Showing posts with label legends about mathematics. Show all posts
Showing posts with label legends about mathematics. Show all posts


Achilles and the tortoise

Achilles and the tortoise. Mathematics For Blondes.
Achilles and the tortoise

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporia, the most famous of which is the Achilles and the Tortoise aporia. Here’s how it sounds:

Suppose Achilles runs ten times faster than a turtle, and is a thousand paces behind it. During the time that Achilles has run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred paces, the tortoise crawls ten more steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.

Another version of the aporia "Achilles and the tortoise":

In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

This reasoning was a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert ... All of them in one way or another viewed the aporia of Zeno. The shock turned out to be so strong that discussions are continuing at the present time; the scientific community has not yet succeeded in reaching a general opinion on the essence of paradoxes. Mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the question; none of them has become a generally accepted solution. Everyone understands that this is a hoax, but no one understands what it is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from quantity to an inverse quantity. This transition implies the use of variable units of measurement instead of constants. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed or has not been applied to the aporia of Zeno. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like a slowing down of time to its full stop at the moment when Achilles approaches the turtle. If time stops, Achilles can no longer overtake the turtle.

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

How to avoid this logical trap? Stay in constant time units and don’t go to inverse quantity. In the language of Zeno, it looks like this:

In the time that Achilles has run a thousand steps, the tortoise will crawl a hundred steps in the same direction. Over the next interval of time equal to the first, Achilles will run a thousand more steps, and the turtle will crawl a hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution. On the aporia of Zeno "Achilles and the Turtle" is very similar to Einstein's statement about the irresistible speed of light. 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 of Zeno tells of a flying arrow:

A flying arrow is motionless, because at each moment of time it is motionless, and since it is motionless at each moment of time, it is always motionless.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment in time a flying arrow is stationary at different points in space, which, in fact, is movement.


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?

Unsolvable equalizations are in preschool. Mixture of grass-snake and hedgehog is a meter of the barbed wire. Mathematics for blondes.
Three-cornered integral De Colletage on the cut of dress not to apply :)))


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.


Multiplication and division by zero

As a zero is not a number, all mathematical operations on multiplying and dividing by a zero take place in area of units of measurements. In relation to the operation of divizion by zero of unit of measuring can be real and virtual. Units of measurements of length behave to the real units of measurements. All other units of measurements, probably, are virtual. Dividing by the zero of virtual units of measuring is impossible, as a result of division by zero of such units of measuring does not make sense.

In the special group it is necessary to distinguish natural unit of measurement of speeds (speed of light) and mathematical unit of corners (corner in 45 degrees). These units of measurements hatch through mathematical methods and their mathematical properties require a further study. More detailed study is required similarly by units of measurements of time.

Virtual units of measurements appear as a result of process that mathematically can be written down as dividing of zero by zero.


where а – is virtual unit of measurement.

The described mathematical properties of virtual units of measurements allow to enter any units of measurements us and use them without influence on the surrounding world. These units of measurements are used both for description of surrounding reality and for everyday needs. The examples of virtual units of measurements can be units of measurements of money, temperature, many physical sizes or applied in a technique and commerce. The process of exit from everyday life of virtual units of measurements of measuring can be mathematically represented as multiplying by a zero. Mathematical properties of similar units of measurements are tested by practice of their use during many millenniums.

In multidimensional space, division by zero increases the amount of the spatial dimensions, multiplcanion by zero diminishes this amount.

In rectangular cartesian coordinates it will look so:

x/0 = xy
xy/0 = (x/0)y = x(y/0) = xyz

At multiplication by zero it is necessary to take into account project properties of space, as a result of such increase depends on that, which one component is multiplied by zero.

xyz*0 = 0 and xy or xz or yz
xy*0 = 0 and x or y

In physical equalizations, division by zero requires introduction of new unit of measurement to examined by equalization the physical co-operation expressed by a mathematical action by an increase (probably, another unit of measurement of length). For example, if to divide unit of measurement of length into a zero, unit of measurement of area will ensue. If to divide unit of measurement of area into a zero, unit of measurement of volume will ensue et cetera.

m/0 = m²
m²/0 = m³

Algebraically it can be presented in a next kind:

a/0 = ab
ab/0 = abc

where а, b, c - mutually perpendicular units of measurements of length.

At multiplication by a zero one of the components of co-operation, described by physical equalization, from co-operation is eliminated. The primary result of co-operation grows into a zero. Remaining components continue to co-operate.

m³*0 = 0 and
m²*0 = 0 and m

Algebraically it can be presented in a next kind:

abc*0 = 0 and ab or ac or bc
ab*0 = 0 and a or b

where а, b, c - mutually perpendicular units of measurements of length.

Expl for blondes: It only began division and multiplication by zero. More in detail we will consider it other time.


Space evolution

Probably, in the process of the evolution, space generates different multidimensional universes with the even amount of dimension. Development originates from spaces with less of dimension to spaces with plenty of dimension. For basis of existence of space it is possible to accept principle of existence of speeds. Our 6-dimension Universe in the chain of space evolution will look like the following.

