Theorems on the largest and smallest integers. Mathematics with a minor

For the state exam in the specialty

1. Linear (vector) space over the field. Examples. Subspaces, simplest properties. Linear dependence and independence of vectors.

2. Basis and dimension of vector space. Coordinate matrix of a vector system. Transition from one basis to another. Isomorphism of vector spaces.

3. Algebraic closedness of the field of complex numbers.

4. Ring of integers. Ordering of integers. Theorems on the “largest” and “smallest” integers.

5. Group, examples of groups. The simplest properties of groups. Subgroups. Homomorphism and isomorphism of groups.

6. Basic properties of divisibility of integers. Prime numbers. The infinity of the set of prime numbers. Canonical decomposition of a composite number and its uniqueness.

7. Kronecker-Capelli theorem (consistency criterion for a system of linear equations).

8. Basic properties of comparisons. Complete and reduced systems of modulo deductions. Modulo residue class ring. Euler's and Fermat's theorems.

9. Application of the theory of comparisons to the derivation of divisibility criteria. Converting a fraction to a decimal and determining the length of its period.

10. Conjugacy of imaginary roots of a polynomial with real coefficients. Irreducible polynomials over the field of real numbers.

11. Linear comparisons with one variable (solvability criterion, solution methods).

12. Equivalent systems of linear equations. Method of sequential elimination of unknowns.

13. Ring. Examples of rings. The simplest properties of rings. Sub-ring. Homomorphisms and isomorphisms of rings. Field. Examples of fields. The simplest properties. Minimality of the field of rational numbers.

14. Natural numbers (basics of the axiomatic theory of natural numbers). Theorems on the “largest” and “smallest” natural numbers.

15. Polynomials over a field. Theorem on division with remainder. The greatest common divisor of two polynomials, its properties and methods of finding.

16. Binary relations. Equivalence relation. Equivalence classes, factor set.

17. Mathematical induction for natural and integer numbers.

18. Properties of relatively prime numbers. The least common multiple of integers, its properties and methods of finding.

19. Field of complex numbers, number fields. Geometric representation and trigonometric form of a complex number.

20. Theorem on division with remainder for integers. The greatest common divisor of integers, its properties and methods of finding.

21. Linear operators of vector space. Kernel and image of a linear operator. Algebra of linear operators in vector space. Eigenvalues ​​and eigenvectors of a linear operator.

22. Affine transformations of the plane, their properties and methods of specifying. Group of affine transformations of the plane and its subgroups.

23. Polygons. Area of ​​a polygon. Existence and uniqueness theorem.

24. Equal size and equal composition of polygons.

25. Geometry of Lobachevsky. Consistency of the system of axioms of Lobachevsky geometry.

26. The concept of parallelism in Lobachevsky geometry. The relative position of lines on the Lobachevsky plane.

27. Movement formulas. Classification of plane movements. Applications to problem solving.

28. The relative position of two planes, a straight line and a plane, two straight lines in space (in analytical presentation).

29. Projective transformations. Existence and uniqueness theorem. Formulas for projective transformations.

30. Scalar, vector and mixed products of vectors, their application to problem solving.

31. The Weyl axiom system of three-dimensional Euclidean space and its content consistency.

32. Movements of the plane and their properties. Group of plane movements. Theorem of existence and uniqueness of motion.

33. Projective plane and its models. Projective transformations, their properties. Group of projective transformations.

34. Plane similarity transformations, their properties. Group of plane similarity transformations and its subgroups.

35. Smooth surfaces. The first quadratic form of a surface and its applications.

36. Parallel design and its properties. Image of flat and spatial figures in parallel projection.

37. Smooth lines. Curvature of a spatial curve and its calculation.

38. Ellipse, hyperbola and parabola as conic sections. Canonical equations.

39. Directorial property of ellipse, hyperbola and parabola. Polar equations.

40. Double ratio of four points on a line, its properties and calculation. Harmonic separation of pairs of points. Complete quadrilateral and its properties. Application to solving construction problems.

