Ordering of impersonal natural numbers. The concept of a natural number and zero. Expression of "equally", "less", "greater" on impersonal natural numbers Understanding nutrition for mathematical analysis
An alternative to the N natural series is an impersonal natural number that does not change the natural number a, so N = (x | x N i x a).
For example, N ce impersonal natural numbers, so do not change 7, so. N = (1,2,3,4,5,6,7).
Significantly two most important powers in the natural series:
1) Be-yaky vіdrіzok N vengeance loneliness. Tsya vlastivistvo viplivaє іz vyznachennya vіdrіzka natural series.
2) If the number x vanishes from the opponent N і x a, then the number x + 1 comes after them and vanishes into N .
Bezlich A is called kіtsevim, as if it were equal to the same counterpart to the N natural series. For example, faceless And the tops of trikutnik, faceless stinks are equal to N = (1,2,3), that is. A~B~N .
Since the number A is non-empty and equal to N, then the natural number a is called the number of elements of the multiplier A and write n(A) = a. For example, if A is the multiplicity of vertices of the tricot, then n(A) = 3.
If it were not empty, the kіtsev bezlіch is equal to one and more than one vіdrіzk of the natural series, tobto. skin endian plural And it can be put in a uniquely equal number a, so that the impersonal A is mutually unambiguous in the number N.
The settling of mutual and one-nobility is the ethics of the unbearabels of the unbearable multi-livo and in the natural row to edible a rakhunka plow A. Zkilka Behind the worships of the same number. In one class, all one-element multiplicands will be reduced, in another - two-element ones, etc. The first number can be seen as the ultimate power of the class of princes of equal strength. In this order, from the theoretical-multiple point of view, a natural number is the main power of the class of terminal multipliers.
The number 0 can also be multiplier-theoretic - it should be set to an empty multiplier: n() = 0.
Also, a natural number as a characteristic of the quantity can be seen from two positions:
1) as the number of elements in the set A, won for a rahunka;
2) how powerful is the power of the class of kіtsevyh equally strong multitudes.
The establishment of links between final multiplies and natural numbers allows us to give a multiplier-theoretical clouding of "less".
If a = n(A), b = n(B), then the number a is less than the number b, even if only if the multiplier A is equal to the power submultiplier of the multiplier, then. A ~ B, de B, B, B (Fig. 1) . Abo if in the natural series N є let's get a lot of power vіdrіzka N, tobto. N N .
Numbers а і b equal, yakscho stinks are equal to equal multiples: a = k А~B de n(A) = a, n (B) = k. For example, 2 = 2, because n(A) = 2, n(B) = 2, A = (a, b), B = (z, x), A~B.
The dominance of the “less” term for natural numbers is also similar to the multiplier-theoretic clouding: transitivity and antisymmetry of this term is related to it, which is transitive and antisymmetric of the term “becomes a multiplier”.
It is shown that the multi-theoretical interpretation of the “less” for natural numbers, which is 2
Let's take the multiplier A, to avenge 2 elements, and the multiplier B, to avenge 5 elements, tobto. n(A) = 2, n(B) = 5. For example, A = (a, b), B = (c, d, e, f, r). From the multiplier B you can see the submultiple, the equal multiplier A: for example B = (c, d) і A ~ B.
Fairness over N
Tsyu nerіvnіst you can look at the little 2. Come on 2 is the number of folds, and 5 is the number of squares. If you put the circles on the squares, then it’s safe to say that part of the squares is left unfinished.
Otzhe, the number of folds is less than the number of squares, tobto. 2
Multiplier-theoretic sense of unevenness 0
The alignment of numbers in the cob course of mathematics is developed in different ways - it is based on all the approaches we have looked at before interpreting the phrase "less".
Theorems about the "largest" and "least" number
Theorem 4 (about the “least” number). If it were not empty, surrounded by the bottom of impersonal numbers, revenge the smallest number. (Here, as in the case of natural numbers, the word "multiple" is replaced by the word "multiple" E
Bringing. Let O A Z i A is fringed from the bottom, tobto. 36? Zva? A(b)< а). Тогда если Ь Е А, то Ь- наименьшее число во множестве А.
Come on now LA.
Todi Ua e Af< а) и, значит, Уа А(а - Ь >O).
Let's make impersonal M of all numbers in the form a - b, de probіgaє impersonal A, tobto. M \u003d (s [c \u003d a - b, a E A)
It is obvious that the impersonal M is not empty, the shards A 74 0
Yak is higher, M C N . Later, following the theorem o r a l n o m h i s l e (54, ch. III), the multiplier M has the least natural number m. A, and shards of t least in M, then Wah? 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). Be something not empty, surround the beast of impersonal numbers, to avenge the greatest number.
Bringing. Let O 74 AC Z i A is surrounded by the beast with the number b, so. ? ZVa e A(a< Ь). Тогда -а >b for all numbers a? BUT.
Later, the multiplier M (z g \u003d -a, a? A) is not empty and is surrounded by the number (-6) below. According to the previous theorem, the multiplier M has the smallest number, that is. ace? ICC? M (z< с).
Tse means what Wah? A(s< -а), откуда Уа? А(-с >a)
Z. Different forms of the method of mathematical induction of whole numbers. Theorem about podіl іz surplus
Theorem 1 (the first form of the method of mathematical induction). Let P(s) - single predicate, assignments to multiples of Z whole numbers., 4 . The same way For the deyaky NUMBER and Z the proposition P (o) і For a sufficient integer number K > a z P (K) slid P (K -4- 1), then the proposition P (g) is correct For all numbers z > a (so on the multiplier Z є the true formula for calculating predicates is:
P(a) cibulya > + 1)) Vus > aP(s)
for any fixed integer a
Bringing. Let the propositions P (c) be true to everything, to go for the mind of the theorem, tobto.
1) P(a) - true;
2) KK SC to + is also true.