Space evolution. Mathematics for blondes. Nikolay Khyzhnjak.
Quite possible, that to Big Bang putting beginning of our Universe, there was a 4-dimension universe in that energy had one dimension of length, and a matter two dimension of length in that part of universe that corresponds our slower-than-light. A black hole, the consequence of that in 6-dimension space was Big Bang, giving beginning of our Universe, appeared in the process of evolution of this 4-dimension universe. A matter with two dimension of length in our Universe grew into 2-dimensional energy. Question about transformation of unidimensional on length energy in the process of transition through a black hole, remains open.

In the process of evolution of our Universe energy with two dimension of length partly passes to the matter with three dimension of length. A matter generates in our Universe black holes that give beginning to the new universes in the eightmeasured space. After Big Bang in the eightmeasured space our matter with three measuring of length grows into energy of the eightmeasured universe. Et cetera. The process of space evolution can develop for ever and ever.

During an evolution one 4-dimension universe generates the great number of 6-dimension universes. In turn every 6-dimension universe generates the great number of the 8-dimension universes. It look like the process of spawning. All 6-dimension universe, generated by one 4-dimension universe, can be in the mutually perpendicular dimension, that eliminates their cross-coupling on each other. Each of universes is mapped to all other universes as a point. The 8-dimension universes can be formed just.

In regard to faster-than-light part of universes of any type the theory of symmetry deserves attention in relation to velocity of light. Energy and matter of slower-than-light part freely move in three dimension of length and hardly fixed in the continuous stream of three dimension of time. Dark energy and dark matter freely move in three dimension of time and hardly fixed in the continuous stream of three dimension of length. Even if it not so, possibility of existence of universes of similar type does not need to be thrown down from accounts.

A transition process from a black hole to Big Bang requires an additional study. Mathematically he can be described by the operations of multiplication and division by zero. There are grounds to suppose that the trigger mechanism of gravitational collapse, resulting in appearance of black hole, is space geometry. This question will be considered additionally.

Trigonometric dependences of corner of scale

We will consider trigonometric dependences of corner of scale in two rectangular triangles - at the increase of scale and at diminishing of scale.

Triangles of corner of scale. Mathematics for blondes. Nikolay Khyzhnjak.
Trigonometric correlations of parties of the got rectangular triangles for diminishing and increase of scale we will take in a table.

Trigonometric dependences of corner of scale. Nikolay Khyzhnjak. Mathematics for blondes.
The got results can be compared to the relativistic radical from the theory of relativity of Einstein. Radicals in these equalizations very look like a sine and secant of diminishing of corner of scale. If to execute not difficult transformations, it is possible to get unit of measurement.

Mathematical transformations. Mathematics for blondes. Nikolay Khyzhnjak.
The conducted transformations specify on that natural unit of measurement of speeds is speed of light. There are no grounds to suppose that there is a change of trigonometric dependences at passing through the point of symmetry of coefficient of scale. On it, it is possible to suppose that for faster-than-light speeds a relativistic radical takes on next values.

Theory of relativity of Einstein for faster-than-light speeds. Nikolay Khyzhnjak. Mathematics for blondes.

On the basis of foregoing it is possible to suggest a next hypothesis about the structure of our Universe. Speed of set is a natural barrier separating slower-than-light part of Universe from faster-than-light part. Slower-than-light part of Universe we have possibility to look after. By virtue of specific properties of velocity of light, faster-than-light part of Universe can not be observed directly. It is possible to suppose that velocity of light is the axis of symmetry of distribution of substance in Universe. Dark matter and dark energy, that render affecting our part of Universe, can be in faster-than-light part of Universe.

If in Universe there are highly developed reasonable civilizations that captured faster-than-light technologies, then for an information transfer they will use not hertzian waves possessing velocity of light, and hard carriers of information on the basis of dark matter, transmissible with speeds, considerably excelling velocity of light.

If to equate trigonometric dependences of corner of scale with the values of trigonometric functions at 90° (it 1; 0 and 1/0), then for diminishing of scale they are taken to equality 0=1, for the increase of scale - to equality of k=0.

Our Universe has three limitations. In space by the border of universe явля-ется area, where velocity of light equals a zero. Outwardly our Universe is a point in space. Mathematical equalization of universe is equality 0=1 - any physical quantity with the unit of measurement in the scales of Universe equals a zero. This law of maintenance is confirmed by some researches of physicists, in particular, about it talked in the lecture of Andrei Linde.

Except spatial limitation, I exist limitations of speed. In slower-than-light part of Universe it is expressed in limitation long - distance between two positions of any point of space in time can not equal a zero. In physics this limitation it is accepted to name the absolute pitch of temperature. Faster-than-light part of Universe has limitation at times - time between two positions of any point in space can not equal a zero. The instantaneous moving in space without moving in time is impossible. Geometrically it can be expressed so: projection of speed on length and for a time can not equal a zero. Implementation of these terms is provided by the presence of rotation on the most different levels: atomic, planetary, galactic. It is possible to suppose that our Universe is similarly revolved in space.

