore for subsequent numbers: after all, for a+b¢ property (S2) must also be satisfied. One might think that this only makes the problem easier by making the inductive step trivial: the statement being proved simply repeats the inductive hypothesis. But the difficulty here is in the proof for the base of induction. For value b= 1, properties (S1) and (S2) must also be satisfied. But property (S2), as shown, presupposes the existence of an operation for all values following 1. This means that checking the base of induction presupposes a proof not for one, but for all numbers, and induction loses its meaning: the base of induction coincides with the statement being proved.

The above reasoning does not mean that recursive definitions are incorrect or require careful justification each time. To justify them, you need to use the properties of natural numbers, which are only being established at this stage. Once these have been established, the validity of the recursive definitions can be proven. For now, let us prove the existence of addition by induction on A: in formulas (S1) and (S2) there is no connection between addition for A And A¢.

Theorem 1. Addition of natural numbers is always feasible, and uniquely so.

Proof. a) First we prove uniqueness. Let's fix it A. Then the result of the operation a+b there is a function from b. Suppose there are two such functions f(b) And g(b), having properties (S1) and (S2). Let's prove that they are equal.

Let M– set of meanings b, for which f(b) = g(b). By property (S1)
f(1) = A + 1 = A¢ and g(1) = A + 1 = A¢ means f(1) = g(1), and 1О M.

Let it now bÎ M, that is f(b) = g(b). By property (S2)

f(b¢) = a+b¢ = ( a+b)¢= f(b)¢, g(b¢) = a+b¢ = ( a+b)¢= g(b)¢ = f(b¢),

Means, b¢Î M. By the axiom of induction M = N. Uniqueness has been proven.

b) Now by induction on A let's prove the existence of the operation a+b. Let M– set of those values A, for which the operation a+b with properties (S1) and (S2) is defined for all b.

Let A= 1. Let us give an example of such an operation. By definition we assume 1 + b == b¢. Let us show that this operation satisfies properties (S1) and (S2). (S1) has the form 1 + 1 = 1¢, which corresponds to the definition. Checking (S2): 1 +b¢ =( b¢)¢ =
= (1+b)¢, and (S2) is satisfied. So, 1О M.

Let it now AÎ M. Let's prove that A¢Î M. We believe by definition
a¢ +b = (a+ b)¢. Then

a¢ + 1 = (a+ 1)¢ = ( A¢)¢,

a¢ +b¢ = ( a+ b¢)¢ = (( a+ b)¢)¢ = ( a¢ +b)¢,

and properties (S1) and (S2) are satisfied.

Thus, M = N, and addition is defined for all natural numbers. The theorem has been proven.

Theorem 2. The addition of natural numbers is associative, that is

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

Proof. Let's fix it A And b and carry out induction on With. Let M- a set of those numbers With, for which the equality is true. Based on properties (S1) and (S2), we have:

(a+b) + 1 = (a+b)¢ = ( a+b¢) = a +(b+ 1) Þ 1О M.

Let it now WithÎ M. Then

(a+b) + c¢ = (( a+b) + c)¢ = ( a +(b + c))¢ = a +(b + c)¢ = a +(b + c¢),

And c¢Î M. According to axiom (N3) M = N. The theorem has been proven.

Theorem 3. The addition of natural numbers is commutative, that is

a + b = b + a. (1)

Proof. Let's fix it A and carry out induction on b.

Let b= 1, that is, it is necessary to prove the equality

A + 1 = 1 + A. (2)

We prove this equality by induction on A.

At A= 1 equality is trivial. Let it be done for A, let's prove it for A¢. We have

A¢ + 1 = ( A + 1) + 1 = (1 + A) + 1 = (1 + A)¢ = 1 + A¢.

The inductive transition is complete. By the principle of mathematical induction, equality (2) is true for all A. This proves the statement of the base of induction on b.

