Two in different degrees.

Alpha stands for real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. Taking as an example the infinite set natural numbers, then the considered examples can be presented as follows:

For a visual proof of their correctness, mathematicians have come up with many different methods. Personally, I look at all these methods as dancing shamans with tambourines. Essentially, they all boil down to the fact that either some of the rooms are not occupied and new guests are moving in, or that some of the visitors are thrown out into the corridor to make room for guests (very humanly). I presented my view on such decisions in the form of a fantastic story about the Blonde. What is my reasoning based on? Relocating an infinite number of visitors takes an infinite amount of time. After we have vacated the first room for a guest, one of the visitors will always walk along the corridor from his room to the next one until the end of the century. Of course, the time factor can be stupidly ignored, but it will already be from the category "the law is not written for fools." It all depends on what we are doing: adjusting reality to match mathematical theories or vice versa.

What is an "endless hotel"? An endless hotel is a hotel that always has any number of free places, no matter how many numbers are busy. If all the rooms in the endless visitor corridor are occupied, there is another endless corridor with the guest rooms. There will be an endless number of such corridors. Moreover, the "infinite hotel" has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians, however, are not able to distance themselves from commonplace everyday problems: God-Allah-Buddha is always only one, the hotel is one, the corridor is only one. Here are mathematicians and are trying to manipulate the serial numbers of hotel rooms, convincing us that it is possible to "shove the stuff in."

I will demonstrate the logic of my reasoning to you on the example of an infinite set of natural numbers. First, you need to answer a very simple question: how many sets of natural numbers are there - one or many? There is no correct answer to this question, since we invented numbers ourselves, in Nature there are no numbers. Yes, Nature is excellent at counting, but for this she uses other mathematical tools that are not familiar to us. As Nature thinks, I will tell you another time. Since we invented the numbers, we ourselves will decide how many sets of natural numbers there are. Consider both options, as befits a real scientist.

Option one. "Let us be given" a single set of natural numbers, which lies serenely on the shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and there is nowhere to take them. We cannot add one to this set, since we already have it. And if you really want to? No problem. We can take one from the set we have already taken and return it to the shelf. After that, we can take a unit from the shelf and add it to what we have left. As a result, we again get an infinite set of natural numbers. You can write all our manipulations like this:

I wrote down the actions in the algebraic notation system and in the notation system adopted in set theory, with a detailed enumeration of the elements of the set. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one subtracts from it and adds the same unit.

Option two. We have many different infinite sets of natural numbers on our shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. We take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. Here's what we get:

Subscripts "one" and "two" indicate that these items belonged to different sets. Yes, if you add one to the infinite set, the result will also be an infinite set, but it will not be the same as the original set. If we add another infinite set to one infinite set, the result is a new infinite set consisting of the elements of the first two sets.

Lots of natural numbers are used for counting in the same way as a ruler for measurements. Now imagine adding one centimeter to the ruler. This will already be a different line, not equal to the original.

You can accept or not accept my reasoning - it's your own business. But if you ever run into mathematical problems, think about whether you are not following the path of false reasoning trodden by generations of mathematicians. After all, doing mathematics, first of all, form a stable stereotype of thinking in us, and only then add mental abilities to us (or, on the contrary, deprive us of free thought).

Sunday, 4 August 2019

I was writing a postscript to an article about and saw this wonderful text on Wikipedia:

We read: "... rich theoretical basis mathematics of Babylon did not have a holistic character and was reduced to a set of disparate techniques, devoid of a common system and evidence base. "

Wow! How smart we are and how well we can see the shortcomings of others. Is it hard for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, I personally got the following:

The rich theoretical basis of modern mathematics is not holistic and is reduced to a set of disparate sections devoid of a common system and evidence base.

I will not go far to confirm my words - it has a language and conventions that are different from the language and conventions of many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole series of publications to the most obvious blunders of modern mathematics. See you soon.

Saturday, 3 August 2019

How do you divide a set into subsets? To do this, it is necessary to enter a new unit of measurement that is present for some of the elements of the selected set. Let's look at an example.