More exact idea about principles of existence of Universe it is possible to get after the detailed study velocities of light as a physical process. For basis it is possible to accept position that velocity of light in our Universe is the result of co-operation of three dimension length with three dimension of time. Mathematically this co-operation is described by an multiplication. Physically our Universe has six dimension - three dimension of length and three dimension of time.

Expl for blondes: Now time to draw our Universe to look, what place she occupies in the space evolution.


Change of corner of scale

The increase of corner of scale can be presented as a change of quantity at unchanging unit of measurement. Diminishing of corner of scale can be presented as a change of unit of measurement at an unchanging quantity. Geometrically in the system of rectangular triangle it will look like the following.

Change of corner of scale. New Math. Mathematics for blondes. Nikolay Khyzhnjak.
An increase and diminishing of scale in the identical amount of one times correspond to one value of corner of scale. On this property of corners of scale trigonometric dependences are based in a rectangular triangle.

From the point of view of mathematical result does not matter, as a change of quantity is described in relation to unit of measurement. A variable quantity at permanent unit of measurement and variable unit of measurement at a permanent quantity will give the identical value of corner of scale.

In a general view the change of scale of quantity can be presented as a turn of unit of measurement on the size of corner of scale. An increase or diminishing of scale depends on that, what position of unit of measurement is taken for basis at comparison and from the type of projection. The increase of scale can be presented as a radial projection on a number ray, perpendicular to the unit of measurement taken for basis at comparison. Diminishing of scale can be presented as a perpendicular projection of the compared unit of measurement on the unit of measurement taken for basis.

Turn of unit of measurement on the corner of scale. Mathematics for blondes. New Math. Nikolay Khyzhnjak.
Expl for blondes: A boring entry is farther closed and interesting begins are trigonometric dependences of corner of scale.


Quantity as basis of mathematics

Co-operation of numbers and units of measurements takes place in a point "unit" and expressed by a mathematical action by an multiplication. Geometrically unit of measurement is perpendicular to the numerical ray. The result of multiplication of numbers on unit of measurement in future will be named "quantity". All quantities are identical mathematical characteristics initially.

Geometrical image of any quantity. Co-operation of numbers and units of measurements. Geometrically unit of measurement is perpendicular to the numerical ray. Mathematics for blondes.
All units of measurements in the surrounding us world it is possible to depict two methods: with a general point "unit" and with a general point "zero". The method of image does not influence on properties of making elements.

If to take unit for a general point, then this will be a circumference with a numerical ray going out the center of circumference. The radiuses of circumference will be units of measurements.

Geometrical image of quantities with a general unit. Represent any universe with all present in her units of measurements foto. Portrait of mathematics. Mathematics for blondes.
Approximately it is so possible to represent any universe with all present in her units of measurements. The image of all units of measurements as radiuses of circumference underlines a that circumstance, that all units of measurements are identical mathematical characteristics initially. (Expl for blondes: And you does not it seem to that this portrait of mathematics very reminds the ancient invention of man - wheel? Are you exactly sure that a wheel was invented exactly by a man? Maybe, did someone try to explain to the man, what mathematics, but he so nothing and did not understand? In memory of meeting with the unknown teachers of mathematics history left us only a wheel... Why did unknown teachers begin the story exactly with it? Because not knowing and not understanding such elementary things, understanding to the mathematician is practically impossible.That the previous generations of mathematicians were brilliantly demonstrated us.)

Geometrical image of quantities with a general zero. The moment of appearance of universe, that it is accepted to name Big Bang. Mathematics for blondes.
Approximately it is so possible to represent the moment of appearance of universe, that it is accepted to name "Big Bang". In this case a numerical ray graphicly can be presented as a numerical cone.

In an algebraic kind any quantity can be presented by multiplying of coefficient of scale by unit of measurement. Numbers come forward as a coefficient of scale.

Equalization of quantity. Any quantity can be presented by multiplying of coefficient of scale by unit of measurement. Mathematics for blondes.
Geometrically any quantity that is the result of multiplication of coefficient of scale on unit of measurement, it is possible to present as a hypotenuse of rectangular triangle the cathetuses of that are unit of measurement and part of numerical ray.

Geometrical image of any quantity of kind ka. Mathematics for blondes.
If the coefficient of scale is equal to unit, then a size is equal to unit of measurement.

Expl for blondes: Here now we got to one of types of mathematical corners we will consider and farther, what corner of scale and as there is a change of corner of scale.


Units of measurements and mathematical actions

Symmetry of addition and deduction in relation to a point a zero testifies that these mathematical actions can be executed only with one unit of measurement. Actually, addition and deduction reflect comparison of three numbers - two present and result. For different unitsof measurements, getting the result of these mathematical actions not maybe, as numbers have different warrants, and their comparison is not possible. The geometrical mapping of addition and deduction will be considered additionally.

Symmetry of multiplication and division in relation to a point "unit" allow to present dividing as multiplying by a number reverse to any number:

а : b = a x 1/b

Just, multiplying by a number reverse to any number, it is possible to present as dividing by any number:

а х 1/b = a : b

Traditional determination of prime fractional number as a result of division of two integers of p and q interchangebly to the result of increase of integer of p on a number reverse to the integer of q:

p : q = p x 1/q

In further exposition term a "multiplication" will imply an increase and division in the generally accepted sense because of their complete symmetry and relativity of these concepts.

