## 9.06.2012

### Degree trigonometric functions formula

Degree trigonometric functions formula - very crib in the study of trigonometric functions. Here are the squares, cubes, and fourth-degree sine, cosine, tangent, cotangent. Still, these formulas can be called the degree of reduction formulas of trigonometric functions. In mathematics, like juggling different formulas, the formulas of trigonometry is just for the occasion.

If you take a particular angle, you will find it to trigonometric functions of the formulas, and then substitute these values into the formula and the appropriate calculations, the left and right sides you should get the same. What, in fact, says equality.

## 9.05.2012

### Matrix in mathematics

Matrix in mathematics perform quite a significant role. With the theory of matrix magic sort it out themselves on your textbooks. I want now to reason about sense of the matrix numbering. Many people know that the matrix used to solve systems of linear equations and matrices used for these purposes. I will not repeat the textbooks, and I suggest you go to kindergarten and there to look for the meaning of mathematical operations with matrices.

So, we will think of the simplest child's task about dogs and cats: "Now in zoosalone 2 cats 3 dogs clipped their wool. While 4 cats and 5 dogs are waiting in the queue. So how many cats and dogs want to change their image?" Simple task on two independent mathematical actions: add up the cats to the cats, add up the dogs to the dogs and get the result two numbers. One of which is the number of the cats, and the second the number of the dogs. That is simply. Now we will solve this task with the use of the matrix numbering on the calculator matrices.

Here's what happened. As creators of calculator consider that operations with row vectors and column vectors are too primitive for mathematical knowledge of such high level, I had to go to a trick. The missing elements of square matrix the second order (this is the minimum size for the calculator matrices) I replaced with zeros.

On a picture it is shown that, I decided this task two methods: first, the values were in the first column, and then I put the same values in the first row of the matrix. Both variants gave the right result. From this we can conclude that the mathematical matrix operations within the meaning of a concurrent execution of mathematical operations with variables that have different units of measure.

In drawing up the matrix to solve a system of linear equations as a unit of measurement to take a unknown. For example, a unit of the decimal number system (just a number, the constant term of the equation), x, y, z, XXL ... Oh! It not from here, this unit is used to measure blouses. By the way, if you write in a matrix the number of your blouses, the matrix of this does not hurt - it does not matter what or where you are writing. But the correct solution to a system of equations with the matrix in the blouse you have not come out.

The principle of a matrix of equations for you take the numerical coefficients before unknown and inscribe them each into the place. That is where the matter of what and where exactly do you fit into the matrix. Then the resulting matrix solving.

Where is yet used principle of mathematical operations with matrices? When you count a whole bunch of money in paper notes and coins. Laid out bills and coins into piles on the dignity and recalculate the number in each pile. On this matrix method ends, farther you bring the got results to a single unit of measurement. Naturally, performing similar actions in everyday life, we never think about how it can be described mathematically.

Remember, we talked about the meters and decimetres? About bows and flounces, on a multi-core processor? There also can be used matrix methods of addition and subtraction. Perform all actions with the same unit of measurement, then all of the other unit of measurement. In the end, if necessary, make conversion between different units of measurement to another.

This method is very good to use in geodesy, when you need to add or subtract a large number of angles in degrees, minutes, seconds. Doing all the actions in degrees, then a minute, then with seconds. And then all received converting seconds into minutes and seconds. To obtain the minute add or subtract those moments obtained after converting seconds. Convert the result to degrees and minutes. To add on the total number of degrees degrees of minutes. All you have turned a corner in degrees, minutes, seconds.

I hope this article will be useful in practice, and you're using methods of matrix computations is in full knowledge of what you are doing and what the result will get.

## 8.26.2012

### Unsolvable equalizations are in preschool

The decision of unsolvable equalization caused some questions. Now we on the other hand will look at unsolvable equalization a number 1:

x + 2 = x

For this purpose we will create task research force and will nose after her behavior. In a research group we will plug:

1) Child from preschool. He only begins to understand the elements of mathematics but already considers well.
2) Blonde, mother of child from preschool. Once at school taught mathematics and knew it well. Now she forgot everything.
3) Professor of mathematics De Colletage. Brilliantly owns a mathematical vehicle, decides the most intricate mathematical problems.
4) Nikolay Khyzhnjak. Engineer, addicts to mathematics, asserts that understands this abstract science.

