无穷大和无穷小-一点数学上的知识

1、无穷大和无穷小-一点数学上的知识（）People usually use it to refer to the zero limit variable (thanks to nineteenth Century that a large number of mathematicians, the word limit is already have a clear definition and is no longer merely a philosophical description), sometimes it was used as a symbol of some calculus

2、in the call, but no matter when people use it when they know exactly what I wanted to say, but the key is that people know that they dont need it, but only occasionally like with a same analogy with it.Then, back to the original meaning of the word: is there such a quantity, smaller than all given p

3、ositive real numbers, but not zero? Or, there is a series of equivalent questions about this problem: there are no two adjacent points on the line Does not exist the smallest unit of length? Wait, wait.Today, we have been able to answer these questions without question: No, no.In fact, the answer to

4、 this question is even later than the time of Cauchy and Weierstrass: it is essentially a question of the understanding of the structure of real numbers. Even Cauchy himself - even though he laid the foundations of modern limit theory - did not really understand the simple question of what real numb

5、ers were. The final establishment about real close theory, that is Piano (Peano), Cantor (Cantor) and Dedekind (Dedekind) that several achievements in the second half of the nineteenth Century mathematician. The so-called Dedekind division is still in the textbook today introduced the concept of rea

6、l standard model. In this model, one can answer all the questions about the structure of real numbers in a logical and self consistent manner, and, as pointed out previously, it completely rejects the existence of infinitesimal.Is it meaningless that mathematicians say that infinitesimal quantities

7、do not exist?This goes back to the question of authority that we have been confronted with in the front of mathematics. If we admit that infinitesimal is a concept of numbers, then the work of mathematicians has told us that there is no infinitesimal position in the theory of real numbers. In fact,

8、I have proved that Cantor is infinitesimal with real Archimedes admitted to the basic principle of contradiction. (Archimedess principle is a basic principle about the nature of real numbers. If Archimedess principle is wrong, the whole mathematics can not be established.) However, if the problem to

9、 the frontiers of mathematics, if that people have the right not to discuss the nature of itself in accordance with the number of mathematicians, then we are facing is entirely another level problem - it would be impossible to get detailed discussed here.- from how to make the length Author: wood.In

10、finity.In the last section, we talked about some figures, many of which were unequivocal big numbers. But these huge numbersThe number of words, such as grain, West class required, although was incredible, but still limitedThat is to say, as long as there is enough time, people can always write them

11、 from beginning to end.However, there are indeed some infinitely large numbers,They are even longer than what we can writeBe big. For example, the number of all integers and the number of all geometric points on a line are obviously infiniteBig. What else can we say about such figures, except that t

12、hey are infinitely large? Dystocia, we canEnough to compare the two infinite numbers above, and see which one is biggerWhat is the greater the number of integers and the number of all the geometric points on an line? - thisDo you have any questions? At first glance, the question really is his head,

13、however, the famous mathematician Cantor(Georg, Cantor) first thought about the problem. So he can really be called infinity arithmetic.Founder.When we compare the large hours of several infinite numbers, we face the problem that these numbers are bothI cant read it. I cant write it. How should I co

14、mpare it? At this point, we ourselves are a bit like oneFind out your belongings, what is the glass beads, or the original tribe of copper. You probably rememberWell, those people can only count to three. Will he give up because of numerous large numbers compared the number of beads and copper play?