Multiplication is this co-operation of two different units of measurements at right angles in a point "zero". As a result of co-operation new unit of measurement appears with beginning in a point "zero", that causes the quality change of interactive units of measurements. A mathematical action opposite on sense to the multiplication is decomposition on factors. Decomposition is executed through trigonometric functions that can have numerical and not numerical (0 and 1/0) values. Simplest similarity of decomposition under a corner in 45 degrees - this square root. Decomposition and trigonometric functions are more detailed will be considered additionally.

An area (for example, area of rectangle) is a result of co-operation of two perpendicular units of measurements of length. The multiplication of parallel units of measurements is not possible (at the multiplication of lengths of two parallel parties of rectangle, measured in meters, it is possible to get meters square, but it is impossible to get an area). Mathematical properties of units of measurements will be considered additionally.

As in mathematics it is accepted to distinguish the separate sets of numbers that is partly included in a concept "Any number", it is at a desire possible to set forth mathematically exact determinations for some from them. For example:

unit and all numbers that can be got addition of units are named natural;

all numbers that can be got addition or deduction of units are named integers (at deduction of the same amount of units, that is present, numbers apply in a zero);

numbers being not whole are named a fractional.

Expl for blondes: Now a turn came to look, as numbers and units co-operate in mathematics. This piece I named quantity.


Relativity of concept is "Any Number"

For the receipt of numerical axis does not matter, what from numbers are taken for any number: positive anymore units, positive less than units negative anymore minus units or negative less than minus units. Imposition reverse and mirror symmetries on any of these groups of numbers results in the receipt of all row of the real numbers.

From the choice of group of numbers as any number the results of mathematical actions will depend are different combinations of increase or diminishing of any number as a result of concrete mathematical action. In a table below the possible variants of concept "Any number" are marked just as there are corresponding to them fragments of numerical axis in traditionally assumed an air. For evidentness the increase of any number is doubled by a sign "+", diminishing - by the sign of "-", corresponding cells are distinguished by a different color.

Relativity of concept is Any Number. Mathematics for blondes.
As be obvious from a table, addition and deduction are mirror symmetric in relation to a point "zero". An increase and division are mirror symmetric relatively two points are points "unit" and points "zero", here reverse symmetry is mirror symmetric in relation to a point "zero". All reasoning about priority and secondaryness of mathematical actions are an error. Symmetry of mathematical actions is considered in the separate article.

Expl for blondes: Farther we will consider units of measurements and mathematical actions.


Number line

In mathematics it is accepted to represent numbers as a number line. We will consider transformation of number ray to the number line.

Reverse symmetry allows to get numbers less unit. As a point of reverse symmetry is unit, this symmetry does not depend on units of measurement. Reverse symmetry reflects relativity of concepts "greater than unit" and "less than unit" at comparison of two numbers. In case of comparison of two numbers without fail it is needed to accept one of these numbers as unit of measurement.

After introduction of unit of measurement we get the absolute system of coordinates for any unit of measurement. Unit of measurement on a picture is represented in the traditionally accepted variant - with imposition on the area of reverse numbers. At imposition of mirror symmetry the point of that is a zero, we enter negative numbers and get the relative system of coordinates. All enumerated transformations are represented on a picture below, where the sign of endlessness is mark any number.

Number line. Number ray. Mathematics For Blondes.
Expl for blondes: Farther we will consider relativity of concept "Any number".


Relativity is in mathematics

All distinctions between two numbers or two units of measurements come to light only at comparison of two numbers or two units of measurements. All results of comparison are relative, as depend on what from two elements takes up basis at comparison. Relativity of results of comparison is represented different kind by symmetries. If up basis of symmetry a point takes "zero", then mirror symmetry ensues. If up basis of symmetry a point takes "unit" - reverse symmetry ensues. For units of measurements of corners reverse symmetry is transformed in perpendicular symmetry that is possessed by the values of trigonometric functions.

All distinctions between two numbers or two units come to light only at comparison of two numbers or two units. All results Comparison of two any numbers is not possible without the presence of the general founding unit comes forward as that. For the location of any numbers in order of growth in modern mathematics as unit of numbers the number systems are used: binary, ternary, octal, decimal, sexadecimal et al. Comparing of two numbers to the different grounds is not possible without bringing them over to the general founding.

Comparison of two numbers at different units of measurements becomes possible at the use of the third unit of measurement - one of the number systems, for example, decimal.

Result of comparison of two numbers is described by concepts "greater than" and "less than". Relativity of comparison of numbers is expressed in that the result of comparison depends on that, what number takes up basis at comparison. For example, if to compare numbers 2 and 3, we will get two results:

2 less than 3
3 greater than 2

On the first place it is accepted to write down a number that takes up basis at comparison, on the second is a that number it is compared to that. The results of comparison possess property of mirror symmetry - at the change of founding a result changes on opposite. The point of mirror symmetry is equality of two compared numbers. The results of comparison of two numbers are analogical to the relative system of coordinates:

less than - equal - greater than
minus - zero - plus

Comparison of two units of measurements is possible at presence of general point "zero". The result of comparison of two different units of measurements can be a conclusion about perpendicularity or parallelism of these units of measurements. Parallelism or perpendicularity of one unit of measurement in relation to other is concepts relative.