Let us have many A consisting of four people. This set was formed on the basis of "people" Let us denote the elements of this set by the letter a, a subscript with a digit will indicate the ordinal number of each person in this set. Let's introduce a new unit of measurement "sex" and denote it by the letter b... Since sexual characteristics are inherent in all people, we multiply each element of the set A by gender b... Note that now our multitude of "people" has become a multitude of "people with sex characteristics." After that, we can divide the sex characteristics into masculine bm and women bw sexual characteristics. Now we can apply a mathematical filter: we select one of these sex characteristics, it does not matter which one is male or female. If a person has it, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we apply the usual school mathematics. See what happened.

After multiplication, reduction and rearrangement, we got two subsets: the subset of men Bm and a subset of women Bw... Mathematicians think about the same when they apply set theory in practice. But they do not devote us to the details, but give a finished result - "a lot of people consist of a subset of men and a subset of women." Naturally, you may wonder how correctly the mathematics is applied in the above transformations? I dare to assure you, in fact, the transformations were done correctly, it is enough to know the mathematical basis of arithmetic, Boolean algebra and other branches of mathematics. What it is? Some other time I'll tell you about it.

As for supersets, you can combine two sets into one superset by choosing the unit of measurement that is present for the elements of these two sets.

As you can see, units of measurement and common mathematics make set theory a thing of the past. An indication that set theory is not all right is that for set theory, mathematicians have come up with own language and own designations. Mathematicians did what shamans once did. Only shamans know how to "correctly" apply their "knowledge". They teach us this "knowledge".

Finally, I want to show you how mathematicians manipulate with.

Monday, January 7, 2019

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". This is how it sounds:

Let's say Achilles runs ten times faster than a turtle and is a thousand steps behind it. During the time it takes Achilles to run this distance, the turtle will crawl a hundred steps in the same direction. When Achilles has run a hundred steps, the turtle will crawl ten more steps, and so on. The process will continue indefinitely, Achilles will never catch up with the turtle.

This reasoning came as a logical shock to all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert ... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them has become a generally accepted solution to the question ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from magnitude to. This transition implies application instead of constants. As far as I understand, the mathematical apparatus for using variable units of measurement either has not yet been developed, or it has not been applied to Zeno's aporia. Applying our usual logic leads us into a trap. We, by inertia of thinking, apply constant units of measurement of time to the reciprocal. From a physical point of view, it looks like time dilation until it stops completely at the moment when Achilles is level with the turtle. If time stops, Achilles can no longer overtake the turtle.

If we turn over the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly catch up with the turtle."

How can you avoid this logical trap? Stay in constant time units and do not go backwards. In Zeno's language, it looks like this:

During the time during which Achilles will run a thousand steps, the turtle will crawl a hundred steps in the same direction. Over the next interval of time, equal to the first, Achilles will run another thousand steps, and the turtle will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the turtle.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insuperability of the speed of light is very similar to the Zeno aporia "Achilles and the Turtle". We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia Zeno tells about a flying arrow:

The flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow rests at different points in space, which, in fact, is motion. Another point should be noted here. From a single photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs are needed, taken from the same point at different points in time, but it is impossible to determine the distance from them. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but they cannot determine the fact of movement (of course, additional data are still needed for calculations, trigonometry will help you). What I want to draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.

Wednesday, 4 July 2018

I have already told you that, with the help of which shamans try to sort "" reality. How do they do it? How does the formation of a set actually take place?

Let's take a close look at the definition of a set: "a set various elements, thinkable as a whole. "Now feel the difference between two phrases:" thinkable as a whole "and" thinkable as a whole. " reality is broken down into separate elements ("whole") of which a multitude will then be formed ("one whole"). At the same time, the factor that allows uniting the "whole" into a "single whole" is carefully monitored, otherwise shamans will not succeed. know in advance which set they want to demonstrate to us.

Let me show you the process with an example. We select "red solid in a pimple" - this is our "whole". At the same time, we see that these things are with a bow, but there are no bows. After that we select a part of the "whole" and form a set "with a bow". This is how shamans feed themselves by tying their set theory to reality.

Now let's do a little dirty trick. Take "solid in a pimple with a bow" and combine these "wholes" by color, selecting the red elements. We got a lot of "red". Now a question to fill in: the resulting sets "with a bow" and "red" are the same set or are they two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so be it.

This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We have formed a set of "red solid into a bump with a bow". The formation took place according to four different units of measurement: color (red), strength (solid), roughness (in a pimple), ornaments (with a bow). Only a set of units of measurement makes it possible to adequately describe real objects in the language of mathematics... This is what it looks like.