41. Theorems of Pascal and Brianchon. Poles and polars.

Sample questions on mathematical analysis

As you know, the set of natural numbers can be ordered using the “less than” relation. But the rules for constructing an axiomatic theory require that this relation be not only defined, but also done on the basis of concepts already defined in this theory. This can be done by defining the “less than” relation through addition.

Definition. The number a is less than the number b (a< b) тогда и только тогда, когда существует такое натуральное число с, что а + с = b.

Under these conditions it is also said that the number b more A and write b > a.

Theorem 12. For any natural numbers A And b one and only one of three relations holds: a = b, a > b, A < b.

We omit the proof of this theorem.. From this theorem it follows that if

a¹ b, either A< b, or a > b, those. the relation “less” has the property of connectedness.

Theorem 13. If A< b And b< с. That A< с.

Proof. This theorem expresses the transitivity property of the “less than” relation.

Because A< b And b< с. then, by the definition of the “less than” relation, there are natural numbers To So what b = a + k and c = b + I. But then c = (a + k)+ / and based on the associativity property of addition we obtain: c = a + (k +/). Because the k + I - natural number, then, according to the definition of “less than”, A< с.

Theorem 14. If A< b, it is not true that b< а. Proof. This theorem expresses the property antisymmetry"less" relationship.

Let us first prove that for not a single natural number A not you-!>! ■ )her attitude A< A. Let's assume the opposite, i.e. What A< а occurs. Then, by the definition of the “less than” relation, there is a natural number With, What A+ With= A, and this contradicts Theorem 6.

Let us now prove that if A< b, then it is not true that b < A. Let's assume the opposite, i.e. what if A< b , That b< а performed. But from these equalities, by Theorem 12 we have A< а, which is impossible.

Since the “less than” relation we defined is antisymmetric and transitive and has the property of connectedness, it is a relation of linear order, and the set of natural numbers linearly ordered set.

From the definition of “less than” and its properties, we can deduce the known properties of the set of natural numbers.

Theorem 15. Of all the natural numbers, one is the smallest number, i.e. I< а для любого натурального числа a¹1.

Proof. Let A - any natural number. Then two cases are possible: a = 1 and a¹ 1. If a = 1, then there is a natural number b, followed by a: a = b " = b + I = 1 + b, i.e., by definition of the “less than” relation, 1< A. Therefore, any natural number is equal to 1 or greater than 1. Or, one is the smallest natural number.

The relation “less than” is associated with the addition and multiplication of numbers by the properties of monotonicity.

Theorem 16.

a = b => a + c = b + c and a c = b c;

A< b =>a + c< b + с и ас < bс;

a > b => a + c > b + c and ac > bc.

Proof. 1) The validity of this statement follows from the uniqueness of addition and multiplication.

2) If A< b, then there is such a natural number k, What A + k = b.
Then b+ c = (a + k) + c = a + (k + c) = a + (c+ To)= (a + c) + k. Equality b+ c = (a + c) + k means that a + c< b + With.

In the same way it is proved that A< b =>ac< bс.

3) The proof is similar.

Theorem 17(the converse of Theorem 16).

1) A+ c = b + c or ac ~ bc-Þ a = b

2) a + c< Ь + с or ac< bcÞ A< Ь:

3) a + c > b+ with or ac > bcÞ a > b.

Proof. Let us prove, for example, that from ac< bс should A< b Let's assume the opposite, i.e. that the conclusion of the theorem does not hold. Then it can't be that a = b. since then the equality would be satisfied ac = bс(Theorem 16); it can't be A> b, because then it would be ac > bc(Theorem!6). Therefore, according to Theorem 12, A< b.

From Theorems 16 and 17 we can derive the well-known rules for term-by-term addition and multiplication of inequalities. We leave them out.