Expl for blondes: we will examine farther, as well as where a Number line appeared from.

Some concepts of mathematics are continuation

A point is this reserved space with the radius of curvature equal to the zero (Expl for blondes: I do not understand clearly, that means this phrase. But I know exactly, that she is correct and very useful for us, when we will begin to examine mathematical principles of teleportation. To ride on an own car even prestige - it not so already prestige. In fact you however will escape farther than this planet. And here with you, even with blondes, the same, that happened to the dinosaurs will happen sooner or later - nature you will kill. Where will you escape from a submarine boat, even if this boat measuring with a planet?). Any space consists of endless amount of points. Through any point of space it is possible to conduct an endless amount mutually perpendicular lines. All points of space possess properties of both zero and unit, that allows without difficulty and arbitrarily to impose any relative system of coordinates and apply any type of symmetry in any point of space. In any point of space equality is executed: a zero is equal to unit. Equalization of point 0 = 1. Properties of zero and unit for one point of space can not show up simultaneously in one system of coordinates.

A line is this open-space with the radius of curvature equal to unit divided by a zero, consisting of separate points.

In mathematics it is necessary to distinguish the next types of corners : corner of scale, trigonometric corner, corner of turn.

Corner of scale is a corner scope from 0 to 90 degrees. The corner of scale can equal a zero, but 90 degrees can not equal. This corner reflects quantitative changes within the limits of one unit of measurement. Any changes of corner of scale can not cause the quality change of unit of measurement.

A trigonometric corner is a corner scope from 0 to 90 degrees. A trigonometric corner can equal both a zero and 90 degrees. This corner reflects dependences between units of measurements (project properties of space) and condition of quality changes of units of measurements. Dependence of quantitative changes of units of measurements on a trigonometric corner is expressed by trigonometric functions. The quality changes of units take place at the values of trigonometric functions equal to the zero and unit divided by a zero.

A corner of rotation is a corner that can have any values. In a range from 0 to 90 degrees the corner of rotation numeral can coincide with a trigonometric corner or corner of scale. The corner of rotation reflects the circular moving without the change of quantitative or quality descriptions of unit.

A direct corner differs from all other corners that the mutual projection of two intersecting lines is a point. For all other values of corner the projection of one line on other is a line. At coal equal to the zero, lines coincide. Mathematical unit of corners is a corner equal 45 degrees. This unit of corners submits to the rules of the binary number system.

Expl for blondes: Thereon the set of clever mathematical words is closed and we pass to consideration of mathematical mechanism - that, as and why works in mathematics. We will begin our excursion with relativity in mathematics.


Some concepts of mathematics

Expl for blondes: "Some concepts of mathematics" are my crib on your own, darling. Deciding some mathematical question, I often had to remember the own non-standard decisions of other questions. What is long not to dig in the memory, I collected all most important moments at the beginning of cycle of reasons of "Bases of mathematics". Some articles from this cycle will bring us over to the conclusions that is already writtenin here.

Equal sign reflects dependence of causality in the surrounding world. (Expl for blondes: is the example of application of the first basic axiom of mathematics.) If 2 х 2 = 4, it not nearly means that 4 = 2 х 2. There is an endless great number of decisions resulting in an exactly such result - four. 2 х 2 - only one of these decisions.

In mathematics there are three basic equalities:

0 = 0
1 = 1
0 = 1

All physical laws and mathematical equalizations are taken to one of these equalities. (Expl for blondes: do you think why I so easily succeeded to find the decision of the undecided equalizations (in Rassian)? Because I beforehand know an answer - the decision of any mathematical equalization is taken to one of these equalities. If know a problem specification and right answer, decision to find much simpler. By the way, here one of remarks of physicists, about that I wrote in the article "Zero is equal to unit": "Sum of energy of substance and gravitational energy is saved, but this law of maintenance is unusual: this sum is equal to ZERO"!. Most strikes me circumstance that physicists result in mathematics, as old jade! But it must be quite the reverse - it mathematicians must explain to the physicists: that, as and why works in this world. Physics is an experimental base of mathematics. If physicists will find some exceptions from mathematical rules, means to the mathematicians it will be needed to correct mathematics.)

In mathematics it is possible to distinguish such basic elements: zero, unit, any number and unit of measurement.

Numbers reflect quantitative description anything. Any number is equal to any number - this property of numbers allows to distinguish them in the special group that it is accepted to designate a word "number". All separately taken numbers possess identical mathematical properties. (Expl for blondes: not surprised, most bad dream of any mathematician (all numbers are equal) is cruel mathematical reality. Do not be afraid, I do not gather to take from mathematicians their favourite toy that are numbers. I simply want to say an obvious thing: all of you know many most different toys (and child, and adults), but all unites them one property - it is possible to play by them.)