Let now formula (1) be satisfied for b. Let's prove it for b¢. We have

a +b¢ = ( a +b)¢ = ( b + a)¢ = b + a¢ = b + (a + 1) = b + (1 + a) = (b + 1) + a = b¢ + a.

Using the principle of mathematical induction, the theorem is proven.

Theorem 4.a + b ¹ b.

The proof is an exercise.

Theorem 5. For any numbers A And b one and only one of the following cases occurs:

1) a = b.

2) There is a number k such that a = b + k.

3) There is a number l such that b = a + l.

Proof. It follows from Theorem 4 that at most one of these cases occurs, since, obviously, cases 1) and 2), as well as 1) and 3), cannot occur simultaneously. If cases 2) and 3) occurred simultaneously, then a = b + k=
= (A + l) + k = A+ (l + k), which again contradicts Theorem 4. Let us prove that at least one of these cases always occurs.

Let a number be chosen A And M – a lot of those b, for each of which, given a case 1), 2) or 3) occurs.

Let b= 1. If a= 1, then we have case 1). If A¹ 1, then by Theorem 1.1.2 we have

a = k" = k + 1 = 1 + k,

that is, we have case 2) for b= 1. So 1 belongs M.

Let b belongs M. Then the following cases are possible:

- A = b, Means, b" = b + 1 = A+ 1, that is, we have case 3) for b";

- A = b+k, and if k= 1, then A = b+ 1 = b", that is, case 1) occurs for b";

if k No. 1, then k = t" And

a = b + t" = b + (t + 1)= b + (1+m) = (b+ 1)+ m = b¢ +m,

that is, case 2) occurs for b";

- b = a + l, and b" =(a + l)¢ = A + l¢, that is, we have case 3) for b".

In all cases b" belongs M. The theorem has been proven.

Exercise. Prove, based on the definition of sum, that 1 + 1 = 2, 1 + 2 = 3, 2 + 2 = 4, 2 + 3 = 5, 2 + 4 = 3 + 3 = 6.

Multiplication of natural numbers

Definition. Multiplication of natural numbers is a binary operation that applies to natural numbers A And b matches the number ab(or a×b), having the following properties:

(P1) A×1 = A for anyone A;

(P2) ab" = ab + a for any A And b.

Regarding the definition of multiplication, all the comments that were made in the previous paragraph regarding the definition of addition remain valid. In particular, it is not yet clear from it that there is a correspondence with the properties given in the definition. Therefore, the following theorem, similar to Theorem 1.2.1, is of great fundamental importance.

Theorem 1. There is only one multiplication of natural numbers. In other words, multiplication is always doable and unambiguous.

The proof is quite similar to that of Theorem 1.2.1 and is offered as an exercise.

The properties of multiplication formulated in the following theorems are easily proven. The proof of each theorem is based on the previous ones.

Theorem 2.(Right distributive law): ( a+b)c = ac + bc.

Theorem 3. Multiplication is commutative: ab = ba.

Theorem 4.(Left distributive law): c(a+b)= сa + сb.

Theorem 5. Multiplication is associative: a(bc) = (ab)c.

Definition. A semiring is a system where + and × are binary operations of addition and multiplication that satisfy the axioms:

(1) is a commutative semigroup, that is, addition is commutative and associative;

(2) – semigroup, that is, multiplication is associative;

(3) right and left distributivity holds.

From an algebraic point of view, the system of natural numbers with respect to addition and multiplication forms a semiring.

Exercise. Prove based on the definition of a product that
2×2 = 4, 2×3 = 6.

Exercises

Prove the identities:

1. 1 2 + 2 2 + ... + n 2 = .

2. 1 3 + 2 3 + ... + n 3 = .

Find the amount:

3. .

4. .

5. .

6. 1x1! + 2x2! + ... + n×n!.

Prove the inequalities:

7. n 2 < 2n для n > 4.

8. 2n < n! For n³ 4.

9. (1 + x)n³ 1 + nx, Where x > –1.

10. at n > 1.

11. at n > 1.

12. .