15、 Not at all. If he is smart enough, he will pass the beads and coins one by one way comparedCome out with the answer. He can put a bead and a copper coin put together another beads and a copper coin placedTogether and always do it. If the beads used, but also left some coins, he knew, copperIf the c

16、oin coins more than beads; first with light, but also beads redundant, he will understand, such as more than copper beads;If both light, he would know, equal to the number of beads and coins.Cantors method of comparing the two infinite numbers is just the same: we can give two sets of infinityAll Sh

17、uyiyi large numbers in pairs. If the last two groups are left intact, the two sets of infinity are equalIf one group is not available, the group is bigger or stronger than the other.This is obviously reasonable and actually the only feasible way to compare the two infinite numbers. howeverWhen you p

18、ut this method to practical use, you have to be prepared for another surprise. For example, all even numbers and allOdd numbers of these two infinite sequences, of course you will intuitively feel that their numbers are equal. The application of the above principles is also completeFully consistent,

19、 because the two sets of numbers can be established as follows one-to-one correspondence.13579.246810.In this table, every even number corresponds to an odd number. Look, its really simple, not naturalPass!But wait a minute. Think again: the number of all integers (odd and even numbers), and the num

20、ber of even numbersWhich one is bigger? Of course, you would say the former is larger, because all integers contain not only all but even numbersPlus all the odd numbers. But thats only your impression. Only applying the above comparison, the two infinityThe laws of large numbers can lead to correct

21、 results.If you apply the law, you will be surprised to find out,Your impression is wrong. As a matter of fact, here is a one-to-one list of all integers and even numbers:12345.246810.In accordance with the above rule of infinite numbers, we have to admit that the number of even numbers is exactly t

22、he sum of all integersEyes are big. This conclusion, of course, seems absurd, since even numbers are only part of all integers.But dont forget that we are dealing with infinity, so we must do the thinking of the unusual natureWant to prepare.In a world of infinity, parts may be equal to all! About t

23、his, the famous German mathematician SilberHilbert (David) has a story that explains nothing better. It is said that one of his discussions is infiniteIn the speech, he used the following words to describe the nature of infinity paradox:We imagine a hotel with a limited room, and all the rooms are f

24、ull. Here comes a bitNew guest. I want to reserve a room. Im sorry, said the owner, all the rooms are full. Now resetAt another hotel, there was an unlimited room, and all the rooms were full, and a new visitor arrived,I want to reserve a room.No problem.! Said the owner. Then he moved the passenger

25、 in room number one to room two, twoRoom number passengers moved to room three, room three passengers moved to room four, and so on, so that the new guestHe was admitted to the vacated room.Lets imagine a hotel with an unlimited number of rooms, and every room is full. At this point, there again?Poo

26、r guests asked for room reservations.Yes, gentlemen, just a moment, please. Said the owner.He put a room to room number two passengers, the passengers moved to room two, room four, room threeThe passengers moved to room six, etc., and so on.Now all the odd numbered rooms have been vacated. The new,

27、infinite number of guests can live in.Because Hilbert told the story during the world war, so even in WashingtonIts not easy to understand. But this example does come up to the point. It makes us understand: infiniteThe nature of large numbers is quite different from the general numbers we encounter

28、 in general arithmetic.By comparing the Cantor laws of the two infinite numbers, we can also prove that all the ordinary scores (such as 3/5, etc.)The number is the same as all integers. Arrange all the scores according to the following rules: first write down the molecules andThe sum of the denomin

29、ator is 2, such a score is only one, that is, 1/1, and then write the sum of for 3, that isAnd 2/1; go down again, the sum of two is 4, that is, 1/3, 2/2, 3/1. In doing so, we can get an infinite fractionThe sequence, which includes all the scores (even repeat - best hsuehchien). Now, write an integ

30、er series next to this sequence, and youll get itThe one-to-one correspondence between infinite fraction and infinite integer. Thus, the number of them is equal!You might say, yes, its wonderful, but does that mean all the infinity?Are all big numbers equal? If so, what can be compared?No, thats not

31、 the case. One can easily find more than all integers and all fractionsAn infinitely large number; an infinite number.If you look at the number of points and integers on a line that appear earlierThe problem, we will find that the two numbers are not the same. The number of points on a line is great

32、er than the number of integersMuch more. To prove this, lets start with a line segment (say, 1 inches long) and an integer seriesCorrespondence.Each point on this line can be represented by the distance from this point to one end of the line, and this distanceCan be written in an infinite decimal fo