Kind of unacceptable. Let's assume that there is such a number
b> a, sho RF) - hello. It is obvious that a, oskіlki R (a) is true. Satisfyingly impersonal M = (z?> a, P (z) - hibno).
Todi bezlich M0, oskіlki L? M and M are bordered below by the number a. Later, after the theorem on na i m e n n m e l e l o m h i sl (Theorem 4, 2), the multiplier M has the least number c. Zvіdsi z\u003e a, sho, my black, pulling s - 1\u003e a.
Let's say that Р(с-1) is true. If c-1 = a, then P (c-1) is true by virtue of mind.
Let c-1 > a. Todi pripuschennya, scho R (s-1) - hibno, pulling behind him the possession of s 1? M, which cannot be but, the number of s is the smallest in M.
In this order, s - 1> a and P (c - 1) - true.
Think of the proposition P((c- 1) + 1) from the proposition P((c- 1) + 1) - that's true. R(s) - true. Tse superechit the choice of the number c, oskіlki? The theorem has been completed.
Respectfully, this theorem is a close consequence of Corollary 1 to Peano's axioms.
Theorem 2 (another form of the method of mathematical induction of integers). Let P (s) - deaky one-m_sny predshsatp, vizna-day) on a multiplicity of Z integers. However, the proposition P (c) is valid For a decimal integer number K and For an adequate integer number s To correct Proposition P (c) For all integer numbers that satisfies the irregularities of K< с < s, слеДует справеДливость этого преДложения Для числа s , то это преДложение справеДливо Для всег целыс чисел с >Before.
p align="justify"> The proof of this theorem is rich, so I repeat the proof of a similar theorem for natural numbers (Theorem 1, 55, Ch.III).
Theorem 3 (the third form of the method of mathematical induction). Let P(s) - one-single predicate, assignments on the multiplier Z cіlіs CHІСі. If P(c) is true For all numbers of the decimal multiplier M of zero natural numbers i For a sufficient integer a C is true P(a) then P(a - 1) is true, then the proposition P(c) is true For all numbers.
The proof is analogous to the proof of the double theorem for natural numbers.
Proponuemo yogo like a cicava right.
It is worthy of note that in practice the third form of mathematical induction is more pronounced, lower and lower. It is explained that for її zastosuvannya it is necessary to know the infinite submultiplier M of the multiplier of natural numbers, it will be clear in the theorem. The knowledge of such a multiplier may appear to difficult tasks.
Ale, the advantage of the third form before the other ones is in the fact that the additional proposition P (c) is brought to all integer numbers.
Below we aim the buttstock of the third-form zastosuvanya ". Ale, back to back, damo is one more respectful understanding.
Appointment. The absolute value of an integer number a is the number assigned according to the rule
0, if a O a, if a > O
A yakscho a< 0.
Otzhe, like a 0, then? N.
It is suggested to the reader that he has the right to bring such power to absolute magnitude:
Theorem (about the overflow). For any number of numbers a i b, de b 0, iсnuє i before that, there is only one pair of numbers q U m such that a r: bq + T L D.
Bringing.
1. Base of the bet (q, t).
Let a, b? Z i 0. It is shown that there is a pair of numbers q i
The proof is carried out by induction in the third form for the quantity a with a fixed number b.
M = (mlm = n lbl, n? N).
It is obvious that M lt is an expression f: N M, which is determined by the rule f (n) = nlbl for any n? N is a bijection. Tse means that M N, that. M-indistinctly.
Let's say that from a certain number a? M (і L-fixed) assertion of the theorem about the basis of the pair of numbers q і t is true.
True, let it be a (- M. Todi a pf! for a real p?
If b > 0, then a \u003d n + O. Considering now q \u003d n and m O, we take the necessary pair of numbers q and m.< 0, то и, значит, в этом случае можно положить q
Zrobimo now induction allowance. Assume that from a sufficient integer number s (and a sufficient fixed b 0) the assertion of the theorem is true, then. is a pair of numbers (q, m) such that
It can be shown that it is more correct i for the number (з 1). Z equals s \u003d bq -4 - viplivaє bq + (t - 1). (one)
Possibly falls.
1) t\u003e 0. Todі 7 "- 1\u003e 0. At this point, having put - t - 1, we take z - 1 - bq + Tl, de para (q, 7" 1,) obviously pleases the mind
0. Todi h - 1 bq1 + 711 de q1
Without practice it is possible that 0< < Д.
In this order, the firmness is true and for a bet of numbers
The first part of the theorem has been completed.
P. Single bet q і etc.
Let's assume that for the numbers a i b 0 it is possible to establish two pairs of numbers (q, m) i (q1, so as to satisfy the minds (*)
Let's see that stinks are escaping. Oh, come on
i a bq1 L O< Д.
Zvіdsi vyplivaє, scho b(q1 -q) t-7 1
Let's now assume that q ql, then q - q1 0, stars lq - q1l 1. - q11 D. (3)
Vodnocha іz nerіvnosti 0< т < lbl и О < < очевидным образом следует - < ф!. Это противоречит (3). Теорема доказана.
U r a f n i n nya:
1. Complete the proofs of Theorems 2 and 3 of 5 1.
2. Complete corollary 2 from Theorem 3, 1.
3. To add, what is the sum of the NS Z, what is added up from the given numbers in the form< п + 1, 1 >(n? N), closed way of folding that multiplication.
4. Let N mean those same impersonal things that you have the right to 3. Bring what you see ј: M pleases the minds:
1) ј - bієktsіya;
2) j(n + m) = j(n) + j(m) and j(nm) = j(n) j(m) for any numbers n, m , i (H, +,).
5. Complete the proof of Theorem 1 of 2.
6. To prove that for any number of numbers a, b, the following implications are valid:
7. Tell a friend that third of the theorem from Z.
8. To prove that the number of Z integers does not avenge the numbers of zero.
Literature
1. Bourbaki N. Theory of multiples. M.: Svit, 1965.
2. Vinogradiv I. M. Fundamentals of number theory. M.: Nauka, 1972. Z. Demidov I. T. Give 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: Vishcha. school, 1979.