Any number is the positive real number more unit. If to one any number to add other any number the first number will increase. Just any number will increase at multiplying of him by other any number. If from any number to subtract other any number, the first number will diminish. If to divide one any number into other any number, the first number will diminish.

Unit is a number, but is not any number, as at multiplying and dividing by unit any number remains unchanging. Unit is a neutral element at an increase and division.

Geometrically any number is represented by a point. All numbers form a numerical ray with beginning in a point "unit". A numerical ray does not have an end. Any number can be designated by a sign "infinity", as any number can be how pleasingly great.

Units of measurement reflect quality description anything. Any unit of measurement is equal to any unit of measurement. All units of measurement possess identical mathematical properties. (Expl for blondes: In mathematics units of measurement symmetric to the numbers, will remember an axiom about symmetry.) For numbers universal units of measurement are the number systems: binary, decimal, sexadecimal to and other. (Expl for blondes: I think, for mathematicians it will be the real discovery. I in any way can not get used to that any writtenin number has a tail of unit of measurement is "abstract unit".)

Geometrically any unit of measurement is represented by a segment (by two points): point "zero" is this beginning of unit of measurement, point "unit" is an end of unit of measurement.

Zero is not a number, as at addition of zero to any number and deduction of zero from any number this number remains unchanging. (Expl for blondes: It there is that simple and elegant decision of problem with zero about that I talked before. I agree, it is another act of mocking above mathematical sacred objects. But, there be nothing to be done - beauty of mathematics requires victims. you only present, how many energy and paper we will economize, if we will not in every example on a division write a "denominator does not equal a zero". Environmentalists will be happy here!) Zero is a neutral element at addition and deduction. Zero is beginning of the absolute system of coordinates. In the relative system of coordinates zero is the point of mirror symmetry.

In a point "unit" takes place connection of unit of measurement with any numbers. Unit is the point of reverse symmetry in the absolute and relative systems of coordinates.

Expl for blondes: It is a yet not end. Tomorrow we will continue to examine some mathematical concepts.


Basic axioms of mathematics

Mathematics is laws there is the surrounding world on that. The laws of mathematics are identical for any universes with any amount of measuring.

Mathematics - it governed without exceptions. If an exception appears in a mathematical rule - this rule must be changed. This statement is the universal formula of the scientific discovery in mathematics.

Mathematics is abstraction. The abstract of mathematics consists in that the laws of mathematics operate always and everywhere identically.

Mathematics is the closed system. If a correct mathematical result is got, then there is an infinite amount of ways resulting in an exactly such result.

Mathematics is symmetry. Absolute symmetry in mathematics is a limit of development of mathematics as sciences.

Mathematics is relativity. Positive and negative numbers do not exist in the wild. Positive and negative numbers are this reflection of our personal opinion in mathematics. A negative number is a sign of the relative system of coordinates, position of that depends exceptionally on our choice of her center. A the same point can have different signs and different numerical values in the different relative systems of coordinates.

Mathematics is basis of commonunication and mutual understanding of reasonable creatures from different civilizations. Geometry translators does not need. Mathematics is closed wherein human logic begins.

Expl for blondes: in more detail we will consider each of these axioms a bit later, and while we will continue an acquaintance with mathematics and will look at some concepts that will be used in future.


Bases of mathematics

Bases of mathematics are a cycle of my reasons. Main task of "Bases of mathematics" - to complement mathematics the absent fragments of mathematical knowledge and set intercommunications between some copy-book maxims already known to us.

Most useful innovations in mathematics will be units of measurement and divizion by zero. Clear that to explain it it will be not simply. For understanding will be thoroughly to understand some generally accepted mathematical concepts to set that there is a true in them, and that is lie.

Why are units of measurement needed in mathematics? Here imagine such situation. Does a child go near you, hands to you an object asks: "That will happen, if to drop this object?" Using principles of modern mathematics, you need to take the list of all great numbers of objects to find, to what great number from existing this object belongs. If this great number of the broken up objects, then this concrete object will be broken up. If this great number of jumpings up objects, then this object will jump up. In the existent lists of great numbers of objects you will have to be long and boring dug, before you will be able to find an answer for a question.

Is it possible to decide a problem simpler? It is possible. If by sight to determine material out of that an object is made and to know properties of this material - then no problems. A glass object will be broken up, a rubber ball will jump up, a ferrous ring will do "drin" and jelly will do "tuff".

Just in mathematics there is business with units of measurement. If you know mathematical properties of unit of measurement, you will say without effort, that can be expected and what it is impossible from a that physical parameter that this unit of measurement belongs to.

Introduction to mathematics of units of measurement as a mathematical element equivalent to the numbers, allows to determine mathematical methods many fundamental properties of the surrounding world.

If you think that as a result of such innovations of mathematician will become yet tangled, you wrong. Mathematics will be simpler, more slender, clearer. Look at the basic axioms of mathematics.


Wunderkindes and cretin with blondes

In comments to my report omeone said of the opinion: "It is the article from heading "Cretins write for blondes", probably. Correct, please". As an author of question felt free in expressions, I also will call a spade a spade and on me for it I ask not offended.