Theorem 18. For any natural numbers A And b; there is a natural number n such that p b> a.

Proof. For anyone A there is such a number P, What n > a. To do this it is enough to take n = a + 1. Multiplying inequalities term by term P> A And b> 1, we get pb > A.

From the considered properties of the “less than” relation, important features of the set of natural numbers follow, which we present without proof.

1. Not for any natural number A there is no such natural number P, What A< п < а + 1. This property is called property
discreteness sets of natural numbers, and numbers A And a + 1 is called neighboring.

2. Any non-empty subset of natural numbers contains
smallest number.

3. If M- non-empty subset of the set of natural numbers
and there is such a number b, that for all numbers x from M not executed
equality x< b, then in abundance M is the largest number.

Let's illustrate properties 2 and 3 with an example. Let M- a set of two-digit numbers. Because M is a subset of natural numbers and for all numbers in this set the inequality x< 100, то в множестве M is the greatest number 99. The smallest number contained in a given set M, - number 10.

Thus, the “less than” relation made it possible to consider (and in some cases prove) a significant number of properties of the set of natural numbers. In particular, it is linearly ordered, discrete, and has the smallest number 1.

Primary schoolchildren become familiar with the “less than” (“greater than”) relation for natural numbers at the very beginning of their education. And often, along with its set-theoretic interpretation, the definition given by us within the framework of axiomatic theory is implicitly used. For example, students can explain that 9 > 7 because 9 is 7+2. The implicit use of the monotonicity properties of addition and multiplication is also common. For example, children explain that “6 + 2< 6 + 3, так как 2 < 3».

Exercises

1, Why can’t the set of natural numbers be ordered using the “immediately follow” relation?

Define attitude a > b and prove that it is transitive and antisymmetric.

3. Prove that if a, b, c are natural numbers, then:

A) A< b Þ ас < bс;

b) A+ With< b + сÞ> A< Ь.

4. What theorems on the monotonicity of addition and multiplication can
use by younger schoolchildren when completing the task “Compare without performing calculations”:

a) 27 + 8 ... 27 + 18;

b) 27-8 ... 27 -18.

5. What properties of the set of natural numbers are implicitly used by primary schoolchildren when performing the following tasks:

A) Write down the numbers that are greater than 65 and less than 75.

B) Name the previous and subsequent numbers in relation to the number 300 (800,609,999).

C) Name the smallest and largest three-digit number.

Subtraction

In the axiomatic construction of the theory of natural numbers, subtraction is usually defined as the inverse operation of addition.

Definition. Subtraction of natural numbers a and b is an operation that satisfies the condition: a - b = c if and only if b + c = a.

Number a - b is called the difference between the numbers a and b, number A– minuendable, number b- deductible.

Theorem 19. Difference of natural numbers A- b exists if and only if b< а.

Proof. Let the difference A- b exists. Then, by the definition of difference, there is a natural number With, What b + c = a, which means that b< а.

If b< а, then, by the definition of the “less than” relation, there is a natural number c such that b + c = a. Then, by definition of the difference, c = a - b, those. difference a - b exists.

Theorem 20. If the difference of natural numbers A And b exists, then it is unique.

Proof. Suppose there are two different values ​​of the difference between numbers A And b;: a – b= s₁ And a - b= s₂, and с₁ ¹ с₂ . Then, by definition of the difference, we have: a = b + c₁, And a = b + c₂ : . It follows that b+ c ₁ = b + c₂ : and based on Theorem 17 we conclude, с₁ = с₂.. We came to a contradiction with the assumption, which means that it is false, but this theorem is correct.

Based on the definition of the difference of natural numbers and the conditions for its existence, it is possible to justify the well-known rules for subtracting a number from a sum and a sum from a number.