The letter "a" with different indices denotes different units of measurement. Units of measurement are highlighted in brackets, by which the "whole" is allocated at the preliminary stage. The unit of measurement, by which the set is formed, is taken out of the brackets. The last line shows the final result - the element of the set. As you can see, if we use units of measurement to form a set, then the result does not depend on the order of our actions. And this is mathematics, not the dancing of shamans with tambourines. Shamans can "intuitively" arrive at the same result, arguing it "by the obviousness", because units of measurement are not included in their "scientific" arsenal.

It is very easy to use units to split one or combine several sets into one superset. Let's take a closer look at the algebra of this process.

Saturday, 30 June 2018

If mathematicians cannot reduce a concept to other concepts, then they do not understand anything in mathematics. I answer: how do the elements of one set differ from the elements of another set? The answer is very simple: numbers and units.

Today, everything that we do not take belongs to some set (as mathematicians assure us). By the way, have you seen on your forehead in the mirror a list of those sets to which you belong? And I have not seen such a list. I will say more - not a single thing in reality has a tag with a list of the sets to which this thing belongs. The multitudes are all the inventions of shamans. How do they do it? Let's look a little deeper in history and see what the elements of a set looked like before shamanic mathematicians pulled them apart into their sets.

A long time ago, when no one had ever heard of mathematics, and only trees and Saturn had rings, huge herds of wild set elements roamed physical fields (after all, shamans had not yet invented mathematical fields). They looked something like this.

Yes, do not be surprised, from the point of view of mathematics, all the elements of the sets are most similar to sea ​​urchins- from one point, like needles, units of measurement stick out in all directions. For those who, I remind you that any unit of measurement can be geometrically represented as a segment of arbitrary length, and a number as a point. Geometrically, any value can be represented as a bunch of segments sticking out in different directions from one point. This point is point zero. I won't draw this piece of geometric art (no inspiration), but you can easily imagine it.

What units of measurement form an element of the set? Anyone describing this element from different points of view. These are the ancient units of measurement that were used by our ancestors and which everyone has long forgotten about. These are the modern units of measurement that we use now. These are also unknown units of measurement that our descendants will invent and which they will use to describe reality.

We figured out the geometry - the proposed model of the elements of the set has a clear geometric representation. What about physics? Units of measurement are the direct connection between mathematics and physics. If shamans do not recognize units of measurement as a full-fledged element of mathematical theories, this is their problem. I personally cannot imagine the real science of mathematics without units of measurement. That is why, at the very beginning of my story about set theory, I spoke of it as the Stone Age.

But let's move on to the most interesting thing - to the algebra of elements of sets. Algebraically, any element of a set is a product (the result of multiplication) of different quantities. It looks like this.

I deliberately did not use the conventions of set theory, since we were looking at a set element in its natural habitat before the emergence of set theory. Each pair of letters in brackets denotes a separate value, consisting of the number indicated by the letter " n"and units of measurement indicated by the letter" a". The indices next to the letters indicate that the numbers and units of measurement are different. One element of the set can consist of an infinite number of quantities (as far as we and our descendants have enough imagination). Each bracket is geometrically depicted as a separate segment. In the example with the sea urchin one bracket is one needle.

How do shamans form sets from different elements? In fact, by units or numbers. Not understanding anything in mathematics, they take different sea urchins and carefully examine them in search of that single needle, along which they form a set. If there is such a needle, then this element belongs to the set, if there is no such needle, it is an element not from this set. Shamans tell us fables about thought processes and a single whole.

As you may have guessed, the same element can belong to very different sets. Further I will show you how sets, subsets and other shamanic nonsense are formed. As you can see, "there cannot be two identical elements in a set", but if there are identical elements in a set, such a set is called a "multiset". Such logic of absurdity will never be understood by rational beings. This is the level of talking parrots and trained monkeys, who lack intelligence from the word "completely". Mathematicians act as ordinary trainers, preaching their absurd ideas to us.

Once the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the incompetent engineer died under the rubble of his creation. If the bridge could withstand the load, a talented engineer would build other bridges.

No matter how mathematicians hide behind the phrase "chur, I'm in the house", or rather "mathematics studies abstract concepts," there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Applicable mathematical theory sets to the mathematicians themselves.

We studied mathematics very well and now we are sitting at the checkout, giving out salaries. Here comes a mathematician to us for his money. We count the entire amount to him and lay out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and hand the mathematician his “mathematical set of salary”. Let us explain the mathematics that he will receive the rest of the bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.