You will say that it is not honest, to include itself in the complement of research group. But, respected, I thought of this group, I offered candidates in her composition, I assert final composition of research group. It is the most democratic principle. I honestly took advantage of democratic right to be select and won in the competition of candidates :)))

And so, we will ask our research group to decide a simple task and give the algebraic variant of her decision:

We have an unknown amount of hedgehogs to that added two grass-snakes. How many hedgehogs did turn out?

As a result a research group gave out next decisions.

1) Child from preschool. As I talked already, a child is clever and able to decide logical tasks, in that it is needed to guess the train of thoughts of author of task and give the same answer that waits from a child author of task. It is such method of training of future zombis, that is well able to guess that is waited from them by owners. A child answered the unknown amount of hedgehogs" will "Turn out. On a request to write down the decision of this task on the sheet of paper, a child answered that to write he is not yet able, but can draw a hedgehog.

2) Blonde, mother of child from preschool. A blonde correctly decided a task after the row of leading questions. On a request to write down the decision through algebra, she began to reason: "Algebra, algebra... Nothing I remember... Algebra means - this unknown... Wow! We will designate the unknown amount of hedgehogs through "х". The amount of grass-snakes is known us and designating him is not needed, simply a number we write... Addition for us - it a sign "plus". Here"!. A blonde stretches out the sheet of paper, it is writtenin on that:

x + 2 = x

3) Professor of mathematics De Colletage. On a request to decide a task about hedgehogs and grass-snakes a famous professor was indignant, that it is ashamed to the great scientist to decide such child's tasks. He sent us after the decision of task in kindergarten. On our request to write down the algebraic decision of this task, a professor непонимающе looked at us, long thought and said: "Well, I will think above this problem". And was farther following...

For the algebraic reflection of decision of this task a professor worked out the special theory of three-cornered integrals, for that him was awarded with rank "academician". To the three-cornered integral the name of author was appropriated and he began to be named "Three-cornered integral De Colletage"...

For the decision of this concrete task the special research institute was created at the head with now already by an academician De Colletage. After a few years of the strained work, multivolume scientific work bore under the name "Practical application of three-cornered integral De Colletage in limits on the radius of cut on a dress"...

All of it nearscientific delirium we we will not study. We are interested only by a result. At once we open the last page of the last volume. There list of some organizations... We leaf pages to beginning of book in search of answer for a task... Somewhere in the middle of the last volume we find a line "Hired executed on account of working off scientific underbacks : .". Oho! This gang of easy riders from science on an even place managed to guzzle the whole lot of moneys! And before this line it is possible to find an answer for our task. "Х" is there written with plenty of reservations.

4) Nikolay Khyzhnjak. This witch-doctor of mathematics profoundly gave out a page under the name "Decision of unsolvable equalizations". According to his opinion, there is other task in that asked, how many hedgehogs will turn out, if to two hedgehogs to add the unknown amount of grass-snakes. He assures that both tasks have an identical decision :)))

By the way, there is another interesting task about grass-snakes and hedgehogs:

Mixture of grass-snake and hedgehog is a meter of the barbed wire. How mathematically to write down this expression?

Three-cornered integral De Colletage on the cut of dress not to apply :)))

## 8.19.2012

### Complex numbers

Complex numbers we we will not examine now with you. If you need to understand complex numbers, take textbooks, swarm in the Internet - information is enough there. I now do not want to engage in complex numbers.

If you want to know my personal opinion about complex numbers, then I will say that all of it is nonsense. On one mathematical forum I read a request about the help of some physicist. He writes there, that he succeeded to decide some equations in an eightmeasure complex space. Now he had a question, as got decision from an eightmeasure complex space to translate in ordinary space. Was it so desirable to say him: "And what fool compels you to decide it in complex space? Do not climb in complex spaces, and you will not have to return back"!

For me the weightiest argument in behalf on fabulosity of complex numbers is circumstance that in all history of existence of complex numbers not a single mathematician got the salary in complex numbers.