Really, normal such will not be written. At most, what the normal are capable on, so it is dull to teach someone once written. And than better they it is quoted then, the are more clever considered. Both religion and science sticks to thereon. Dull untalented mediocrity which considers itself a norm and which is managed by more sharp untalented to mediocrity turns out in the total. Any bureaucratic vehicle consists of them, from normal. All, who though by something from a norm differs, are considered fools and fools. What, I agree to be a fool or cretin which writes for fools and blondes.

I would not begin to watch out for this comment, if he was not the object-lesson of other problem. One scientific site on which solid scientists communicate and one of them formulated such question was here remembered me: "MANY CLEVER COLLECTIONS of TASKS And GUYS WHICH DECIDE THEM. WHY THEN SO FEW DISCOVERIS"?. Farther a few quotations are for illustration of problem:

I was always surprised by one circumstance. When look tasks which are offered in our collections of tasks on physics and on mathematics for high school and institute of higher, on school olympiads, at entering university and so ддалее, then there is the impression, that they are counted on supermen. In any event, suppose the very high level of possession material. And some tasks in a mathematical magazine "Quantum" - it in general, to my mind, whole research, counted on the experienced specialist. Moreover, appears, there are guys (and I personally by a sign not with one!) which all these the tasks decide easily.

Certainly, we have very clever and capable young people. Why then, if do we decide intricate problems so easily, we so few accomplish discoveris?

... By the way, the same phenomenon I look after in the West. Take in hands collection of tasks on physics for the graduate students of Massachusetts Institute of Technology or collection of tasks on a gravitation and theory of relativity edited from Saul Teukolsky. They on someone are counted. Them someone decides. But where is discoveris equivalent to the tasks, published in books?

It is a problem, lying inplane division of people on fools and normal. First let us understand with the decision of tasks. What task? It by someone the made set of basic data and question on which it is needed to give an answer. Problem definition supposes application already of the known method of decision. For the decision of task it is enough to learn material and apply the got knowledge. An ordinary calculator turns out. What quicker such calculator decides tasks, the he is considered more clever. There are even wunderkindes which tasks decide - as nuts break. All this system of tasks and decisions differs small what from the system of training of animals. For a wunderkind-calculator, instead of command, it is enough to formulate a task, id est to say what where it is needed to find.

All of it results in that at taught standard character of thought is produced. Standard tasks decide standard methods easily and simply. It is needed only to deviate from standards - problems begin here. Let us try to decide a task, which it is scientifically well-proven for, that this task can not be decided. Who will engage in the decision of such task? Only fools. Normal, and the more so clever, never such task to decide will not become. They know an answer, by someone once written: "Task does not have a decision". Will fools be able to decide this task? Improbably, because and clever, and fools use identical standard principles of decision. Who can decide a similar task? That, whoever knows that this task does not have a decision and whoever uses standard principles of decision. Quite naturally, what a decision will be acknowledged as a discoveri.

Here historical fact. When the American mathematician George Dantzig was the student of university, then was once late to a lesson and counted up the equalizations written on a board a homework. Equalizations seemed to him more difficult, than set usually, but in a few days he did a homework however. Appeared, that these were tasks on statistics, on the decision of that many scientists worked and that at that time were considered "undecided".

And now we will go back to fools-blondes. All consider them fools because:

at first, their character of thought differs from standard;
secondly, they memorize standards badly.

Based on aforesaid, I have all grounds to assert that one blonde has many more chances to do a discoveri, what at all wunderkind-calculators, together taken. You will memorize: among normal there are not genii.

Why is the factorial of zero equal to unit?

A factorial appeared in mathematics near 1800. In mathematics a factorial is name work of all natural numbers, including indicated. Designate a factorial an exclamation mark, written after a number.

5! = 1 х 2 х 3 х 4 х 5 = 120

The official version of appearance of factorial in mathematics I do not want to search, because I know perfectly, how it was actually. And there was all so.

Completing all letups on a theory and practice of factorial calculations, Scientist bore the creation on the court of Saint Mathematical Inquisition. Functions of supervision after mathematicians on behalf of Saint Mathematical Inquisition in this locality executed Saint Scientific Infirmities. Responsibility on him lay enormous, work was very much, more precisely, to do it was in general nothing. Therefore Saint Scientific Infirmities selflessly dug up in a nose. After this employment he was found by Scientist.

Executed in due form bureaucratic art, a folder with the theory of factorials lay down on feet before the eyes of Saint Scientific Infirmities. Infirmities were taken out by a finger from a nose, fastidiously made a wry face and began the same finger to leaf a folder. He checked content of folder for accordance to "Law on registration of papers, presented to Saint Mathematical Inquisition". Obvious occasion to say no to Scientist in consideration of his papers was not. To enormous regret of Saint Scientific Infirmities. Vexation affected his official. An at this time leading hand reached the form of Official Statement, Book of Registration of Visitors, Book of Registration of Incoming Documents, Book of Registration of Get-away Visitors and other treasures of Responsible Leading already.