First of all, the logic of the deputies will work: "You can apply it to others, you can not apply it to me!" Further, we will begin to assure us that there are different banknote numbers on bills of the same denomination, which means that they cannot be considered the same elements. Okay, let's count the salary in coins - there are no numbers on the coins. Here the mathematician will start to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms in each coin is unique ...

And now I have the most interest Ask: where is the line beyond which the elements of the multiset turn into elements of the set and vice versa? Such a line does not exist - everything is decided by shamans, science did not lie anywhere near here.

Look here. We select football stadiums with the same pitch. The area of ​​the fields is the same, which means we have got a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How is it correct? And here the mathematician-shaman-shuller takes a trump ace out of his sleeve and begins to tell us either about the set or about the multiset. In any case, he will convince us that he is right.

To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you, without any "thinkable as not a single whole" or "not thinkable as a whole."

Table of degrees of numbers from 1 to 10. Online degree calculator. High quality interactive table and grade table images.

Degree calculator

Number

Degree

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With this calculator, you can online calculate the power of any natural number. Enter the number, degree and click the "calculate" button.

Grade table from 1 to 10

Grade table from 1 to 10

Theory

Degree of Is an abbreviated notation for the operation of multiple multiplication of a number by itself. The number itself in this case is called - basis degree, and the number of multiplication operations is exponent.

a n = a × a ... × a

the entry is read: "A" to the "n" power.

"A" is the base of the degree

"N" - exponent

4 6 = 4 × 4 × 4 × 4 × 4 × 4 = 4096

This expression is read: 4 to the power of 6 or the sixth power of the number four or raise the number four to the sixth power.

Download the table of degrees

• Click on the picture for a larger view.
• Click on the "download" sign to save the picture to your computer. The image will be with high resolution and in good quality.

Enter the number and degree, then press =.

Degree table

Degree properties - 2 parts

Table of basic degrees in algebra in a compact form (picture, convenient to print), on top of the number, on the side of the degree.

Let's consider a sequence of numbers, the first of which is equal to 1, and each subsequent one is twice as large: 1, 2, 4, 8, 16, ... Using exponents, it can be written in the equivalent form: 2 0, 2 1, 2 2, 2 3, 2 4, ... It is called quite expectedly: a sequence of powers of two. It would seem that there is nothing outstanding in it - consistency as a sequence, no better and no worse than others. However, it has some very remarkable properties.

Undoubtedly, many readers have met her in the classic story about the inventor of chess, who asked the ruler as a reward for the first square of the chessboard for one grain of wheat, for the second - two, for the third - four, and so on, all the while doubling the number of grains. It is clear that their total number is

S= 2 0 + 2 1 + 2 2 + 2 3 + 2 4 + ... + 2 63 . (1)

But since this amount is incredibly large and many times exceeds the annual grain harvest around the world, it turned out that the sage tore off the ruler like a sticky one.

However, let us now ask ourselves another question: how to calculate the value S? Owners of a calculator (or, moreover, a computer) may well perform multiplications in the foreseeable future, and then add the resulting 64 numbers, getting the answer: 18 446 744 073 709 551 615. And since the amount of calculations is considerable, the probability of an error is very high.

Who is more cunning can see in this sequence geometric progression... Those who are not familiar with this concept (or those who have simply forgotten the standard formula for the sum of a geometric progression) can use the following reasoning. Let's multiply both sides of equality (1) by 2. Since doubling the power of two increases its exponent by 1, we get

2S = 2 1 + 2 2 + 2 3 + 2 4 + ... + 2 64 . (2)

Now we subtract (1) from (2). On the left side, of course, you get 2 S – S = S... On the right side, there will be a massive mutual annihilation of almost all powers of two - from 2 1 to 2 63 inclusive, and only 2 64 - 2 0 = 2 64 - 1 will remain. So:

S = 2 64 – 1.

Well, the expression has become noticeably simpler, and now, having a calculator that allows you to raise to a power, you can find the value of this value without the slightest problem.

And if there is no calculator - what to do? Column multiply 64 deuces? What else was missing! An experienced engineer or applied mathematician, for whom the main factor is time, would be able to quickly estimate the answer, i.e. find it approximately with acceptable accuracy. As a rule, in everyday life (and in most natural sciences) an error of 2–3% is quite acceptable, and if it does not exceed 1%, then this is just great! It turns out that it is possible to calculate our grains with such an error without a calculator at all, and in just a few minutes. How? You will see now.