Theorem 21. Let A. b And With- integers.

and if a > c, then (a + b) - c = (a - c) + b.

b) If b > c. then (a + b) - c - a + (b - c).

c) If a > c and b > c. then you can use any of these formulas.
Proof. In case a) the difference of numbers A And c exists because a > s. Let us denote it by x: a - c = x. where a = c + x. If (A+ b) - c = y. then, by definition of the difference, A+ b = With+ at. Let us substitute into this equality instead A expression c + x:(c + x) + b = c + y. Let's use the associativity property of addition: c + (x + b) = c+ at. Let us transform this equality based on the property of monotonicity of addition and obtain:

x + b = u..Replacing x in this equality with the expression a - c, will have (A - G) + b = y. Thus, we have proven that if a > c, then (a + b) - c = (a - c) + b

The proof is carried out similarly in case b).

The proven theorem can be formulated in the form of a rule that is convenient for remembering: in order to subtract a number from a sum, it is enough to subtract this number from one term of the sum and add another term to the resulting result.

Theorem 22. Let a, b and c - integers. If a > b+ s, then A- (b + c) = (a - b) - c or a - (b + c) = (a - c) - b.

The proof of this theory is similar to the proof of Theorem 21.

Theorem 22 can be formulated as a rule: in order to subtract the sum of numbers from a number, it is enough to subtract from this number each term one by one.

In primary mathematics teaching, the definition of subtraction as the inverse of addition is, as a rule, not given in general form, but it is constantly used, starting with performing operations on single-digit numbers. Students should clearly understand that subtraction is related to addition and use this relationship in calculations. Subtracting, for example, the number 16 from the number 40, students reason like this: “Subtracting the number 16 from 40 means finding a number such that when added to the number 16, the result is 40; this number will be 24, since 24 + 16 = 40. So. 40 - 16 = 24."

The rules for subtracting a number from a sum and a sum from a number in an initial mathematics course are the theoretical basis for various calculation techniques. For example, the value of the expression (40 + 16) - 10 can be found not only by calculating the sum in parentheses and then subtracting the number 10 from it, but also in this way;

a) (40 + 16) - 10 = (40 - 10) + 16 = 30 + 16 = 46:

b) (40 + 16) - 10 = 40 + (16- 10) = 40 + 6 = 46.

Exercises

1. Is it true that each natural number is obtained from the immediate next one by subtracting one?

2. What is special about the logical structure of Theorem 19? Can it be formulated using the words “necessary and sufficient”?

3. Prove that:

and if b > c, That (a + b) - c = a + (b - c);

b) if a > b + c, That a - (b+ c) = (a - b) - c.

4. Is it possible, without performing calculations, to say which expressions will have equal values:

a) (50 + 16)- 14; d) 50 + (16 -14 ),

b) (50 - 14) + 16; e) 50 - (16 - 14);
c) (50 - 14) - 16, f) (50 + 14) - 16.

a) 50 - (16 + 14); d) (50 - 14) + 16;

b) (50 - 16) + 14; e) (50 - 14) - 16;

c) (50 - 16) - 14; e) 50 - 16-14.

5. What properties of subtraction are the theoretical basis for the following calculation techniques studied in the initial mathematics course:

12 - 2-3 12 -5 = 7

b) 16-7 = 16-6 - P;

c) 48 - 30 = (40 + 8) - 30 = 40 + 8 =18;

d) 48 - 3 = (40 + 8) - 3 = 40 + 5 = 45.

6. Describe possible ways to evaluate the value of an expression of the form. a - b- With and illustrate them with specific examples.

7. Prove that when b< а and any natural c the equality is true (a – b) c = ac - bc.

Note. The proof is based on axiom 4.

8. Determine the value of an expression without performing written calculations. Justify your answers.

a) 7865 × 6 – 7865 ×5: b) 957 × 11 – 957; c) 12 × 36 – 7 × 36.

Division

In the axiomatic construction of the theory of natural numbers, division is usually defined as the inverse operation of multiplication.

