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.


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.

Blocks with an alphabet. Mathematics for blondes.
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.

Decision of unsolvable equations. Mathematics for blondes.
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.


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.

Ordinary circumference. What trigonometric circle. Mathematics for blondes.
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.

What trigonometric circle. All elements we divide by a radius. Mathematics for blondes.
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.

What trigonometric circle.  Mathematics for blondes.
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.

Trigonometric functions determination through a triangle

Trigonometric functions can be determined on a rectangular triangle. On a picture you can see determinations of all trigonometric functions through the elements of rectangular triangle.

Trigonometric functions determination through a triangle. Mathematics for blondes.
It is another icon for hanging out on a wall and overlearnings of text. We this we will not engage in. We will look better, as possible to combine a rectangular triangle and circumference. For this purpose we will combine two pictures: circumference from classic determination of trigonometric functions and our triangle. We will place a triangle next to a circumference. That is it looks.

Triangle and circumference. Mathematics for blondes.
As see, pictures are practically identical, and component elements are named differently in them. Identically mark only corner alpha and point of "B". Now we will impose a triangle straight on a circumference. All graphic images accepted for a circumference we will save, and a rectangular triangle we will underline red lines on the inside (type we will lead around the contour of triangle lipstick).

A triangle is in a circumference. Mathematics for blondes.
As be obvious from a picture, the hypotenuse of rectangular triangle grows into the radius of circumference, cathetuses become equal to the coordinates of point. It as in a human language - a the same concept in different languages is designated by different words. This distinction in a pronunciation does not give to us to understand foreign languages. Approximately the same takes place in mathematics. Some consider determination of trigonometric functions on a rectangular triangle primitive. If you want to understand mathematics, will memorize the following: there are not primitive things in mathematics. There are primitive creatures counting itself very clever. Exactly desire to seem more clever than other, resulted in that mathematicians almost nothing is understood in mathematics.

Specially I will quote a phrase from Russian-language Wikipedia, where talked about determination of trigonometric functions on a triangle: "This determination has some pedagogical advantage, because does not require introductions of concept of the system of coordinates, but also such large defect, that it is impossible to define trigonometric functions even for obtuse angles that must be known at the decision of elementary tasks about amblygons". Honestly speaking, like thick ignorance of mathematicians simply shocks me. What triangle are trigonometric functions determined for? Correctly, for RECTANGULAR. Let though one mathematician will show to me RECTANGULAR TRIANGLE With OBTUSE ANGLE. I swear to you, as soon as I will see this geometrical miracle, I here will invert the portrait of tangent heels over head and I will hang oneself on a cosine. Not a single mathematician understands really, that the decision of "elementary tasks about amblygons" through trigonometric functions is ALWAYS taken to breaking up of one amblygon on two rectangular triangles?

Now I will show the most primitive example of existence of trigonometry you in life. More primitive, probably, is not already.

Trigonometry is in life. Mathematics for blondes.
Look at a photo - how many human bodies serenely stationed oneself on the horizontal plane of rock, at the same time there is nobody on a vertical plane. Why? And because trigonometric functions for perpendicular direction have quite another values. For us with you basic sense of trigonometric functions is that exactly trigonometric functions determine our possibilities.


Trigonometric functions are determination

Determination of trigonometric functions I took from Russian-language part of Wikipedia and designed as a picture. You can look and read.

Trigonometric functions are determination about the coordinates of point and radius of circumference. Sine, cosine, tangent, cotangent, secant, cosecant, radius of circumference, coordinate of point. Mathematics for blondes.
Picture, naturally, I a bit did. In Wikipedia text of determination of trigonometric functions is written about one - about the coordinates of point and radius of circumference, and on a picture quite another is drawn. Probably, an artist from Wikipedia looked not in the that page of textbook on mathematics. Well yet, that he the textbook of mathematics with a textbook on biology did not entangle. Here that turned out for him.