13. Find the error in the proof by induction that all numbers are equal. We prove an equivalent statement: in any set of n numbers, all numbers are equal to each other. At n= 1 statement is true. Let it be true for n = k, let's prove it for n = k+ 1. Take a set of arbitrary
(k+ 1) numbers. Let's remove one number from it A. Left k numbers, by inductive hypothesis they are equal to each other. In particular, two numbers are equal b And With. Now let's remove the number from the set With and turn it on A. In the resulting set there is still k numbers, which means they are also equal to each other. In particular, a = b. Means, a = b = c, and that's all ( k+ 1) the numbers are equal to each other. The inductive transition is completed and the statement is proven.

14. Prove the enhanced principle of mathematical induction:

Let A(n) is a predicate on the set of natural numbers. Let A(1) true and from truth A(k) for all numbers k < m follows truth A(m). Then A(n) true for everyone n.

Ordered sets

Let us recall the basic definitions associated with the order relation.

Definition. Relation f (“above”) on a set M called order relation, or simply in order, if this relation is transitive and antisymmetric. System b M, fñ is called ordered set.

Definition. strict order, if it is anti-reflexive, and loose order, if reflexive.

Definition. A relation of order f is called a relation linear order, if it is connected, that is a ¹ bÞ a f bÚ b f a. An order that is not linear is called partial.

Definition. Let á M A– subset M. Element T sets A called the smallest, if it is less than all other elements of the set A, that is

("XÎ A)(X ¹ T® X f T).

Definition. Let á M, fñ – ordered set, A– subset M. Element T sets A called minimal, if in a set A there is no smaller element, that is (" XÎ A)(X ¹ T® Ø T f X).

The largest and maximum elements are determined similarly.

Exercises

1. Prove that the transitive and anti-reflexive relation is an order relation.

2. Prove that the divisibility relation M on the set N there is a partial order relation.

3. Prove that a set can have at most one largest and at most one smallest element.

4. Find all the minimum, maximum, greatest and smallest elements in the set (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) for the divisibility relation.

5. Prove that if a set has a smallest element, then it is the only minimal one.

6. In how many ways can we define linear order on a set of three elements? linear and strict? linear and lax?

7. Let á M, fñ is a linearly ordered set. Prove that the relation > defined by the condition

a > b Û a f b & a¹ b

is a relation of strict linear order.

8. Let á M, fñ is a linearly ordered set. Prove that the relation ³ defined by the condition

a ³ b Û a f b Ú a= b,

is a relation of non-strict linear order.

Definition. Linearly ordered set b M, fñ, in which every non-empty subset has the smallest element is called quite orderly. The relation f in this case is called the relation complete order.

According to Theorem 1.4.6, the system of natural numbers is a completely ordered set.

Definition. Let á M An interval separated by element a, called a set R a all elements below A and different from A, that is

R a = {x Î Mï a f x, x¹ a}.

In particular, if A is the minimum element, then R a = Æ.

Theorem 1.(Principle of transfinite induction). Let á M, fñ is a completely ordered set and A Í M. Let for each element A from M from belonging to A all elements of the interval R a follows that AÎ A. Then A = M.

Proof.

Let A" = M\A is the set-theoretic difference of sets M And A. If A"= Æ, then A = M, and the theorem is true. If A"¹ Æ , then, since M is a completely ordered set, then the set A" contains the smallest element T. In this case, all elements preceding T and different from T, don't belong A" and therefore belong A. Thus, Р m Í A. Therefore, by the conditions of the theorem T Î A, and therefore T Ï A", contrary to the assumption.

Let á A; fñ is an ordered set. We will assume that A– a finite set. With every element A sets A let's compare some point T (A) of a given plane so that if an element A immediately follows the element b, then point T (a) we will place above the point T(b) and connect them with a segment. As a result, we obtain a graph corresponding to this ordered set.