So, we need to find the product of 64 twos as accurately as possible (we will discard the unit at once due to its insignificance). Let's divide them into a separate group of 4 twos and another 6 groups of 10 twos. The product of twos in a separate group is equal to 2 4 = 16. And the product of 10 twos in each of the other groups is equal to 2 10 = 1024 (make sure who doubts!). But 1024 is about 1000, i.e. 10 3. That's why S should be close to the product of 16 by 6 numbers, each of which is equal to 10 3, i.e. S ≈ 16 10 18 (for 18 = 3 6). True, the error here is still too large: after all, 6 times, when replacing 1024 by 1000, we were mistaken by a factor of 1.024, and in total we were mistaken, as is easy to see, by a factor of 1.024 6 times. So now, multiply 1.024 six times by itself additionally? No, we’ll do it! It is known that for the number NS, which is many times less than 1, the following approximate formula is valid with high accuracy: (1 + x) n ≈ 1 + xn.

Therefore 1.024 6 = (1 + 0.24) 6 ≈ 1 + 0.24 6 = 1.144. Therefore, it is necessary to multiply the number 16 · 10 18 we found by the number 1.144, resulting in 18 304 000 000 000 000 000, and this differs from the correct answer by less than 1%. Which is what we wanted!

In this case, we were very lucky: one of the powers of two (namely, the tenth) turned out to be very close to one of the powers of ten (namely, the third). This allows us to quickly estimate the value of any power of two, not necessarily the 64th. Among the powers of other numbers, this is rare. For example, 5 10 differs from 10 7 also by a factor of 1.024, but ... in a smaller direction. However, this is the same field as a berry: since 2 10 5 10 = 10 10, then how many times 2 10 surpasses 10 3, the same number of times 5 10 smaller than 10 7.

Another interesting feature of the sequence under consideration is that any natural number can be constructed from various powers of two, in a unique way. For example, for the current year number we have

2012 = 2 2 + 2 3 + 2 4 + 2 6 + 2 7 + 2 8 + 2 9 + 2 10 .

To prove this possibility and uniqueness does not amount to special labor... Let's start with possibilities. Suppose we need to represent in the form of a sum of various powers of two some natural number N... First, we write it as a sum N units. Since one is 2 0, then initially N there is an amount the same powers of two. Then let's start pairing them. The sum of two numbers equal to 2 0 is 2 1, so the result is obviously less the number of terms equal to 2 1, and, possibly, one number 2 0, if he did not find a pair. Next, we combine the same terms 2 1 in pairs, obtaining an even smaller number of numbers 2 2 (here, too, the appearance of an unpaired power of two 2 1 is possible). Then we again combine equal terms in pairs, and so on. Sooner or later, the process will end, because the number of equal powers of two after each union decreases. When it becomes equal to 1, it's over. It remains to add up all the resulting unpaired powers of two - and the presentation is ready.

As for the evidence uniqueness representation, the method "by contradiction" is well suited here. Let the same number N managed to be presented in the form two sets of different powers of two that do not completely coincide (that is, there are powers of two that are included in one set, but not included in another, and vice versa). First, discard all coinciding powers of two from both sets (if any). You get two representations of the same number (less than or equal to N) as a sum of various powers of two, and all degrees in representations different... In each of the representations, select the greatest degree. By virtue of the above, for two representations these degrees different... The representation for which this degree is greater will be called the first, another - second... So, let in the first representation the greatest degree is 2 m, then in the second it obviously does not exceed 2 m-1 . But since (and we have already encountered this above, counting the grains on the chessboard), the equality is true

2m = (2m –1 + 2m –2 + ... + 2 0) + 1,

then 2 m strictly more the sum of all powers of two not exceeding 2 m-1 . For this reason, already the greatest power of two included in the first representation is probably greater than the sum of all powers of two included in the second representation. Contradiction!

In fact, we have just justified the possibility of writing numbers in binary number system. As you know, it uses only two digits - zero and one, and each natural number is written in the binary system in a unique way (for example, the above-mentioned 2012 - as 11 111 011 100). If we number the digits (binary digits) from right to left, starting from zero, then the numbers of those digits in which the ones stand will just be the exponents of the twos included in the representation.