Definition. The division of natural numbers a and b is an operation that satisfies the condition: a: b = c if and only if To when b× c = a.

Number a:b called private numbers A And b, number A divisible, number b- divisor.

As is known, division on the set of natural numbers does not always exist, and there is no such convenient sign of the existence of a quotient as exists for a difference. There is only a necessary condition for the existence of the particular.

Theorem 23. In order for there to be a quotient of two natural numbers A And b, it is necessary that b< а.

Proof. Let the quotient of natural numbers A And b exists, i.e. there is a natural number c such that bc = a. Since for any natural number 1 the inequality 1 £ With, then, multiplying both its parts by a natural number b, we get b£ bc. But bc = a, hence, b£ A.

Theorem 24. If the quotient of natural numbers A And b exists, then it is unique.

The proof of this theorem is similar to the proof of the theorem on the uniqueness of the difference of natural numbers.

Based on the definition of the quotient of natural numbers and the conditions for its existence, it is possible to justify the well-known rules for dividing a sum (difference, product) by a number.

Theorem 25. If the numbers A And b divisible by a number With, then their sum a + b divided by c, and the quotient obtained by dividing the sum A+ b per number With, equal to the sum of the quotients obtained by dividing A on With And b on With, i.e. (a + b):c = a:c + b:With.

Proof. Since the number A divided by With, then there is a natural number x = A; s that a = cx. Similarly, there is such a natural number y = b:With, What

b= su. But then a + b = cx+ cy = - c(x + y). It means that a + b is divided by c, and the quotient obtained by dividing the sum A+ b by the number c, equal to x + y, those. ax + b: c.

The proven theorem can be formulated as a rule for dividing a sum by a number: in order to divide the sum by a number, it is enough to divide each term by this number and add the resulting results.

Theorem 26. If natural numbers A And b divisible by a number With And a > b, then the difference a - b is divided by c, and the quotient obtained by dividing the difference by the number c is equal to the difference of the quotients obtained by dividing A on With And b on c, i.e. (a - b):c = a:c - b:c.

The proof of this theorem is similar to the proof of the previous theorem.

This theorem can be formulated as a rule for dividing the difference by a number: For In order to divide the difference by a number, it is enough to divide the minuend and the subtrahend by this number and subtract the second from the first quotient.

Theorem 27. If a natural number A is divisible by a natural number c, then for any natural number b work ab divided by s. In this case, the quotient obtained by dividing the product ab to number s , equal to the product of the quotient obtained by dividing A on With, andnumbers b: (a × b):c - (a:c) × b.

Proof. Because A divided by With, then there is a natural number x such that a:c= x, where a = cx. Multiplying both sides of the equality by b, we get ab = (cx)b. Since multiplication is associative, then (cx) b = c(x b). From here (a b):c = x b= (a:c) b. The theorem can be formulated as a rule for dividing a product by a number: in order to divide a product by a number, it is enough to divide one of the factors by this number and multiply the resulting result by the second factor.

In elementary mathematics education, the definition of division as the inverse operation of multiplication is, as a rule, not given in general form, but it is constantly used, starting from the first lessons of familiarization with division. Students should clearly understand that division is related to multiplication and use this relationship when doing calculations. When dividing, for example, 48 by 16, students reason like this: “Divide 48 by 16 means finding a number that, when multiplied by 16, results in 48; such a number would be 3, since 16×3 = 48. Therefore, 48: 16 = 3.

Exercises

1. Prove that:

a) if the quotient of natural numbers a and b exists, then it is unique;

b) if the numbers a and b are divided into With And a > b, That (a - b): c = a: c - b: c.
2. Is it possible to say that all these equalities are true:
a) 48:(2×4) = 48:2:4; b) 56:(2×7) = 56:7:2;

c) 850:170 =850:10:17.

What rule generalizes these cases? Formulate it and prove it.