Exercises

9. Let á M, fñ is a completely ordered set, b Î M, sÎ M. Prove that or Pb = R s, or Pb Ì R s, or R s Ì Pb.

10. Let á M, f 1 с and b L, f 2 ñ are completely ordered sets such that
M Ç L=Æ . In abundance M È L Let us define a binary relation f by the following conditions:

1) if a, bÎ M, That, a f b Û a f 1 b;

2) if a, bÎ L, That, a f b Û a f 2 b;

3) if AÎ M,bÎ L, That, a f b.

Prove that system b MÈ L, fñ is a completely ordered set.

Ordered semigroups

Definition.Semigroup called algebra á A, *ñ, where * is an associative binary operation.

Definition. Semigroup á A, *ñ is called a semigroup with reduction if it satisfies the properties

a*c = b*c Þ a = b;c*a = c*b Þ a = b.

Definition.Ordered semigroup called system b A, +, fñ, where:

1) system b A, +ñ – semigroup;

2) system b A, fñ – ordered set;

3) the relation f is monotone with respect to the semigroup operation, that is
a f b Þ a+c f b + c, c + a f c+b.

Ordered semigroup á A, +, fñ are called ordered group, if system b A, +ñ – group.

In accordance with the types of order relations are determined linearly ordered semigroup, linearly ordered group, partially ordered semigroup, strictly ordered semigroup etc.

Theorem 1. In an ordered semigroup á A, +, fñ inequalities can be added, that is a f b, c f d Þ a+c f b+d.

Proof. We have

a f b Þ a+c f b + c, c f d Þ b+c f b + d,

from where by transitivity a+c f b+d. The theorem has been proven.

Exercise 1. Prove that the system of natural numbers is a partially ordered semigroup with respect to multiplication and divisibility.

It is easy to see that the system b N, +, >ñ – strictly ordered semigroup, b N, +, ³ñ is a non-strictly ordered semigroup. We can give an example of such ordering of the semigroup á N, +ñ, in which the order is neither strict nor non-strict.

Exercise 2. Let us define the order f in the system of natural numbers as follows: a f b Û a ³ b & a¹ 1. Prove that b N, +, fñ is an ordered semigroup in which the order is neither strict nor non-strict.

Example 1. Let A– a set of natural numbers not equal to one. Let us define the ratio f in A in the following way:

a f b Û ($ kÎ N)(a = b+k) & b No. 3.

Prove that system b A, +, fñ is a partially and strictly ordered semigroup.

Proof. Let's check transitivity:

a f b, b f c Þ a = b + k, b No. 3, b = c + l, c¹ 3 Þ a = c +(k+l), c¹ 3 Þ a f c.

Because a f b Þ a > b, then anti-reflexivity is satisfied. From Exercise 2.1.1 it follows that f is a relation of strict order. The order is partial because elements 3 and 4 are not in any relation.

The relation f is monotonic with respect to addition. Indeed, the condition a f b Þ a+c f b+c could only be violated when
b+c= 3. But the sum can be equal to 3, since it is possible A no unit.

A group of two elements cannot be linearly and strictly ordered. In fact, let 0 and 1 be its elements (0 is the zero of the group). Let's assume that 1 > 0. Then we get 0 = 1 + 1 > 0 + 1 = 1.

Theorem 2. Every linearly ordered cancellable semigroup can be linearly and strictly ordered.

Proof. Let á A, +, fñ is an ordered semigroup. The strict order relation > is defined as in Exercise 2.1.5: a > b Û a f b & a¹ b. Let us show that condition 3) from the definition of an ordered semigroup is satisfied.

a > b Þ a f b, a¹ bÞ a+c f b+c.

If a+c = b+c then, reducing, we get a = b, which contradicts the condition
A > b. Means, a+c ¹ b+c, And a+c > b+c. The second part of condition 3) is checked similarly, which proves the theorem.

Theorem 3. If b A, +, fñ is a linearly and strictly ordered semigroup, then:

1) A + With = b + c Û a = b Û c + a = With + b;

2) A + With f b + c Û A f b Û With + a f With + b.