7. Kurosh A.G. The course of the most advanced algebra. M: Nauka, 1971.
8. Lyubetsky V. A. Basic concepts of school mathematics. M: Prosvitnitstvo, 1987.
9. Lyapin ES. that in. Right from the theory of groups. M: Nauka, 1967.
10. Maltsev A. I. Algebraic systems. M: Nauka, 1970.
11. MenDelson Ege. Introduction to mathematical logic. M: Nauka, 1971.
12. Nechaev V. I. Numerical systems. M: Prosvitnitstvo, 1975.
13. Novikov P.S. Elements of mathematical logic. M.. Nauka, 1973.
14. Petrova V. T. Lectures on Algebra and Geometry.: U 2 year.
CHL. M: Vlados, 1999.
15. Sochasni ambush school mathematics course Avt. credit: Vilenkin N.Ya., Dunichev K.I., Kalltzhnin LA Stolyar A.A. M: Prosvitnitstvo, 1980.
16. L. A. Kushnir, Elements of Algebra. M: Nauka, 1980.
17. Stom R.R. Impersonality, logic, axiomatic theories. M.; Osvita, 1968.
18. Stolyar A. A. Logical introduction to mathematics. Minsk: VISCHII. school, 1971.
19. V. P. Filippov, Algebra and Number Theory. Volgograd: VGPІ, 1975.
20. Frenkel A., Bar-Hilel I. Give the theory of multiples. M: Svit, 1966.
21. Fuchs L. Chastkovo ordering systems. M.: Svit, 1965.
Initially seen
Volodymyr Kostyantinovich Kartashov
INTRODUCTORY COURSE OF MATHEMATICS
Chief help
Editorial preparation O. I. Molokanova Original layout designed by O. P. Boshchenko
„PR 020048 dated 20.12.96
Signed to each other on 28.08.99. Format 60x84/16. Druk office. boom. type of. M 2. Uel. pich. l. 8.2. Uch.-view. l. 8.3. Circulation 500 copies. Enchantment 2
Vidavnitstvo "Zmina"
A natural number is the whole number, as if winning for a rahunka of objects. Vono viniklo z practical needs of the people. The development of understanding the natural number can be divided into a number of stages: 1. old people, in order to overcome the insignificance, established the essentials: for example, the insoles, the fingers on the hands. Nedolik - por_vnyuvani mnozhini vinni buli but one hour available for inspection. 2. Bezlich - intermediaries, for example, stones, turtles, sticks. The concept of kіlkіst is more folded. І numbers tied to specific subjects. 3. Appearance of a number (designation of a number by the visible digits). The birth of mathematics. Arithmetic as a science originated in the lands of the Ancient Descent - China, India, Egypt, distant development in Greece. The term "natural number" was first used by the Roman teachings of Boetius. Rakhunok is necessary to designate a lot of money. Rozіb'єmo all kіlkіsnі multipliers on the class of equivalence, for example, in one class of equivalence. to see the faceless tops of trikutniks, the sides of the square, the faceless letters of the word light. If you continue this process, then through those that have equivalence - everything is equally strong. Kіntsevі multiplied vyyavlyatsya for classes. That. theoretically - the plurality of the kіlkіsnogo natural number - є zagalna vlastіvіst class kіncevih equally strong plurals. The skin class has its own number. Zero is set to empty multiplier.
The numbers A and B are called equal, because they are equal in number.
Such a method stagnates in cob classes.
The technique of working on tasks that reveal the specific meanings of arithmetic diy.
Arithmetic tasks in the course of mathematics occupy a significant place. Mayzhe half an hour before an hour of mathematics lessons to be introduced to the completion of the task. All the great spiritual and illuminating roll, that the stench plays under the hour of children's education. Virishennya arithmetic tasks help to reveal the basic math of arithmetic actions, concretize them, and relate to the singing life situation. Zavdannya to take over math understand, Vіdnosin, laws. When the task is fulfilled, children develop quite respect, caution, more logical thought, Mova, kmіtlivist. The goal is to develop such processes of cognitive activity as analysis, synthesis, alignment, and refinement.
In the process of solving arithmetic tasks, the learners learn to plan and control their activities, open up acceptance, self-control (re-verification of tasks, estimation of tasks then) they sway in their arrogance, will, develop interest to the point of solving tasks. Great is the role of the virishennya zavdan in preparing children for life, for the future labor activity. When solving the plot tasks, the learners begin to shift between objects and values to the “language of mathematics”. In the arithmetic tasks, numerical material is victorious, which inspires the success of the country in the various galleries of the people's state, culture, and science. Tse spryaє expand the horizons of the students, enriched with new knowledge about the topical action. Uminnyam vyrishuvati arithmetic zavdannya uchnі opanovuyut with great difficulties.
The reasons for the pardoning tasks of children are crying out for us in front of the peculiarities of their minds. In the process of navchannya rozvyazannyu tasks should be uniquely stretched at the top of the task of the first mind, it is necessary to take into account the approach to the rozvyazannya of tasks, to orientate at the simple life situation, the descriptions of the task, the consideration of the task, the consideration of the given vision. In the process of working on any arithmetic problem, you can see the following stages:
1. Work on the task manager.
2. Poshuk problem solving.
3. Problem solving.
4. Formulation of the opinion.
5. Revising the problem solving.
6. Away from the robot over the top tasks.
I mean the respect of the next to attach the robots over the zmist of the factory, tobto. over the understanding of the situation in the tasks, the establishment of fallows between danim and shukanim. The sequence of work on the conquest of the task;
a) analysis of ignorant words and virazivs;
b) reading the text given by the teacher and learning;
c) a record of minding the task;
d) repetition of the food task.
Vyraznym reading the text of the head of the next study. It is necessary to remember that children especially need to read a promotional reading, they cannot read the task correctly on their own, cannot arrange logical voices, etc.