Less well known is the following property of the set of non-negative integer powers of two. Let's arbitrarily assign a minus sign to some of them, that is, make positive ones negative. The only requirement is that, as a result, both positive and negative numbers it turned out infinite amount. For example, you can assign a minus sign to every fifth power of two, or, say, leave positive only the numbers 2 10, 2 100, 2 1000, and so on - there are as many options as you like.

Surprisingly, but any whole the number can (and, moreover, in the only way) be represented as the sum of various terms in our "positive-negative" sequence. And it's not very difficult to prove it (for example, by induction on exponents of twos). main idea proof - the presence of arbitrarily large in absolute value of both positive and negative terms. Try to complete the proof yourself.

It is interesting to observe the last digits of the members of the sequence of powers of two. Since each subsequent number in the sequence is obtained by doubling the previous one, the last digit of each of them is completely determined by the last digit of the previous number. And since there are a limited number of different digits, the sequence of the last digits of powers of two is simply is obliged be periodic! The length of the period, of course, does not exceed 10 (since this is how many numbers we use), but this is a greatly overestimated value. Let's try to evaluate it without writing out the sequence itself. It is clear that the last digits of all powers of two, starting with 2 1, even... In addition, there cannot be zero among them - because a number ending in zero is divisible by 5, in which one cannot suspect powers of two. And since there are only four even digits without zero, then the length of the period does not exceed 4.

Verification shows that it is so, and the periodicity appears almost immediately: 1, 2, 4, 8, 6, 2, 4, 8, 6, ... - in full accordance with the theory!

The length of the period of the last pair of digits of the sequence of powers of two can be estimated no less successfully. Since all powers of two, starting with 2 2, are divisible by 4, then the numbers formed by their last two digits are divisible by 4. There are no more than two-digit numbers divisible by 4, there are only 25 (for single-digit numbers we consider zero as the penultimate digit), but from them it is necessary to throw out five numbers ending in zero: 00, 20, 40, 60 and 80. So the period can contain no more than 25 - 5 = 20 numbers. The check shows that this is so, the period begins with the number 2 2 and contains pairs of numbers: 04, 08, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72 , 44, 88, 76, 52, and then 04 again, and so on.

Similarly, one can prove that the length of the period of the latter m digits of the sequence of powers of two does not exceed 4 5 m-1 (moreover - in fact, she is equal to 4 5 m-1, but it is much more difficult to prove it).

So, quite strict restrictions are imposed on the last figures of the powers of two. and how about the first numbers? Here the situation is practically the opposite. It turns out that for any of a set of numbers (the first of which is not zero) there is a power of two starting with this set of numbers. And such powers of two infinitely many! For example, there is an infinite number of powers of two starting with 2012 or, say, 3 333 333 333 333 333 333 333.

And if we consider only one very first digit of various powers of two - what values ​​can it take? It is easy to make sure that any are from 1 to 9 inclusive (of course, there is no zero among them). But which ones are more common and which ones are less common? Somehow, one cannot immediately see the reasons why one number should occur more often than another. However, deeper reflections show that one should not expect exactly the same occurrence of numbers. Indeed, if the first digit of any power of two is 5, 6, 7, 8 or 9, then the first digit of the next power of two will be mandatory unit! Therefore, there must be a "bias", at least towards unity. Consequently, the rest of the figures are unlikely to be "equally represented."

Practice (namely, direct computer calculation for the first several tens of thousands of powers of two) confirms our suspicions. Here is the relative fraction of the first digits of powers of two rounded to 4 decimal places:

1 - 0,3010
2 - 0,1761
3 - 0,1249
4 - 0,0969
5 - 0,0792
6 - 0,0669
7 - 0,0580
8 - 0,0512
9 - 0,0458

As you can see, with the growth of the numbers, this value decreases (and therefore the same unit is about 6.5 times more likely to be the first digit of powers of two than nine). It may seem strange, but practically the same ratio of the numbers of the first digits will take place for almost any sequence of degrees - not only two, but, say, three, five, eight, and in general almost any numbers, including non-integer ones (the only exceptions are some "special" numbers). The reasons for this are very deep and complex, and to understand them, you need to know the logarithms. For those who are familiar with them, let's open the curtain: it turns out that the relative fraction of powers of two, the decimal notation of which begins with a digit F(for F= 1, 2, ..., 9) is lg ( F+ 1) - lg ( F), where lg is the so-called decimal logarithm, equal to the exponent to which the number 10 must be raised to get the number under the sign of the logarithm.