Proof. Let A + With = b + c. If a ¹ b, then due to the connection A f b or
b f a. But then accordingly A + With f b+ c or b + With f a+ c, which contradicts the condition A + With = b + c. Other cases are dealt with similarly.

So, every linearly and strictly ordered semigroup is a cancellable semigroup.

Definition. Let á A, +, fñ is an ordered semigroup. Element A sets A called positive (negative) if a + a¹ A And a+a f A(respectively A f a + a).

Example 2. Prove that an element of an ordered commutative semigroup with cancellation greater than a positive element is not necessarily positive.

Solution. Let's use example 1. We have 2 + 2 f 2, which means 2 is a positive element. 3 = 2 + 1, which means 3 f 2. At the same time, the relation 3 + 3 f 3 does not hold, which means 3 is not a positive element.

Theorem 4. The sum of positive elements of a commutative semigroup with cancellation is positive.

Proof. If a + a f A And b+b f b, then by Theorem 1

a + a+ b+b f a + b Þ ( a + b)+ (a+b)f a + b.

It remains to check that ( a + b)+ (a+b)¹ a + b. We have:

b+b f b Þ a+b+b f a+b(1)

Let's pretend that ( a + b)+ (a+b)=a + b. Substituting into (1), we get

a+b+b f a+b+a+b Þ a f a+a.

Due to antisymmetry a = a + a. This contradicts the fact that the element A positive.

Theorem 5. If A is a positive element of a linearly and strictly ordered semigroup, then for any b we have a+b f b, b + a f b.

Proof. We have a+ a f A Þ a+ a+ b f a+ b. If it is not true that a+ b f b, then, due to linearity, it holds a+b=b or b f a+ b. Adding from the left A, we get accordingly a+ a+ b= a+ b or a+ b f a+ a+ b. These conditions contradict the antisymmetry and strictness of the order relation.

Theorem 6. Let á A, +, fñ – linearly and strictly ordered semigroup, AÎ A And A+ A¹ a. Then the elements:

A, 2*A, 3*A, ...

everyone is different. If in this case the system b A, +, fñ is a group, then all elements are different:

0, A,–A, 2*A, - 2*a, 3*a, –3*A, ...

(under k*a, kÎ N , aÎ A, means the amount a+ …+ a, containing k terms)

Proof. If a + A f A, That a + A + A f a + a, etc. As a result, we get a chain ... f ka f… f 4 A f3 A f2 A f A. Due to transitivity and antisymmetry, all elements in it are different. In a group, the chain can be continued in the other direction by adding an element - A.

Consequence. A finite semigroup with cancellation, if the number of its elements is at least 2, cannot be linearly ordered.

Theorem 7. Let á A, +, fñ is a linearly ordered group. Then

a f a Û b f b.

The proof is an exercise.

Thus, every linearly ordered group is either strictly or non-strictly ordered. To denote these orders we will use the signs > and ³, respectively.

Exercises

3. Prove that the sum of positive elements of a linearly and strictly ordered semigroup is positive.

4. Prove that every linearly and strictly ordered element of a semigroup greater than a positive element is itself positive.

5. Prove that an ordered semigroup is linearly ordered if and only if any finite set of its elements has only one greatest element.

6. Prove that the set of positive elements of a linearly ordered group is not empty.

7. Let á A, +, fñ is a linearly and strictly ordered group. Prove that the element A systems A if and only if is positive if A > 0.

8. Prove that there is only one linear and strict order in the additive semigroup of natural numbers, in which the set of positive elements is not empty.

9. Prove that the multiplicative semigroup of integers cannot be linearly ordered.

Ordered rings

Definition. System b A, +, ×, fñ is called ordered semiring, If

1) system b A, +, ×ñ – semiring;

2) system b A, +, fñ – ordered semigroup with non-empty set A+ positive elements;

3) monotonicity holds with respect to multiplication by positive elements, that is, if WithÎ A+ and A f b, That ac f bc, ca f cb.