Trigonometric functions are determination. Mathematics for blondes.
Many of you can say: "I do not understand trigonometry. That to do"? Do the same, that do all mathematicians - bluntly teach determinations and, if necessary, pronounce them aloud, as talking parrots. For this purpose unseal my picture, hang up it above a bed and every evening before a dream read determination of trigonometric functions instead of prayer. I will open to you one great secret of mathematicians : those, who rewrites the determination given above from a textbook in a textbook, very small understand in mathematics in general, and quite nothing is understood in trigonometric functions in particular. Those, who understands, what trigonometric functions and able to use them, on all Universe travel in flying piattis. We hardly tear away the priests from a terrene through airplanes and rockets. By the way, would you like to exchange it stinking petrol car on an elegant flying object of UFO type?

If did you already read my notes How to memorize trigonometric functions? and the Trigonometric functions, then to you teaching nothing is not necessary. All, that it is needed for formulation of any determination of trigonometric functions, you know already. Later I will show you, as to use these knowledge. And while we will keep reading Wikipedia, where talked about determination of trigonometric functions for a rectangular triangle.


Trigonometric functions

Trigonometric functions is simply. At the last time we considered the portrait of tangent, today we will expose the secrets of practical application of this masterpiece of painting.

Any trigonometric function is the result of division. It needs to be memorized forever. We divide something by something and as a result get a trigonometric function. That on what divided in trigonometric functions? Variants are different here. I suggest not to learn by rote you on memory all, in respect of trigonometric functions. Let us try simply to apply the portrait of tangent and minimum set of necessary knowledge.

And so, a tangent is a sine on a cosine. As we said hardly higher, in a tangent a sine is divided by a cosine. A division in mathematics can mark a fractional line or two points. We will transform our portrait of tangent in the record of partaking with two points. Here that turned out for us.

Trigonometric functions. Tangent it is a sine on a cosine. Mathematics for blondes.
Most important, what we are taught by this picture, so it to those places, where it is needed to search sines and cosines. In the system of denotations accepted by mathematicians, sines always are situated on a vertical line - it upwards-downward. Cosines always are situated on a horizontal - it to the left-right (or aside). That is a complete set will look from 6 trigonometric functions in our variant of denotations. For evidentness a sine and cosine are depicted by a different color and we will add fractional hyphens wherein without them not to do.

Trigonometric functions. Sine, cosine, tangent. Mathematics for blondes.
Trigonometric functions. Cotangent, secant, cosecant. Mathematics for blondes.
Do not be frightened such amount of pictures. That it is represented in overhead part, above a red line, we almost learned already. We will repeat:

a sine - it upwards

a cosine - is aside

a tangent - it to divide a sine into a cosine

Do you see? The half of trigonometric functions we learned already. With the second half, that under a red line, yet simpler - invert that you learned already heels over head. Remaining trigonometric functions are shots reverse already known to us. We swap a numerator and denominator - all is ready already for us.

With a cotangent all is simple. A cotangent is this attitude of cosine toward a sine. With gimbleting of tangent no problems arise up - he has both a numerator and denominator. But as to be with a sine and cosine? We do not have denominators of these shots in fact! Do not worry, any number has a denominator that in mathematics, writing is not accepted is unit. In a secant and cosecant exactly this unit appears in a numerator, and a cosine and sine are written in a denominator.

It is important to memorize! In the pairs of the names of trigonometric functions a secant is a cosine and cosecant - a sine can be only one prefix to on two names. If we take a sine for basis, then a reverse shot will be named cosecant. If we take a cosine, then a reverse shot is named secant, because one prefix to we already used up on a cosine.

It is important to memorize! In the pairs of the names of trigonometric functions a secant - is a cosine and cosecant - a sine can be only one prefix co on two names. If we take a sine for basis, then a reverse shot will be named cosecant. If we take a cosine, then a reverse shot is named secant, because one prefix co we already used up on a cosine.

In principle, it fully sufficiently, in determinations of trigonometric functions to feel, close as an oyster in water. A question remains not found out, that at a sine and cosine to write in a denominator instead of unit. In fact we said in beginning, that trigonometric functions are a result of division(relation) or shot. Do not worry, mathematicians will prompt us, that exactly they want there to see. For this purpose there are determinations of trigonometric functions. We will talk next time about them.