The order of concretization of the assignment for additional subjects, stencils and little ones in the practice of robots in schools of wide breadth has been formed in such a form to write down the assignment of the task:
1. The form of the note is shortened, with the text of the task, write down numerical data and only a few words and words, as necessary for understanding the logical sense of the task.
2. A short-structural form of writing, if the skin logical part of the task is written from a new row.
3. Schematic form of the record.
4. Graphical form of writing.
Since the function of control in children is weakened, then the re-examination of the order may be illumined, but it is also significant. In younger classes it is necessary:
1. Verbally formulate the tasks, roaming over the objects.
2. Reconsider the reality of the situation.
3. Reconsider the adequacy of the mind and the food of the plant. Rechecking the solution of tasks in other ways її vyshennya is possible from the 4th class.
In order to control the correctness of the development of the task, it is necessary to select and act on the elements of the programmed learning. This element is even more corny tim, that I will once again take into account the correctness of the chi and the pardon of my own actions. For the pardon of the decision of wines, there are new ways of cherry.
The teacher at the school is most likely to be sung that the rozvyazannya avdannya was enlightened by the teachings. It’s better for him to carry out the work of fixing the completion of this task. The work of fixed tasks can be carried out in different ways.
1. Set up a university food to save the day.
2. Proponuetsya rozpovіsti all rozvyazannya zadovі z obґruntuvannyam vyboru dіy.
3. Put food up to okremih diy chi food. For students, the number of variances of analogous tasks is important, and the understanding of the subject situation is important between them. Tsіy metі і to serve afar as a robot over the tasks of the task, as you can see how important it is to form the beginnings of solving tasks of this type. To a better understanding of the subject matter, the task, the fallows between the data and the shukani, the perfection of the task from the daily numerical data, written not in numbers, but in words. Be careful to show that the best teachers are widely victorious as one of the methods of teaching the tasks of arranging the task of organizing the tasks themselves.
The ordering of the task helps children to better understand the life-practical significance of the task, to better understand its structure, and to learn to differentiate the task of different species, to understand the decision. The ordering of the tasks is carried out in parallel with the decisions of the prepared tasks. Dosvid that caution will show that it is easier for the uchnіv chastkovo folded task. Slid to stimulate the formation of the teachings of the heads of the various plots. Tse spryaє razvitku їhnyoї vyavlyaet clemency, іnіtsiativi. It’s more embarrassing, if for the storage of the head of the school they get the material that they “obtain” for an hour of excursions, from dovіdnikіv, newspapers, magazines, etc. Students of the senior classes need to learn how to write and write business documents related to these and other rosrahunka. For example, write a letter of approval, fill in the form for a penny order just fine. All higher appointments can be widely used at the celebration of all kinds of tasks.
A simple arithmetic task is called a task, as if one arithmetic task is to be solved. Forgive the zavdannya to play the super-primary role of the hour of teaching mathematics. The simplest tasks allow you to expand the basic knowledge and concretize arithmetic functions, formulate those and other mathematical concepts. Forgive the order of the warehouse folding order, later, shaping the vminnya virishuvati їx, the teacher prepares the students to the opening of the folding order.
On the basis of dermal priming, learn to learn about new types of the simplest tasks. The step-by-step introduction of them is explained by the different stages of the problem of mathematical understanding, the process of cultivating quiet arithmetic processes, the specific solution of such stench is revealed. Not less respect for the teacher when choosing the leader of which kind of merit and concretization of that honor. Nareshti, reader to concretize the zmіst zavdannya, rozkrivayuchi zalezhnistі mіzh dannymi that shukanimi for additional forms of short recording.
The completion of the work of the best readers shows that the preparation for the completion of arithmetic tasks should be started from the improvement of the development of practical knowledge of learning, orientation of them at the necessary efficiency. Having learned it is necessary to lead in that life situation, in which it is possible to improve, revise arithmetic tasks, work to change. Moreover, these situations are not the next thing to create piece by piece, they are less likely to turn around and take the respect of the students. The teacher organizes guarding for the changing number of elements in the subject multitudes instead of vessels. bud., sho priyaє razvitku yavlen uchnіv pro kіlkіst to znajomstvo їх іz sing termіnologiєyu, yak zstrіnetsya with the verbal formulation of the task: it became, everything was lost, they took it, it increased, it changed, etc. It is necessary to organize such a playful and practical activity of the students, so that, being uninterrupted participants in this activity, as well as posterigayuchi, the students themselves could work the visnovka at the skin’s creamy drop; the number of elements of the multiplier has increased or the number of elements of the multiplier has changed, and some operation that verbal viraz shows the increase or change. This stage of preparing the work starts with the cob of work on the numbers of the first ten and familiarity with arithmetic actions, with solutions and folding applications of operations from subject plurals.
First of all, the beginning of learning the arithmetic tasks, the teacher is guilty of clearly revealing himself, like knowledge, it is necessary to give those skills to the students. To solve the task, learn the duties of arithmetic arithmetic, listen, and then read the task, repeat the task from food, for a short note, from memory, see the warehouse components in the problem, check the task, and reverse the correctness of the breakdown. At the 1st class, the learners begin to check the task of rebuking the bag and the excess. The qi of the task are entered before the beginning of the hour of the beginning of the numbers of the first ten. At the beginning of the rozvyazannya, the task was to change the sum of the same dodankivs, on the bottom on the equal part of the chi went on for the silver, followed by spiraling on the understanding of the daily arithmetic processes of the multiplication and the bottom. Before the opening of the order of the difference between the teachings, it is necessary to give an understanding of the order of the objects in one totality, two objective totalities, sizes, numbers, setting the s-similarity of them in the same line of equivalence and nervousness. Let's put it together, or put it together, arithmetic tasks are called tasks, like two people can't more arithmetic processes. Psychological studies of the development of the features of the arithmetic warehouse tasks show that children do not recognize simple tasks in the context of a new warehouse task. Preparing the work until the completion of the warehouse tasks is to be done by the system of rights, admissions, and directives of the educational institutions until the completion of the warehouse tasks. Before the completion of the warehouse manager, you can go over to the same place, if you change your mind, that the scientists mastered the arrangement of simple tasks with the help of tricks, if you go to the warehouse manager, you yourself can put together a simple task of a singing mind. When rozv'yazannі warehousing zavdan uchnі povinnі or to danih put food or food to get data. Also in the preparatory period, tobto. by stretching the last one of the first fate, that on the cob of another fate, learning, following the teachings of the task:
1. Wash your food before it’s ready.
2. From the food, add up the task, picking up the daily numerical data.
Folding simple and warehouse tasks, learning step by step to learn from the warehouse tasks is simple, even if you have completed them even more correctly, you have the right to fold the folding tasks. Tse accept the shortest mastering of the views of simple tasks, smarten them up to distinguish them from warehouse tasks, and help learners to analyze the tasks. When vyrіshennі warehouse zavdan uchnіv sled nauchit zagalnyh priyom_v work z zavdannyam; vminnyu to analyze zmist tasks, seeing in the given data, shukane (to establish what is necessary to be recognized in the task), depending on which data are not used for review on the head of the nutrition in the task. In practice, the work of the school is true to itself by the use of work with cards, tasks in which the sequence of work on tasks is laid down. When the order is completed, the decision is written down with nutrition, or the skin action is recorded and explained. The variation of the specified method of arranging tasks of a given type is ensured by the variant arranging of tasks with different types, plots, solutions prepared and folded by the students themselves, tasks of a given type with types of tasks that were previously solved, and so on.
1. Explain the counting method for vipadkіv 40 + 20, 50-30, 34 + 20, 34 + 2, 48-30, 48-3 must be counted with a hundred concentration.
1) 40+20= 4d+2d=6d=60
2) 50-30 = 5d-3d = 2d = 20
3) 34+20= 3d+4od+2d=5d 4ed=54
4) 34+2 \u003d 3d + 4od + 2od \u003d 3d 6od \u003d 36
5) 48-30 \u003d 4d + 8od-3d \u003d 1d 8ed \u003d 18
6) 48-3= 4d+8od-3d=4d 5d=45
Usі priyomi and counting usnі and vykonuyutsya on the basis for the ranks of folding and vіdnіmannya.
As it turns out, the numberless natural numbers can be put in order for an additional "less" expression. But the rules of the axiomatic theory should be emphasized, so that the goal was not only determined, but also improved on the basis of the already assigned ones in this theory to understand. You can do more by making the payment "less" through the addition.
Appointment. 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.
For tsikh minds to say the same, scho number b more a she write b > a.
Theorem 12. For any natural numbers aі b may be one and only one of the three viables: a = b, a > b, a < b.
The proof of this theorem is omitted. Z ієї of the theorem is obvious, what is it
a ¹ b, te chi a< b, or a > b tobto. vіdnoshennia "less" may be the power of pov'yazanostі.
Theorem 13. Yakscho a< b і b< с. then a< с.
Bringing. This theorem expresses the power of transitivity by suggesting “less”.
so yak a< b і b< с. then, for the purpose of naming "less", there are such natural numbers before and what b \u003d a + i c \u003d b + I. Ale todi h = (a + k)+ / і on the basis of the associativity of the folding is taken: h \u003d a + (to +/). Oskilki to + I - is a natural number, then a< с.
Theorem 14. Yakscho a< b, it's not true that b< а. Bringing. Tsya theorem expresses power antisymmetry vodnosini "less".
Let us start from the beginning, what for any natural number a don't wi-!>! ■ ) її resignation a< a. Let's not accept it, tobto. what a< а maє mistse. Todi, for the purposes of the blue "less", there is such a natural number With, what a+ h= a, and not to supersede Theorem 6.
Now let's say that yakscho a< b, then it's not true that b < a. Let's not accept it, tobto. what yakscho a< b , then b< а win. A list of equalities in Theorem 12 a< а, which is impossible.
So, as we say, “less” is antisymmetric and transitive and may have power in relation to the linear order, but the impersonality of natural numbers linearly ordered without a face.
From the designation "less" that yoga of power can be introduced in the house of power of a multiplier of natural numbers.
Theorem 15. Of all natural numbers, one is the smallest number, tobto. I< а для любого натурального числа a¹1.
Bringing. Come on a - be a natural number. Then there are two possibilities: a = 1 ta a ¹ 1. Yakscho a = 1, then it is a natural number b, for which one follows a: a \u003d b " \u003d b + I = 1+ b, tobto, for the purpose of the vodnosini "less", 1< a. Otzhe, be it natural more 1 chi more than 1. Abo, loneliness is the smallest natural number.
The introduction of “less” is connected with folding and multiplication of numbers by the power of monotony.
Theorem 16.
a \u003d b => a + c \u003d b + c that a c \u003d b c;
a< b =>a + c< b + с и ас < bс;
a > b => a + c > b + c and ac > bc.
Bringing. 1) The justice of this firmness is evident from the unity of folding and multiplication.
2) Yakscho a< b,
then it is a natural number k, what a
+ k = b.
Todi b+ c = (a + k) + c = a + (k + c) = a + (c+ to)= (a + c) + k. Equity b+ c = (a + c) + to means that a + c< b
+ With.
So it goes without saying that a< b =>ace< bс.
3) Be brought in the same way.
Theorem 17(Reverse Theorem 16).
1) a+ c = b + c or ac ~ bc-Þ a = b
2) a + c< Ь + с or ace< bcÞ a< Ь:
3) a + c > b+ w o ac > bcÞ a > b.