Positive element ordered semiring A is any positive element of an ordered semigroup á A, +, fñ.

Ordered semiring b A, +, ×, fñ is called ordered ring (field), if the semiring b A, +, ×ñ – ring (respectively field).

Definition. Let á A, +, ×, fñ – ordered semiring. Order f of the system A called Archimedes, and the system A - Archimedean ordered, if, whatever the positive elements A And b systems A, you can specify such a natural number P, What na f b.

Example 1. A semiring of natural numbers with the relation > (greater than) is a linear, strictly and Archimedean ordered semiring.

For a linearly ordered ring b A, +, ×, 0, fñ system b A, +, 0, fñ is a linearly ordered group. This implies, according to Theorem 2.2.7, that the order of f is either strict or non-strict. In abundance A you can introduce (Exercises 2.1.5. and 2.1.6) a new linear order, which will be strict if the order f is non-strict, and non-strict if the order f is strict. In connection with this remark, in a linearly ordered ring A Usually two binary order relations are considered, one of which, strict, is denoted by the sign >, and the second, non-strict one, is marked with ³.

For what follows, it is useful to recall that in a linearly ordered ring the element A is positive if and only if A> 0 (Exercise 2.2.7).

Theorem 1. Let system b A,+,×,0,>ñ – linearly ordered ring. Then for any element A from A or A = 0, or A> 0, or – A > 0.

Proof. Due to linearity and strictness between elements
a+ a And A one and only one of the relations holds a+ a>a, a+ a = a, a+ a < a. In the first case A– positive element. In the second we add to both parts - A and we get A= 0. In the third case, we add to both sides – a – a – a and we get –a < -a-a, where –a– positive element.

Theorem 2. The sum and product of positive elements of a linearly ordered ring are positive.

The proof is an exercise.

Theorem 3. In a linearly ordered ring, the square of any nonzero element is positive.

The proof is an exercise.

Theorem 4. In a linearly ordered field if a> 0, then a –1 > 0.

The proof is an exercise.

Theorem 5. ( Order criterion) . Ring á A, +, ×, 0ñ if and only then can be linearly and strictly ordered (i.e., introduce a linear and strict order) if the set A has a subset A+ , satisfying the conditions:

1) AÎ A + Þ A¹ 0 & – AÏ A + ;

A¹ 0 Þ AÎ A + Ú – AÎ A + ;

2)a, bÎ A + Þ a+ bÎ A + & abÎ A + .

Proof. Let first á A,+,×,0,>ñ – linearly ordered ring. As the desired subset A+ in this case, by virtue of Theorems 1 and 2, many positive elements of the system can appear A.

Let it now A+ is a subset of the ring b A,+,×,0ñ, satisfying the conditions of the theorem. Let's try to introduce a linear order > in the ring á A,+,×,0ñ. Let's define this relationship as follows:

A > b Û a – b Î A + .

It is easy to check that the relation we introduced is connected, anti-reflexive, antisymmetric, transitive, and monotone with respect to addition and multiplication by any element from A + .

A bunch of A+ with the properties mentioned in the conditions of Theorem 4 are called positive part of the ring á A,+,×,0ñ. In the future, when introducing order in any ring, we will look for the “positive part” in it. If such a part exists in the ring, then the ring can be ordered; if not, then it cannot; if there are several such non-coinciding positive parts, then it can be ordered in several ways.

From the above it follows that when defining a linearly ordered ring, instead of the binary relation >, one can take the unary relation “positive part” as the main relation.

Theorem 6. ( Criterion for uniqueness of linear order) . Let A+ and A++ – positive parts of ring b A,+,×,0ñ. Then

A + = A ++ Û A + Í A ++ .