I consider that complex numbers are a first-ever virtual game for adults. Game it so pleased all, that in her with enthusiasm play until now. If to look at complex numbers from the point of view of Code of criminal procedure, then this will be an usual swindle. Сomplex numbers can be named a scale swindle in history of science. Notice, we thought of a rule about that, whatever ignorance of law releases from responsibility. I will add from itself, that the laws of mathematics I do not consider an exception. "One law for all" - is mathematics. "One law for all, except me and I will say, to whom the action of law does not spread yet" - it all other.

Due to existence of complex numbers in that nobody understands nothing, mathematicians occupy the same position today, what in ancient times was occupied by priests. Only mathematicians are known all secrets of complex numbers and only mathematicians know that it is needed to do with these complex numbers.

The not last role in development of imaginaries was played by the bureaucratic system. When nobody of simple people understands nothing, they can talk whatever. So hypercomplex numbers appeared. If you want to succeed in the bureaucratic system, then imaginaries for you are the real treasure. You easily can think of some super-puper-bam-trah-plus hypercomplex number. Thus every new prefix to the complex numbers gives an opportunity to write practically an infinite amount to nobody not clear dissertations and to get for it real, but not complex, money. Then you to the pension will teach other that in what understand nothing. All very simply. An even talking parrot can be taught to pronounce mathematical determinations. Any train animal is able to execute certain actions in a certain sequence. People yield to training much better of any animal, therefore they handle complex numbers so adroitly.

Complex numbers are an example that can turn out, if problem to decide not correctly. Those problems that mathematicians decide in relative stem of coordinates with the use of complex numbers must decide in the absolute system of coordinates, where complex numbers can not arise up in principle, as there are not negative numbers from that it was possible to extract a root square and to get an complex number. Search these decisions and you them will find necessarily. Transport companies perfectly managed with the decision of similar problem, in a difference from mathematicians. Whatever direction you went in, in direct or reverse, you always pay and never pay to you, even if you countermarch. I think that in the future expression "complex numbers" will be used approximately in the same sense, in what we today apply expression "three whale". For example, if a student will go out to the board and will say "I can not decide" it, a teacher him will advise "You enter complex numbers".

We will look now, where complex numbers appeared from. Show to me those real circumstances mathematical description of that results in appearance of equation

x² + 1 = 0

I can dream up. We it admits, someone printed great symbols on the sheet of paper equation in that a sign costs minus. With this sheet he went for a walk on a zoo and by chance dropped a sheet check to the monkey. A monkey heaved up this sheet, conducted on it the dirty finger. On a clean chance this dirty finger converted a sign minus in a sign plus. A monkey looked on the got result, thought: "Nonsense some, in jungles such equations are not" and threw out this sheet from a cage. At this time by a cage with a monkey a mathematician passed and picked up an ill-starred sheet. "What interesting equation" - a mathematician thought and began him to decide. Imaginaries appeared as a result of decision. Told a story fully applies on realisticness, but to mathematics she has no relation.

Think, I clear enough expounded the point of view on complex numbers. I hope, nobody of believing in existence complex numbers will begin to consider this article as insult of feelings of believers. You can believe in whatever: in God, in Devil or in complex numbers. But it is not necessary to compel me to believe in the same.

That to do to you, did eat you compel to teach complex numbers? Teach - such are bureaucratic rules of playing education. Do you want to get the scrap of paper, confirmative, that for you clever brain on shoulders? Here when you will take the deserving place in the bureaucratic system of science or education, then will decide, continue these playing complex numbers or give up them.

## 8.15.2012

### Decision of unsolvable equations

In comments to the article infant "Wunderkindes and cretin with blondes" (in Russian language) me offered to decide a few equations it is "known from a digit, that a decision is not present". These equations look like the following:

1) x + 2 = x

2) √x = -1

3) x/0 = x

Probably, on mathematicians these unsolvable equations operate, as a boa on rabbit and infuse with in their souls the awesome trembling. I am deprived such superstitions. Moreover, the author of this idea, Vag, tossed up to me a magic stick:

"Decisions does not have" means that it is WELL-PROVEN that such circumstances at that problem abode by ARE not.

Today we will consider circumstances at that terms at least one of equations observed. Consequently, at least one unsolvable equation we will decide now.