Bringing. We bring, for example, what ace< bс next a< b Let's not accept it, tobto. that the theorem is not victorious. Todi can't buti, scho a = b. to the fact that even then jealousy would be victorious ac = bc(Theorem 16); can't be i a> b, either way ac > bc(Theorem!6). Therefore, as far as Theorem 12, a< b.
From Theorems 16 and 17, one can introduce the rule of term-by-term addition and multiplication of irregularities. We omit it.
Theorem 18. For any natural numbers aі b; is also a natural number n, which p a.
Bringing. For be-whom a find such a number P, what n > a. For whom is enough to take n = a + 1. Multiplying term by term unevenness P> aі b> 1, acceptable pb > a.
From looking at the authorities, one can see the blue “less” to sing out the important singularities of the multiplier of natural numbers, which we induce without proof.
1. Ні for one natural number a no such natural number P, what a< п < а +
1. Tsya power is called in power
discreteness impersonal natural numbers, and numbers aі a + 1 name judicial.
2.
Be-yak not empty submultiplier of natural numbers to take revenge
least number.
3. Yakscho M- Empty number of impersonal natural numbers
and is the same number b, what for all numbers x s M won't win
equanimity x< b, then in the faceless Mє most.
Illustrating the power of 2 and 3 on the butt. Come on M- anonymous two-digit numbers. so yak Mє submultiplier of natural numbers і for all numbers< 100, то в множестве Mє the largest number is 99. M, - Number 10.
In this manner, the introduction of "less" allowed to look at (and bring in a row of vipadkiv) the significance of the power of a multiplier of natural numbers. Zokrema, it is linearly arranged, discrete, at least 1.
With the setting “less” (“more”) for natural numbers, young schoolchildren are familiar with the very beginning of learning. And often, in order of yogo multiplier-theoretic interpretations, the definition given by us within the framework of the axiomatic theory is implicitly vindicated. For example, students can explain that 9 > 7, shards 9 - not 7 + 2. Often and implicitly victorious power monotony folding and multiplication. For example, children explain that “6 + 2< 6 + 3, так как 2 < 3».
right
1, Why can’t the impersonal natural numbers be ordered after the help of the blue “behind the line”?
Formulate a vision a > b and prove that it is both transitive and antisymmetric.
3. Tell me what it is a, b, c- natural numbers, then:
a) a< b Þ ас < bс;
b) a+ h< b + su> a< Ь.
4. Some theorems about the monotonicity of addition and multiplication can
vykoristovuvaty young schoolchildren, vykonuyuchi zavdannya "Porіvnya, do not vykonuyuchi calculate":
a) 27+8...27+18;
b) 27-8...27-18.
5. Like the power of the multiplier of natural numbers, young schoolchildren implicitly win, win the same task:
A) Write down the numbers, like bigger, lower 65, smaller, lower 75.
B) Name the next number according to the date before the number 300 (800,609,999).
C) Name the smallest and largest three-digit number.
Vidnimannya
At axiomatic motivation The theory of natural numbers is known to sound like an operation that returns to stock.
Appointment. Considering the natural numbers a and b, the operation is called, which pleases the mind: a - b = s only and only a few, if b + c = a.
Number a - b called the difference of numbers a i b, number a- change, and the number b- seen.
Theorem 19. Variation of natural numbers a- bіsnuє tоdі і less than tоdі, if b< а.
Bringing. Let retail a- bІсnuє. Todi, for the designated retail, there is such a natural number With, what b + c = a, and tse means that b< а.
Yakshcho b< а, then, for the purpose of naming "less", it is also a natural number that b + c = a. Todi, for the appointed retail, c \u003d a - b, tobto. retail a - bІсnuє.
Theorem 20. What is the difference between natural numbers aі b I'm sure, there's only one.
Bringing. It is acceptable that there are two different values difference of numbers aі b;: a - b= c₁і a - b= c₂, moreover c₁ ¹ c₂ . Todi for designated retailers, maybe: a = b + c₁,і a = b + c₂ : . See what follows b+ s ₁ \u003d b + c ₂ : and on the basis of Theorem 17 it is possible to fit c₁ = c₂. They came to the point of omission, then, it’s wrong, but the theorem is correct.
Vyhodyachi z vznachennya raznitsі natural numbers that mind її іsnuvannya, you can follow the rules of vіdnimannya numbers from sumi and sumi from numbers.
Theorem 21. Come on a. bі h- natural numbers.
but yakscho a > c, then (a + b) - c \u003d (a - c) + b.
b) Yakscho b > c. then (a + b) - h - a + (b - c).
c) Yakscho a > c and b > c. then you can vikoristovuvati whether-yaku from these formulas.
Bringing. In times a) difference in numbers aі cіsnuє, oskelki a > c. Significantly її through x: a - c \u003d x. stars a = c + x. Yakscho (a+ b) - c \u003d y. then, for the appointed price, a+ b = h+ at. We represent in qiu equanimity zamіst a viraz h + x:(h + x) + b = c + y. We are speeding up the power of associativity to add: c + (x + b) = c+ at. Let's change this equanimity on the basis of the power of monotony, adding, we take:
x + b = y.. Replaced in Danish equivalence x with viraz a - c, let's mother (a - G) + b = y. In this rank, we were brought, scho yakscho a > c, then (a + b) - c = (a - c) + b
Similarly, the proof is carried out in case b).
The result of the theorem can be formulated as a rule that is easy to remember: in order to take the number from the sum, it is enough to take the number from one warehouse sum and to the result of adding more supplements.
Theorem 22. Come on a, b i c - natural numbers. Yakscho a > b+ c, then a- (b + c) = (a - b) - c or a - (b + c) \u003d (a - c) - b.
The proof of this theory is similar to the proof of Theorem 21.
Theorem 22 can be formulated as a visual rule, in order to consider the sum of numbers from the number, it is sufficient to consider the number of consecutive skin additions one by one.