3. What properties of division are the theoretical basis for
completing the following tasks offered to primary school students:

Is it possible, without performing division, to say which expressions will have the same meaning:

a) (40+ 8):2; c) 48:3; e) (20+ 28):2;

b) (30 + 16):3; g)(21+27):3; f) 48:2;

Are the equalities true:

a) 48:6:2 = 48:(6:2); b) 96:4:2 = 96:(4-2);

c) (40 - 28): 4 = 10-7?

4. Describe possible ways to calculate the value of an expression
type:

A) (A+ b):c; b) A:b: With; V) ( a × b): With .

Illustrate the proposed methods with specific examples.

5. Find the meaning of the expression in a rational way; their
justify your actions:

a) (7 × 63):7; c) (15 × 18):(5× 6);

b) (3 × 4× 5): 15; d) (12 × 21): 14.

6. Justify the following methods of dividing by a two-digit number:

a) 954:18 = (900 + 54): 18 = 900:18 + 54:18 =50 + 3 = 53;

b) 882:18 = (900 - 18): 18 = 900:18 - 18:18 = 50 - 1 =49;

c) 480:32 = 480: (8 × 4) = 480:8:4 = 60:4 = 15:

d) (560 × 32): 16 = 560(32:16) = 560×2 = 1120.

7. Without dividing with a corner, find the most rational
in a quotient way; Justify the chosen method:

a) 495:15; c) 455:7; e) 275:55;

6) 425:85; d) 225:9; e) 455:65.

Lecture 34. Properties of the set of non-negative integers

1. The set of non-negative integers. Properties of the set of non-negative integers.

2. The concept of a segment of a natural series of numbers and counting elements of a finite set. Ordinal and cardinal natural numbers.

Theorems on the “largest” and “smallest” integers

Theorem 4 (about the “smallest” integer). Every non-empty set of integers bounded from below contains the smallest number. (Here, as in the case of natural numbers, the word “set” is used instead of the word “subset” E

Proof. Let O A C Z and A be bounded below, i.e. 36? ZVa? A(b< а). Тогда если Ь Е А, то Ь- наименьшее число во множестве А.

Let now b A.

Then Ua e Af< а) и, значит, Уа А(а - Ь >ABOUT).

Let us form a set M of all numbers of the form a - b, where a runs through the set A, i.e. M = (c [ c = a - b, a E A)

Obviously, the set M is not empty, since A 74 0

As noted above, M C N . Consequently, by the theorem of natural numbers (54, Ch.III) in the set M there is the smallest natural number m. Then m = a1 - b for some number a1? A, and since m is the smallest in M, then Ua? A(t< а - Ь) , т.е. А (01 - Ь < а - Ь). Отсюда Уа е А(а1 а), а так как ат (- А, то - наименьшее число в А. Теорема доказана.

Theorem 5 (about the “largest” integer). Every non-empty, limited set of integers contains the greatest number.

Proof. Let O 74 A C Z and A be limited from above by the number b, i.e. ? ZVa e A(a< Ь). Тогда -а >b for all numbers a? A.

Consequently, the set M (with r = -a, a? A) is not empty and is bounded below by the number (-6). Hence, according to the previous theorem, the smallest number occurs in the set M, i.e. ace? MUs? M (s< с).

Does this mean Wah? A(c)< -а), откуда Уа? А(-с >A)

H. Various forms of the method of mathematical induction for integers. Division theorem with remainder

Theorem 1 (first form of the method of mathematical induction). Let P(c) be a one-place predicate defined on the set Z of integers, 4. Then if for some NUMBER a Z the proposition P(o) and for an arbitrary integer K > a from P(K) follows P(K -4- 1), then the proposition P(r) is valid for all integers with > a (i.e., the following predicate calculus formula is true on the set Z:

Р(а) bow > + 1)) Ус > аР(с)

for any fixed integer a

Proof. Let everything that is said in the conditions of the theorem be true for the sentence P (c), i.e.