Carefully collating everything, that was written by Scientist, with the "Explanatory dictionary for muddle-headed Bureaucrats. Rules of writing of Words and Letters" (this masterpiece was hidden from the eyes of visitors under a vulture "For the official use"), said Saint Scientific Infirmities:

"Your materials will be considered in the term set by Law".

His leading finger again submerged in a nose, continuing the interrupted work. It meant the end of audience.

The time of consideration of document fixed by law passed. A scientist again came to Saint Scientific Infirmities. Discontentedly made a wry face infirmity, scratched the back of head, feeding speech goes to remember about what, then got a folder with a factorial and began attentively it to study. An answer he must give today because the time taken on bureaucratic procedure made off already. Twisted infirmities backward, checking, firmly how enough Leading Arm-chair sticks to under them. Leading Arm-chair squeaked treacherously.

"You, Scientist, must know that a zero is a natural number", - said Infirmities and with relief breathed, - "All your factorials will equal a zero. On determination."

I want to remind readers, that events took place on a wild west, where a zero all consider a natural number.

"Your work very interesting. It will be really it is sorry me, if she will remain known to nobody", - continued Infirmities - "I would assist to the publication of your work, if you will add me in coauthors."

It is a widespread in science reception. Through him hacks do to itself a career in science. Suggestion of Infirmity did not surprise Scientist. He answered:

"I will be happy to be the coauthor of such prominent scientist, as you. But how to be with a zero?"

"I see no problems," - Infirmities demonstrated a complacency and grandeur, - "It is said In Saint Mathematical Limning, that any number, increased on a zero, equals a zero. But in this Limning there is not a single word about the factorial of zero. I will write an address to Saint Mathematical Inquisition with a request making alteration in text of Saint Mathematical Limning. Let them write, that the factorial of zero is equal to unit."

An agreement was attained. A scientist here entered a coauthor in the work. Wrote saint Scientific Infirmities appeal. This advanced study was sent for consideration of higher scientific leaders.

Higher scientific leaders knew the rules of bureaucratic games well. The name of Scientist they did not touch. The name of the inferior every higher scientific leader wiped and inscribed the name into place of coauthor. As a result of it a hole appeared in the coauthors of Scientist.

From the same pores in Saint Mathematical Limning there is a masterpiece of scientific thought: Gospel says of from Rules of Increase, that a zero, increased on unit, will be equal to the zero

0 х 1 = 0

A Gospel asserts from Factorial, that a zero, increased on unit, is equal to unit

0 х 1 = 1

Whew mathematics grows into a marasmus.


Legend about Sine and Cosine (completion)

In decision of Mathematics, Sine and Cosine appeared forever bound in a direct corner. Exactly from a that mournful day, Day of Sentence, Sine and Cosine live on different parties of one direct corner. Unit untiringly watches after execution of sentence. Where was not one of brothers-twins, the second always will be in the distance, equal to Unit. If Sine will consider itself equal to Unit, Cosine grows into a zero. If Cosine becomes equal to Unit, Sine grows into a zero. Such is Sentence of Mathematics. In the different worlds he is named variously, Gods of him call "Sorrow of Sentence". In your world him it is accepted to name "Theorem of Pythagoras".

From the same pores Sine and Cosine never met and will no longer meet. There will not be more disputes between them about that, who of them main. Mathematics deprived them even rights to communicate inter se. As a mediator at the relations of Sine and Cosine Reverse Symmetry was appointed - for the unsurpassed talent of diplomat. When she socializes with Sine, then appears Tangent. She assures Sine, that he is more main, as is in a numerator, and Cosine - in the denominator of shot. When Reverse Symmetry socializes with Cosine, she sets up for Cotangent and assures Cosine, that he is more main, as takes seat in a numerator, and somewhere there, in to the bottom, in a denominator, there is Sine. After such socializing with Reverse Symmetry both Sine and Cosine feel incredibly happy, in fact the cherished dream of each came true of them - to be main.

For socializing with the surrounding world Sine and Cosine have a single circle. Any persons interested can pass from this circle either to Cosine or to Sine. They are always very glad to the guests and certainly the Sacred Books show to them, each it. Yes, exactly that book which it is written in, that he - main. With care wrapped in shreds, these books are kept each of them, as the most expensive relicts.

But rarely who of Gods dare glance in dwelling of Sine or Cosine. Gods know how jealously these two mites watch after guests, especially after those guests which go not to them. A guest failed to appear in what place of single circle, Sine knows exactly, what distance a guest will pass to Cosine. Cosine knows just, what distance dissociates a guest from Sine. Therefore swingeing majority of guests straight from a single circle ask interesting them distance and disappear on the businesses.

Fuss of single circle is almost continuous. Here always many visitors, not giving to miss to Sine and Cosine. But are in their life such minutes, on which even Gods try not to look...

In rare periods of calm, when the last visitor disappears from a single circle, Sine and Cosine go out to the corner in forty five degrees, again becoming indistinguishable. They turn the little persons to each other, trying to look over though something through endless distance of Unit... Almost not visible tears flow down On their cheeks... A Sine and Cosine with an inexpressible melancholy remember that distant and happy time. Time, when they were together...