At cob mathematicians vyznachennya vіdnіmannya yak dії, zvorotnogo dodavannya, at the sight, sound, do not give, but they are constantly koristuyutsya, pochinayuchi z vykonannya dіy over single-digit numbers. Learn to owe a good understanding of what you have to say about the folds, and win over the interrelationships when calculating. See, for example, from the number 40 the number 16, learn to mark like this: “Look at the number 16 from the 40 - which means to know such a number, when folding it with the number 16, enter 40; this number will be 24, so 24 + 16 = 40. Mean. 40 - 16 = 24".
Rules for interpreting numbers from sum and sum from numbers in the cob course of mathematics є theoretical basis Calculate other incomes. For example, the value of a virase (40 + 16) - 10 can be known, not only by counting the sum in the arms, but then by counting the number 10 from it, but in such a rank;
a) (40 + 16) - 10 = (40 - 10) + 16 = 30 + 16 = 46:
b) (40 + 16) - 10 = 40 + (16-10) = 40 + 6 = 46.
right
1. Chi is correct, what is a natural number of skin to come out of an uninterruptedly advancing loneliness?
2. Why is the logical structure of Theorem 19 special? Can you її formulate, victoriously, the words “necessary that sufficient”?
3. Bring what:
but yakscho b > c, then (a + b) - c \u003d a + (b - c);
b) yakscho a > b + c, then a - (b+ c) = (a – b) – p.
4. Chi can, without counting, say, the meaning of such virazіv dorivnyuvatimut:
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. Yakі power vіdnіmannya є theoretical basis of advancing priyomіv calculus, scho vychayutsya at the cob course of mathematics:
12 - 2-3 12 -5 = 7
b) 16-7 \u003d 16-6 - P;
c) 48 - 30 \u003d (40 + 8) - 30 \u003d 40 + 8 \u003d 18;
d) 48 - 3 = (40 + 8) - 3 = 40 + 5 = 45.
6. Describe the possible methods for calculating the value by sight. a - b- h and illustrate them on specific butts.
7. Tell me what b< а and be any natural c virna equanimity (a - b) c \u003d ac - bc.
Vkazivka. The proof is based on axiom 4.
8. Calculate the value of the virazu, without counting the letters. Vidpovidi wrap.
a) 7865 × 6 - 7865 × 5; b) 957 × 11 - 957; c) 12×36 - 7×36.
Podil
Under the axiomatic theory of the natural numbers, the rozpodil sounds like an operation, turned to a multiplication.
Appointment. The subdivision of natural numbers a and b is an operation that satisfies the mind: a: b \u003d s todi and only todi, before if b× h = a.
Number a:b called private numbers aі b, number a dilimim, number b- dilnik.
As it seems, it is not necessary to distinguish natural numbers on impersonal natural numbers, and there are no such obvious signs of a private basis, as it is necessary for retail. Є tilki necessary mind the basis of the private.
Theorem 23. In order to create privately two natural numbers aі b necessary b< а.
Bringing. Keep private natural numbers aі b I know that. is such a natural number c that bc = a. Oskіlki for any natural number 1 is valid nerіvnіst 1 £ With, then, multiplying the offending part by a natural number b, taken b£ bc. ale bc \u003d a, otzhe, b£ a.
Theorem 24. How private natural numbers are aі bіsnuє, there is only one.
The proof of the theorem is similar to the proof of the theorem about the unity of the difference of natural numbers.
Vyhodyachi z vyznachennya parts of natural numbers that mind yogo іsnuvannya, you can turn around the rules for subіlu sumi (retail, create) on the number.
Theorem 25. What are the numbers aі b divide by number With, then that amount a + b share with, and more privately a+ b per number With, one sum of private ones a on the hі b on the h, then. (a + b):c = a: c + b:With.
Bringing. Oskіlki number a be divided into With, then this is a natural number x = a; h, sho a = cx. Similar to the existing natural number y = b:With, what
b= su. Ale todi a + b = cx+ su \u003d - s (x + y). Tse means what a + b divided by c, moreover, it is more private, which is taken away when spreading sumi a+ b to the number c, which is more expensive x + y, tobto. ax + b: c.
The result of the theorem can be formulated using the rule of subdividing the sum by the number: in order to divide the sum by the number, it is sufficient to divide the sum by the number of skin additions and subtract the results.
Theorem 26. Like natural numbers aі b divide by number hі a > b then retail a - b be divided by c, moreover, it is private, won when the difference is divided by the number c, more private, won when the difference is divided a on the hі b to c, tobto. (a - b): c \u003d a: c - b: c.
The proof of this theorem is carried out similarly to the proof of the previous theorem.
This theorem can be formulated as a rule for subdivision of the difference on the number: for In addition, to divide the difference by number, it is enough to divide by the whole number, which changes and is seen from the first private sighting of a friend.
Theorem 27. What is a natural number a be divisible by a natural number c, then for any natural number b tvir ab share on p. In case of any privacy, what is taken away when you spread creativity ab to the number z , one dobutka of a private a on the With, i number b: (a × b): c - (a: c) × b.
Bringing. so yak a be divided into With, then there is a natural number x that a:s= x, stars a = cx. Having multiplied the offending parts of jealousy by b, taken ab = (cx) b. Oskіlki plural associatively, then (cx) b = c(x b). Zvіdsi (a b): c \u003d x b \u003d (a: c) b. The theorem can be formulated as a rule for subdividing a number into a number: divide the number by a number, divide the number by one of the multipliers, and subtract the result, multiply the other multiplier.
For the cob-savvy mathematician, the podil is assigned as the operation of the turnaround, for the savage look, it doesn’t give a sound, but they are constantly koristuyutsya, starting from the first lessons of knowledge of the podil. Learn to blame good reason, that he gave the reasons for the multiplications and victorious interrelationships during the calculations. For example, he divided 48 by 16, learners say this: “To divide 48 by 16 means to know such a number, when multiplying it by 16, we will make 48; this number will be 3, shards 16 × 3 = 48. Also, 48: 16 = 3.
right
1. Bring what:
a) just a fraction of natural numbers a b if it is, then there is only one;
b) like numbers a b subscribe to hі a > b then (a - b): c \u003d a: c - b: c.