Department of Education of the Administration of the Kirov District of Volgograd

Municipal educational institution

gymnasium No. 9

Mathematics section

On this topic:Integers

6th grade students

Shanina Lisa

Supervisor:

Mathematic teacher

Date of writing:

Manager's signature:

Volgograd 2013

Introduction page 3

§1. Basic concepts and definitions p.4

§2. Axiomatics of natural numbers page 5

§3. “ABOUT SOME SECRETS THAT NUMBERS KEEP” p.8

§4. Great mathematicians page 10

Conclusion page 12

References page 13

Introduction

What are natural numbers? All! Oh how good. And who can explain? Hm, hm, “positive integers”, no, that won’t do. We'll have to explain what "integers" are, and that's more difficult. Are there any other versions? 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, introduce operations on them (addition, multiplication), we need some kind of formal definition. Otherwise, the operation of adding will remain the same informal, at the level of “there were two piles of apples, we put them into one.” And it will become impossible to prove theorems that use addition, which 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 we can rely on in proofs. For example, “through any two points on a plane you can draw a straight line and, moreover, only one.” Etc. I would like something like this.

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

Relevance and novelty of the 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 studying the question of natural numbers and the secrets of numbers.

The main hypothesis of the work is Peano's axioms and the secrets of numbers.

§1. Basic concepts and definitions

Number - it is an expression of a certain quantity.

Natural number element of an indefinitely continuing sequence.

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

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

listing (numbering) items (first, second, third, ...);

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

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

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

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

§2. Axiomatics of natural numbers

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

The five axioms can be considered 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 A follows a natural number b and beyond the natural number With, That b And With are identical;

Unit– this is the first number of the natural series , as well as one of the digits in the decimal number system.

It is believed that the designation of a unit of any category with the same sign (quite close to the modern one) appeared for the first time in Ancient Babylon approximately 2 thousand years BC. e.

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

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

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

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

2) combinational 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) distributive law (distributivity), - a property that connects the addition and multiplication of numbers and is expressed by identities:

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

§3. .“About SOME SECRETS THAT NUMBERS KEEP”

Mersenne numbers.

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

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

A composite number is a number that has more than two divisors. Here's an example: the French monk Marin Mersenne (1 year) wrote down the formula for numbers “for simplicity”, which were called the Mersenne number.

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

I checked whether this formula is valid for all prime numbers

To date, numbers greater than 2 have been tested for primeness for all p up to 50000.E.” As a result, more than 30 Mersenne prime numbers have been discovered.

3.1 Perfect numbers.

Among the composite numbers, there is a group of numbers that are called ■ perfect if the number is 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 became acquainted with eastern mathematics. Pythagoras believed that the secret of the world is hidden in numerical patterns; numbers have their own special life meaning. Among 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 used a calculator to find a couple more friendly numbers.

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 and. etc.

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

Friendly numbers

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

§4. Great mathematicians

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

After Grassmann was educated in Stetin, he entered the University of Berlin, the Faculty of Theology. Having successfully passed both exams in theology, for a long time he did not give up the thought of devoting himself to the work of a preacher, and retained his desire 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 formal logical justification of mathematics. His axiomatics of the natural series of numbers has come into general use. Known his example of a continuous (Jordan) curve that completely fills a certain square.

Sir Isaac Newton (eng. Sir Isaac Newton, December 25, 1642 - March 20, 1727 according to the Julian calendar, which was 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 creators of classical physics . The author of the fundamental work “Mathematical Principles of Natural Philosophy,” in which he outlined the law of universal gravitation and the three laws of mechanics, which became the basis of classical mechanics. He developed differential and integral calculus, color theory and many other mathematical and physical theories.

Maren Mersenne (outdated transliteration 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 encounter numbers at every step and have become so accustomed to it that we hardly realize how important they are in our lives. Numbers are part of human thinking.

Having completed this work, I learned the axioms of natural numbers, great mathematicians, and 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 and theories that explain unsolved natural phenomena.

Bibliography

1. Kordemsky mathematics for schoolchildren: (Material for classroom 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.: JSC Stoletie, 1994. – 164 p.

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

5. , Shevkin. – M.: UC Pre-University Education 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. – 352 p.

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