33、rm, as0.7350624780056.perhaps0.38250375632.Now all we have to do is compare the numbers of all integers and all the infinite decimals that may existThe number of. So whats the difference between the infinite decimals and the ones that are written?One must remember the rule in arithmetic: each common

34、 score can be broken upInfinite recurring decimal. Such as. We have proved that the number of all points is equal to the number of all integersThe number of all recurring decimals must be equal to the number of all integers. However, the point on a line is notCan be completely represented by recurri

35、ng decimals, most of which are represented by decimals. So its very niceIt is easy to prove that in this case, one-to-one correspondence can not be established.Suppose someone claims that he has established such a correspondence, and that the correspondence has the following form:N10.38601256854.20.

36、52315566584.30.25896499872.40.99586499859.Of course, since it is impossible to write an infinite number of integers and an infinite number of decimals without writing a light,Statements simply imply that the person has discovered some universal law (similar to the law we use to rank fractions)Under

37、the guidance of this rule, he made the list, and any decimal, or sooner or later, would be on this listAppear on.However, it is easy to prove that any such claim is untenable because we mustYou can also write an infinite number of decimals not included in this infinite table. How do you write it? It

38、s simpler. The first decimal digit of this decimal number is different from the first decimal number in the table, secondThe decimal bits are different from the second smallest digits of number second in the table, and so on. Thats probably the numberIt looks different (and possibly something else):

39、The number is not to be found in the chart at any rate. If the author of this table says to you, your number is hereHis watch is on line 137th (or any other number),You can immediately answer, no!Im not the number for you, because the 137th digits of this number and the first of your numberBaisanshi

40、qi different decimal.As a result, the one-to-one correspondence between points on the line and integers can not be established. In other words，The infinite number of points on a line is greater than (or stronger than) the infinity of all integers or fractions.The line segment just discussed is 1 inc

41、hes long. However, it is easy to prove that according to infinity arithmetic ruleNo matter how long the line is the same. In fact, 1 inches long line, or 1 feet long line, or 1 liThe length of the line is good, and the points above are the same. Just look at figure 6,AB and AC are two line segments

42、of different lengths, and now compare their points. Every point in the line of AB makes parallel lines of BC, which are in phase with the ACIntersection, thus forming a set of points. Such as D and D, E and E, F and F, and so on. On any point on the AB, there is AC on itA point corresponding to it,

43、and vice versa. In this way, we establish one-to-one correspondence. Visible, in accordance with ourRule, these two infinite numbers are equal.By this analysis of infinite numbers, a more surprising conclusion can be obtained: all on the planeThe number of points in the line is equal to the number o

44、f points on the line. To prove this, lets consider a line 1 inches long, AThe number on B and the number of square CDEF on the 1 inch side.Assume that a point on a line is 0.7512036. We can divide this number into odd and even fractionsBits separated into two different decimals:0.7108.and0.5236.Take

45、 the two numbers to measure the horizontal and vertical directions of the square, and draw a point. This point is calledMake the dual point of the point on the original line. Any point in a square, for example, from a square0.4835, 0.9907, these two numbers describe the points, we put the two number

46、s together, we get the lineThe corresponding dual point 0.49893057.It is clear that this approach establishes a one-to-one correspondence between the two sets of points. Each point on the line is flatThere is a corresponding point on the surface, each point on the plane has a corresponding point on

47、the line segment, and there is no leftPoint. Therefore, according to Cantors criterion, the infinity of all points in the square and the point on the lineThe infinity of numbers equal.In the same way, we can easily prove that all the points in the cube and the square or the line segment areThere are equal numbers of points, just divide the infinite decimals representing a point on the line segment into three parts and use the three new decimalsFind the dual point in the cube. Like two different length line segments, squares and cubesThe number of points in the body is independent of