1) P(a) - true;

2) UK Shch k + is also true.

From the opposite. Suppose there is such a number

b > a, that RF) is false. Obviously b a, since P(a) is true. Let us form the set M = (z ? > a, P(z) is false).

Then the set M 0, since b? M and M- are limited from below by the number a. Consequently, by the theorem on the least integer number (Theorem 4, 2), there is a least integer c in the set M. Hence c > a, which, in turn, implies c - 1 > a.

Let us prove that P(c-1) is true. If c-1 = a, then P (c-1) is true by virtue of the condition.

Let c- 1 > a. Then the assumption that P(c- 1) is false entails belonging to 1? M, which cannot be, since the number c is the smallest in the set M.

Thus, c - 1 > a and P(c - 1) is true.

Hence, by virtue of the conditions of this theorem, the sentence P((c- 1) + 1) is true, i.e. R(s) - true. This contradicts the choice of number c, since c? M The theorem is proven.

Note that this theorem generalizes Corollary 1 of Peano's axioms.

Theorem 2 (second form of the method of mathematical induction for integers). Let P(c) be some one-place predicate defined on the set Z of integers. Then if the proposition P(c) is valid for some integer K and for an arbitrary integer s K from the validity of the proposition P(c) For all integers satisfying the inequality K< с < s, слеДует справеДливость этого преДложения Для числа s , то это преДложение справеДливо Для всег целыс чисел с >TO.

The proof of this theorem largely repeats the proof of a similar theorem for natural numbers (Theorem 1, 55, Chapter III).

Theorem 3 (third form of the method of mathematical induction). Let P(c) be a one-place predicate defined on the set Z of integer NUMBERS. Then if P(c) is true for all numbers of some infinite subset M of the set of natural numbers and for an arbitrary integer a, the truth of P(a) implies the truth of P(a - 1), then the proposition P(c) is valid for all integers.

The proof is similar to the proof of the corresponding theorem for natural numbers.

We offer it as an interesting exercise.

Note that in practice, the third form of mathematical induction is less common than the others. This is explained by the fact that to apply it, it is necessary to know the infinite subset M of the set of natural numbers, which is discussed in the theorem. Finding such a set can be a difficult task.

But the advantage of the third form over the others is that with its help the proposition P(c) can be proven for all integers.

Below we will give an interesting example of the application of the third form." But first, let's give one very important concept.

Definition. The absolute value of an integer a is a number determined by the rule

0, if a O a, if a > O

And, if a< 0.

Thus, if a 0, then ? N.

We invite the reader, as an exercise, to prove the following properties of absolute value:

Theorem (about division with remainder). For any integer numbers a and b, where b 0, there exists and, moreover, only one pair of numbers q U m such that a r: bq + T L D.

Proof.

1. Existence of a pair (q, m).

Let a, b? Z and 0. Let us show that there is a pair of numbers q and satisfying the conditions

We carry out the proof by induction in the third form on the number a for a fixed number b.

M = (mlm= n lbl,n? N).

It is obvious that M C is a mapping f: N M, defined by the rule f(n) = nlbl for any n? N, is a bijection. This means that M N, i.e. M- infinitely.

Let us prove that for an arbitrary number a? The M (and b-fixed) statement of the theorem about the existence of a pair of numbers q and m is true.

Indeed, let a (- M. Then a pf! for some n? N.

If b > 0, then a = n + O. Now setting q = n and m O, we obtain the required pair of numbers q and m. If b< 0, то и, значит, в этом случае можно положить q

Let us now make an inductive assumption. Let us assume that for an arbitrary integer c (and an arbitrary fixed b 0) the statement of the theorem is true, i.e. there is a pair of numbers (q, m) such that

Let us prove that it is also true for the number (with 1). From the equality c = bq -4- it follows that bq + (t - 1). (1)

There may be cases.