Using the above-mentioned connection between powers of two and five, A. Kanel discovered an interesting phenomenon. Let's select several digits from the sequence of the first digits of powers of two (1, 2, 4, 8, 1, 3, 6, 1, 2, 5, ...) contract and write them in reverse order... It turns out that these numbers will certainly meet also in a row, starting from some place, in the sequence of the first digits of powers of five.

The powers of two are also a kind of "generator" for the production of well-known perfect numbers, which are equal to the sum of all their divisors, except for itself. For example, the number 6 has four divisors: 1, 2, 3 and 6. Discard the one that is equal to the number 6. There are three divisors, the sum of which is exactly equal to 1 + 2 + 3 = 6. Therefore, 6 is a perfect number.

To obtain a perfect number, take two consecutive powers of two: 2 n–1 and 2 n... Reducing the largest of them by 1, we get 2 n- 1. It turns out that if this is a prime number, then, multiplying it by the previous power of two, we form the perfect number 2 n –1 (2n- 1). For example, for NS= 3 we get the initial numbers 4 and 8. Since 8 - 1 = 7 is a prime number, then 4 · 7 = 28 is a perfect number. Moreover - at one time Leonard Euler proved that all even perfect numbers look exactly like this. Odd perfect numbers have not yet been discovered (and few people believe in their existence).

Power of two has a close relationship with the so-called by Catalan numbers, the sequence of which has the form 1, 1, 2, 5, 14, 42, 132, 429 ... They often arise when solving various combinatorial problems. For example, in how many ways can a convex n-gon into triangles with disjoint diagonals? All the same Euler found out that this value is equal to ( n- 1) th Catalan number (we denote it K n-1), and he also found out that K n = K n-fourteen n – 6)/n... The sequence of Catalan numbers has many interesting properties, and one of them (just related to the topic of this article) is that the ordinal numbers of all odd Catalan numbers are powers of two!

Powers of two are often found in various problems, and not only in conditions, but also in answers. Take, for example, the once popular (and still not forgotten) Tower of hanoi... This was the name of the puzzle game invented in the 19th century by the French mathematician E. Lucas. It contains three rods, one of which is equipped with n discs with a hole in the middle of each. The diameters of all discs are different, and they are arranged in descending order from bottom to top, that is, the largest disc is at the bottom (see figure). It turned out like a tower of disks.

It is required to transfer this tower to another rod, observing the following rules: shift the discs strictly one at a time (removing the upper disc from any rod) and always put only the smaller disc on the larger one, but not vice versa. The question is: what is the smallest number of moves required for this? (By a move we mean removing a disc from one rod and putting it on another.) Answer: it is equal to 2 n- 1, which can be easily proved by induction.

Let for n disks, the required minimum number of moves is X n... Find X n+1. In the process of work, sooner or later, you will have to remove the largest disc from the rod, on which all the discs were originally put on. Since this disc can only be put on an empty rod (otherwise it will "press down" the smaller disc, which is prohibited), then all upper n disks must first be transferred to the third rod. This will require at least X n moves. Next, we transfer the largest disc to an empty rod - here's another move. Finally, in order to "squeeze" it from above with smaller n disks, again you will need at least X n moves. So, X n +1 ≥ X n + 1 + X n = 2X n+ 1.On the other hand, the actions described above show how you can cope with the task exactly 2 X n+ 1 in moves. Therefore finally X n +1 =2X n+ 1. A recurrence relation has been obtained, but in order to bring it to a "normal" form, one must also find X 1 . Well, it's as easy as shelling pears: X 1 = 1 (there simply cannot be less!). It is not difficult, based on these data, to find out that X n = 2n– 1.

Here's another interesting challenge:

Find all natural numbers that cannot be represented as the sum of several (at least two) consecutive natural numbers.

Let's check the smallest numbers first. It is clear that the number 1 in the indicated form is not representable. But all odd ones that are greater than 1 can, of course, be imagined. Indeed, any odd number greater than 1 can be written as 2 k + 1 (k- natural), which is the sum of two consecutive natural numbers: 2 k + 1 = k + (k + 1).