2. What can be confirmed that all data are correct:
a) 48:(2×4) = 48:2:4; b) 56:(2×7) = 56:7:2;
c) 850: 170 = 850: 10:17.
What is the rule to aggravate these vipadkіv? Formulate yoga and bring it.
3. Yakі power podіlu є theoretical basis for
vikonanna coming days, preached to schoolchildren cob classes:
How can you, without depending on the bottom, say that the meanings of such words will be the same:
a) (40 + 8): 2; c) 48:3; e) (20 + 28): 2;
b) (30 + 16): 3; d) (21 +27): 3; f) 48:2;
Chi vіrnі іvnostі:
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 the virus
mind:
a) (a+ b):c; b) a:b: With; in) ( a × b): s .
Suggested methods and illustrate on specific butts.
5. Find out the meaning of expression in a rational way; own
dії wrap:
a) (7 × 63): 7; c) (15 × 18):(5× 6);
b) (3 × 4× 5): 15; d) (12 × 21): 14.
6. Round the next steps and the bottom on a double 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) = 560x2 = 1120.
7. Don't beat yourself up under the couch, find the most rational
in a private way; choose a way to prime:
a) 495:15; c) 455:7; e) 275:55;
6) 425:85; d) 225:9; e) 455:65.
Lecture 34
1. Anonymous number of unknown numbers. The power of a multiplicity of tsilih nevid'emnyh numbers.
2. Understanding the natural series of numbers and elements of the final multiplier. Ordinal and kіlkіsnі natural numbers.
Up to sovereignty of specialty
1. Linear (vector) space over the field. apply. Under space, the simplest power. Linear and independent vectors.
2. Basis and peace vector space. The matrix of coordinates of the system of vectors. Transition from one basis to another. Isomorphism of vector space.
3. Algebraic closure of the field of complex numbers.
4. A ring of whole numbers. Ordering of whole numbers. Theorems about the "largest" and "least" number.
5. Group, apply group. The simplest power groups. Subgroups. Homomorphism and isomorphism of groups.
6. The main power of the fake numbers. Forgive the numbers. Infinity of impersonal prime numbers. The canonical layout of the stock number is that uniqueness.
7. The Kronecker-Capelli theorem (criterion for the integrity of the system linear rivers).
8. Main characteristics of the roads. Povna that is induced by the system v_drahuvan modulo. Kіltse kіltse v_drahuvan for the module. Euler's theorem and Fermat.
9. The addendum of the theory of porіvnyan to vysnovka is a sign of falsity. Zvernennya zvichaynogo fraction to the tenth and the appointment of the last yogo period.
10. Success of an explicit root of a polynomial with effective coefficients. Happened over the field of real numbers with rich terms.
11. Linear alignment with one change (criterion of rozvyaznosti, ways of rozvyazannya).
12. Equal systems of linear alignments. The method of subsequent exclusion is unknown.
13. Kiltse. Apply a keel. The simplest power of the kіlets. Pidkiltse. Homomorphisms and isomorphisms of the ring. Field. Irrigation example. The simplest power. Minimality of the field of rational numbers.
14. Natural numbers (foundations of the axiomatic theory of natural numbers). Theorems about the "greatest" and "least" natural number.
15. Rich segments over the field. Theorem about podіl іz surplus. The biggest collaborative dilnik of two rich members, the power of that way of knowing.
16. Binary blues. Suggestion of equivalence. Classes of equivalence, factor multiplier.
17. Mathematical induction for natural and integer numbers.
18. The dominance of mutually prime numbers. The least significant multiple of the numbers, the power of that way of knowing.
19. Field of complex numbers, numerical fields. Geometric appearance trigonometric form complex number.
20. The theorem about podіl іz surplus for whole numbers. The largest collection of numbers of numbers, the power of that way of knowing.
21. Linear operators of vector space. Kernel and image of a linear operator. Algebra of linear operators in vector space. Power values and power vectors of a linear operator.
22. Athenian transformation of the flat, their dominion is the way of zavdannya. A group of Athenian transformations of the plane and її subgroups.
23. Bagatokutniki. Bagatokutnik square. The theorem of reason and unity.
24. Equivalentness and evenness of bagatokutnikiv.
25. Geometry of Lobachevsky. Nonsuperity of Lobachevsky's system of axioms of geometry.
26. The concept of parallelism in the geometry of Lobachevsky. Mutual expansion of the straight Lobachevsky area.
27. Formulas ruhіv. Classification of the ruins of the area. Dodatki to rozvyazannya tasks.
28. Mutual expansion of two flats, straight flats, two straight flats near the expanse (in an analytical presentation).
29. Projective transformation. The theorem of reason and unity. Formulas of projective transformations.
30. Scalar, not vector create zmіshane vectors, їх additions to the development of tasks.
31. Weyl's system of axioms of trivi- metrical Euclidean space and її zmistovna non-superity.
32. Ruhi of the area and yoga of power. Group of ruins flat. The theorem of foundation and unity of movement.
33. The projective plane of that її model. Projective transformation, power. Group of design changes.
34. Reformation of likeness to the flat, their dominion. A group of transformations similar to the plane and її subgroups.
35. Smooth surfaces. The first quadratic form of the surface is zastosuvannya.
36. Parallel projecting that yoga of power. Images of flat and spacious figures in a parallel projection.
37. Smooth lines. The curvature of the space curve is the same.
38. Elips, hyperbola and parabola as a finite parabola. Canonical equality.
39. Directorial power of the ellipse, hyperbole and parabola. Polar alignment.
40. Under the influence of some points of the straight line, the power of that calculation. Harmonious split steam dots. Povniy chotirikutnik and yoga of power. An addendum to rozvyazannya tasks on pobudova.
41. Pascal's and Brianchon's theorems. Poles and polars.
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