But to begin would I like not from a decision, and from an answer to the question: "Where these unsolvable equations appeared from?" Me it seems to, here from where...

...A mother bought to the child a new toy are blocks with an alphabet. She collected a few words from blocks, read them and gave blocks to the child. A mother took up the everyday businesses.

A child with enthusiasm began to fold blocks. Finishing to lay out series from the blocks taken by chance, he put question to the mother:

- That is here written?

- Nothing. You laid down letters wrong, - a mother explained, - When will learn letters, then you will be able correctly to fold blocks.

A child was very offended and burst into tears. What a more mother tried to quiet him, the he cried stronger. Then a mother said:

- Well, try once again, I necessarily will read you.

Bustling a nose, a child at random took a few blocks and laid down them in series.

- What did I write? - he asked with tears.

Farther not to disorder a child, a mother answered:

- In mathematics a certain integral is so designated in limits from Old Testament to nine evening, taken on the surface of asynchronous point.

A child quieted down puzzled. He understood nothing, but something disturbed him in this phrase. A bit thinking, he asked:

- And will a fairy-tale be at nine o'clock of evening?

- It will be necessarily. Where will she go? - a mother answered very confidently.

Very attentively looking to the mother in eyes, a child by touch took a few blocks and laid down another abracadabra.

- And it what did I write? - he asked with distrust, not tearing away a look from mother eyes.

If a mother will not look at blocks will say that is written there, she means cheats him. If will look - she means that is written there really reads.

A mother looked another portion of incoherent words at blocks and crfpfkf. A child was happy. He has the most clever mother! Farther this game proceeded until child not tired of. He really had a clever mother that knew many clever words.

A child grew up then, learned letters, began independently to fold words from blocks. When he became adult, he forgot playing blocks, but faith that for him the most clever mother, remained...

Just as a child made blocks, mathematicians worked out unsolvable equations from mathematical symbols. A child grew and became adult, mathematicians so remained in pampers of the determinations.

Now we will pass to business.

Decision of the first unsolvable equation

x + 2 = x

This equation traditionally can be taken to next equality

2 = 0

As this equality did not look wildly, but in mathematics such quite possible. I will say anymore, the first equation is only one equation from the system of two equations, that has one common decision. The second equation looks so:

2 + х = 2

This equation is usually taken to equality

х = 0

The decision of this system of equations looks so:

х ⊥ 2

It means that the numbers taken by you are on perpendicular numerical axes, therefore the rules of ordinary arithmetic give such result.

I especially want to underline that all the known rule "from transposition of elements a sum does not change" in this case stops to work. A result depends on that, what number you take for basis at implementation of action of addition.

The further decision of this system of equalizations is possible two methods resulting in different results.

First method. Turn of one of numerical axes on 90 degrees and passing to the rules of ordinary addition. Undeniable equality will ensue:

x + 2 = x + 2

n this case the laws of symmetry of mathematical actions begin to work.

Second method. Remaining in the rectangular system of coordinates, to apply the methods of vectorial algebra and find a sum on the theorem of Pythagoras, where х and 2 are the cathetuses of rectangular triangle, and result of addition - by a hypotenuse. If you consider that application of facilities of vectorial algebra is impossible, when on the ends of sticks the tips of pointers are absent, then it is your personal problems already. In this case both equalizations of the system are taken to the identical decision:

x + 2 = √(x² + 2²)

It is done away with the decision of this unsolvable equalization. you need to be only determined with the desires - what result needs you.

Decision of the second unsolvable equalization.

√x = -1

This decision caused most difficulties from the simplicity. We erect both parts of equalization in a square and get an answer:

x = 1

This equalization is taken to equality:

+1 = -1

In the parallel instances of mathematics usually begin to dissert upon the modules of numbers. I will say that before to put signs before numbers, it is needed to understand sense of positive and negative numbers.

Decision of the third unsolvable equalization

x/0 = x

The decision of this equalization is taken to equality:

1 = 0

Personally for me there is nothing unusual in this equality. This one of basic equalities of mathematics, without that mathematics in principle is impossible. If our mathematicians until now do without this equality, then only due to the pampers. At exposition of mathematics on the pages of this web-site I will repeatedly call to this equality.