1) m > 0. Then 7" - 1 > 0. In this case, putting - m - 1, we obtain c - 1 - bq + Tl, where the pair (q, 7"1,) obviously satisfies the condition

0. Then c - 1 bq1 + 711 , where q1

We can easily prove that 0< < Д.

Thus, the statement is also true for a pair of numbers

The first part of the theorem has been proven.

P. Uniqueness of the pair q, etc.

Suppose that for the numbers a and b 0 there are two pairs of numbers (q, m) and (q1, then, satisfying the conditions (*)

Let us prove that they coincide. So let

and a bq1 L O< Д.

This implies that b(q1 -q) m- 7 1 1. From this equality it follows that

If we now assume that q ql, then q - q1 0, whence lq - q1l 1. Multiplying these inequalities term by term by the number lbl, we obtain φ! - q11 D. (3)

At the same time, from inequalities 0< т < lbl и О < < очевидным образом следует - < ф!. Это противоречит (3). Теорема доказана.

Exercises:

1. Complete the proofs of Theorems 2 and 3 from 5 1.

2. Prove Corollary 2 from Theorem 3, 1.

3. Prove that the subset H C Z, consisting of all numbers of the form< п + 1, 1 >(n? N), closed under addition and multiplication.

4. Let H mean the same set as in Exercise 3. Prove that the mapping ј : M satisfies the conditions:

1) ј - bijection;

2) ј(n + m) = ј(n) + j(m) and j(nm) = ј(n) j(m) for any numbers n, m (i.e. ј carries out an isomorphism of the algebras (N, 4, and (H, + ,).

5. Complete the proof of Theorem 1 of 2.

6. Prove that for any integers a, b, c the following implications hold:

7. Prove the second and third theorems from Z.

8. Prove that the ring Z of integers does not contain zero divisors.

Literature

1. Bourbaki N. Set theory. M.: Mir, 1965.

2. Vinogradov I. M. Fundamentals of number theory. M.: Nauka, 1972. Z. DemiDov I. T. Foundations of arithmetic. M.: Uchpedgiz, 1963.

4. Kargapolov M.I., Merzlyakov Yu.I. Fundamentals of group theory.

M.: Nauka, 1972.

5. Kostrikin A.I. Introduction to algebra. M.: Nauka, 1994.

b. Kulikov L. Ya. Algebra and number theory. M.: Higher. school, 1979.

7. Kurosh A.G. Higher algebra course. M.: Nauka, 1971.

8. Lyubetsky V. A. Basic concepts of school mathematics. M.: Education, 1987.

9. Lyapin EU. and others. Exercises on group theory. M.: Nauka, 1967.

10. Maltsev A.I. Algebraic systems. M.: Nauka, 1970.

11. MenDelson E. Introduction to mathematical logic. M.: Nauka, 1971.

12. Nechaev V.I. Numerical systems. M.: Education, 1975.

13. Novikov P.S. Elements of mathematical logic. M.. Science, 1973.

14. Petrova V. T. Lectures on algebra and geometry.: At 2 hours.

CHL. M.: Vlados, 1999.

15. Modern foundations of the school mathematics course Auth. Col.: Vilenkin N.Ya., Dunichev K.I., Kalltzhnin LA Stolyar A.A. M.: Education, 1980.

16. Skornyakov L. A. Elements of algebra. M.: Nauka, 1980.

17. Stom R.R. Set, logic, axiomatic theories. M.; Enlightenment, 1968.

18. Stolyar A. A. Logical introduction to mathematics. Minsk: HIGHEST. school, 1971.

19. Filippov V.P. Algebra and number theory. Volgograd: VGPI, 1975.

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21. Fuchs L. Partially ordered systems. M.: Mir, 1965.

Educational publicationEdition

Vladimir Konstantinovich Kartashov

INTRODUCTORY MATHEMATICS COURSE

Tutorial

Editorial preparation by O. I. Molokanova The original layout was prepared by A. P. Boshchenko

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