What about even numbers? It is easy to see that the numbers 2 and 4 cannot be represented in the required form. Maybe this is the case for all even numbers? Alas, the next even number refutes our assumption: 6 = 1 + 2 + 3. But the number 8 again defies. True, the following numbers are again inferior to the onslaught: 10 = 1 + 2 + 3 + 4, 12 = 3 + 4 + 5, 14 = 2 + 3 + 4 + 5, but 16 is again unimaginable.

Well, the accumulated information allows us to draw preliminary conclusions. Please note: could not be presented in the specified form powers of two only... Is this true for the rest of the numbers? It turns out, yes! Indeed, consider the sum of all natural numbers from m before n inclusive. Since all of them, by condition, are at least two, then n > m... As you know, the sum of consecutive terms arithmetic progression(and we are dealing with it!) is equal to the product of the half-sum of the first and last terms by their number. The half-sum is ( n + m) / 2, and the number of numbers is n – m+ 1. Therefore, the sum is ( n + m)(n – m+ 1) / 2. Note that the numerator contains two factors, each of which strictly more 1, and their parity is different. It turns out that the sum of all natural numbers from m before n inclusive is divisible by an odd number greater than 1, and therefore cannot be a power of two. So now it is clear why it was not possible to represent the powers of two in the required form.

It remains to make sure that not powers of two you can imagine. As for the odd numbers, we have already dealt with them above. Take any even number that is not a power of two. Let the greatest power of two, by which it is divisible, is 2 a (a- natural). Then if the number is divided by 2 a, it will turn out already odd a number greater than 1, which we will write in the familiar form - as 2 k+ 1 (k- also natural). So, on the whole, our even number, which is not a power of two, is equal to 2 a (2k+ 1). Now let's look at two options:

1. 2 a+1 > 2k+ 1. Take the sum 2 k+ 1 consecutive natural numbers, the average of which equals 2 a... It is easy to see that then least of which equals 2 a - k, and the largest is 2 a + k, and the smallest (and, therefore, all the others) are positive, that is, really natural. Well, and the sum, obviously, is exactly 2 a(2k + 1).
2. 2 a+1 < 2k+ 1. Take the sum 2 a+1 consecutive natural numbers. You cannot specify here the average number, because the number of numbers is even, but indicate a couple of medium numbers you can: let them be numbers k and k+ 1. Then least of all numbers is k+ 1 – 2a(and also positive!), and the greatest is k+ 2a... Their sum is also equal to 2 a(2k + 1).

That's all. So the answer is: non-representable numbers are powers of two, and they are the only ones.

And here is another problem (it was first proposed by V. Arbitrary, but in a slightly different formulation):

The garden area is surrounded by a solid fence of N planks. According to Aunt Polly's order, Tom Sawyer whitewashes the fence, but according to his own system: moving clockwise all the time, he first whitens an arbitrary board, then skips one board and whitens the next, then skips two boards and whitens the next, then skips three boards and whitens the next one. and so on, each time skipping one more board (while some boards can be whitewashed several times - this does not bother Tom).

Tom believes that with such a scheme, sooner or later, all the boards will be whitewashed, and Aunt Polly is sure that at least one board will remain unbleached, no matter how much Tom works. Under what N is Tom right, and under what N is Aunt Polly right?

The described whitewashing system seems to be quite chaotic, so at first it may seem that for anyone (or almost any) N each board will someday get its share of lime, that is, mostly, Tom is right. But the first impression is deceiving, because in fact, Tom is right only for the meanings. N, which are powers of two. For others N there is a board that will remain forever unbleached. The proof of this fact is rather cumbersome (although, in principle, it is not difficult). We suggest the reader to do it himself.

This is what they are - powers of two. It looks as easy as shelling pears, but as you dig ... And here we have not touched all the amazing and mysterious properties of this sequence, but only those that caught the eye. Well, the reader is given the right to independently continue research in this area. They will undoubtedly prove fruitful.

Zero number).
And not only deuces, as noted earlier!
Those hungry for details can read the article by V. Boltyansky "Do powers of two often begin with one?" ("Quantum" No. 5, 1978), as well as an article by V. Arnold "Statistics of the first digits of powers of two and the redistribution of the world" ("Quant" No. 1, 1998).
See the problem M1599 from the "Problem book" Quantum "(" Quantum "No. 6, 1997).
Currently, 43 perfect numbers are known, the largest of which is 2 30402456 (2 30402457 - 1). It contains over 18 million digits.