To accept the solutions offered by me or not accept is this your personal file. Monkeys too nobody compelled to go down on earth. Many of them until now on branches bruise along and fully happy without pampers))))

### Theorem about parallel lines, thought of in motion

While I wrote about the types of triangles, I was visited by one idea - I thought of a theorem about parallel lines. She sounds so:

distance between parallel lines (by planes, volumes etc.) it is impossible to define mathematical methods

To engage in proof of this theorem about parallel lines we will not be now with you. We will leave proof on the future. If are wishing to prove or refute this theorem - occupy.

You can argue that there are very much tasks in the textbooks of mathematics, where it is needed to find distance between parallel lines and all of them decide mathematical methods. I agree, but... In the textbooks of mathematics I am much such, what is not present in mathematics. Yet more in mathematics what is not present in the textbooks of mathematics. Little by little we will understand with you, who is who in mathematics and where it undertakes from. And we will begin from unsolvable equalizations.

## 7.08.2012

### What trigonometric circle?

What trigonometric circle? Yet he is named trigonometric circumference. Question interesting enough. We will not go into the details of distinctions of concepts "circle" and "circumference". Although, for blondes I will explain that a circle is a coin or hole from a bagel, a circumference is this wedding ring or bagel. Coming from classic determination of trigonometric functions, more correct to talk "trigonometric circumference", as the question is about the set of points and their coordinates, but not about a flat figure and her area. But in the Internet much more often search a "trigonometric circle" exactly, probably because collecting so less letters is necessary.

Before we will pass to formulation of determinations of trigonometric functions with the use of portrait of tangent, we need to understand, what a trigonometric circle differs from all other circles. In reasoning we will use one mathematical focus sense of that is based exactly on these differences.

We will take the picture of circumference from determination of trigonometric functions. He looks here so.

On a picture the ordinary circumference of arbitrary radius is represented. As in the center of circumference cheekily fell to pieces the cartesian coordinate system, then all points of this circumference have vertical and horizontal coordinates. Radius of circumference and coordinate of points expressed by numbers. The intersections of axes of coordinates and circumference are determined by the size of radius of circumference and numeral him equal to.

In an order to convert an ordinary circumference into trigonometric one, it is needed to divide all elements of circumference into the radius of circumference. Letters of "X" and "Y" are the tags of the system of coordinates, designating her obtrusive presence on our picture. These letters by a radius are not divided, as well as letter of "B", designating a point on a circumference. Point "0"(zero) as a result of dividing by a radius so will remain a point "0"(remember, the zero divided by any number equals a zero?). Corner alpha it is possible to divide into a number, but it is impossible to divide into a radius circumferences(by the way, here to you the prepared proof that a division does not make sense) :))), therefore a corner it is better not to touch, in order to avoid large troubles(from all present person on a picture, a corner is the steepest element, he is in all trigonometric functions in the natural kind). For us remain only radius of circumference and coordinate points of "B", that we can divide into the size of radius of circumference.

Clear that at dividing of radius of circumference by an exactly such radius, we will get unit as a result. And at dividing of coordinates of point of circumference by a radius we will get the values of sine and cosine of corner alpha(on determination of trigonometric functions). That is our transformations look on a picture.

Now we need only to substitute all radiuses by units and coordinates of point on a sine and cosine of corner alpha and a trigonometric circumference will turn out for us. We even can set forth determination of single circumference (let child prodigy teach):

a circumference with a radius equal to unit is named a trigonometric circumference, if there is a center of the cartesian system of coordinates in the center of circumference.

This determination far not complete and does not take into account many nuances on that mathematicians take no notice usually. But about it we better will talk at the study of properties of trigonometric functions. And while we will look at a picture, as a trigonometric circumference looks in all beauty.

Actually, this is a trigonometric circle. If on him to place the values of different corners and corresponding to them values of sine and cosine, then we will get the trigonometric circle of sine and cosine.

Now about that focus that we will do during formulation of determinations of trigonometric functions. Here we substituted the radius of circumference by unit, and there we will act vice versa - we will replace unit the radius of circumference. Id est, from a trigonometric circumference to pass to the circumference of arbitrary radius, as that is required by a textbook on mathematics.