Showing posts with label interestingly. Show all posts
Showing posts with label interestingly. Show all posts


Squaring the circle

Squaring the circle. The solution of the problem. Almost accurate. Math draw.

Squaring the circle. The solution of the problem. Almost accurate. Mathematics For Blondes.
Squaring the circle. The solution of the problem. Almost accurate.


This construction is not a solution to the problem of squaring the circle, but it forced me to get my hands on the math and do all the verification calculations. If the correct solution should give a segment with a length of 1.77245 radii, then the length of the red segment in the figure is 1.78885 radii. A little more than necessary. How much more? The segment exceeds the required length by 0.9253% or by 0.01640 of the radius. The area of a square with sides equal to the length of this segment is 1.8592% greater than the area of a circle.

How to use it? For a friendly prank of math lovers. Build a square with sides equal to ten centimeters. Inscribe in a square a circle with a radius of 5 centimeters. Draw a line as shown in the figure and offer a ruler to measure the length of the resulting segment. It will be equal to 8.9 centimeters. Using a calculator, calculate the area of a circle, which is equal to 78.54 square centimeters. The area of a square with sides of 8.9 centimeters is 79.21 square centimeters. This discrepancy can be explained by the low accuracy of measurements using a ruler. Personally, after such rough calculations, I sat down to check the solution.

I don't recommend this kind of joke with my teachers or math teachers - they can force you to do all the verification calculations to refute this solution. The unsolvability of this problem was proved by the German mathematician Lindemann back in 1882. However, I have long been very distrustful of many proofs of mathematicians.


Hilbert's sixth problem

Here I have in the comments appeared one modest request:

Help solve the problem associated with the extend of Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field.

Such a number of clever words in one sentence, I have not met. I immediately suspected that not all of the school textbook is taken, and from some other place. Type in the search phrase Wikipedia "Kronecker–Weber theorem" and look at the result. That's right, I slipped twelfth problem of Hilbert, has not been solved so far. Suddenly her uncle decides - all would be funny!

You should not expect. For me, the phrase "abelian extensions", "the rational numbers to any base number field" is no more than a collection of letters. In higher mathematics, I did not go through dog training and have no idea what to do, when he heard these phrases. Moreover, I do not even have a clue what the root of the unit n-th degree is different from the unit itself. In mathematics, the same thing can have different names, the same name can mean a variety of things. Hence, problems arise.

More than a hundred years ago David Hilbert formulated twenty-three problems of mathematics. Some problems have been resolved, some partially solved two problems remain unsolved to this day. There are also a number of problems that are simply hushed for clarity: "too vague" or "requires clarification of the phrase". Among these "jammed" caught my attention problems Hilbert's sixth problem, which is: Mathematical treatment of the axioms of physics. Mathematics jumped from the solution of this problem with the phrase "too vague". This is where I disagree with them.

Formulation of Hilbert very clearly, that's just to express nothing - not taken root in physics ideas about mathematics axioms, as his time in the chemistry did not take the idea of negative numbers. This mathematics all their theories verify with the axioms of physics as his theories verify with the experimental results. There physics postulates, but it's just a temporary patch on the white spots of our knowledge. Sooner or later, the postulates are replaced by physical laws. Temporary postulates physicists do not go to any comparison with the monumental firmness axioms of mathematics.

Here and there is a very sudden decision Hilbert's sixth problem - the language of mathematics is much easier to state "axioms" of religion than the axioms of physics. Looks mathematical presentation of the fundamentals of religion something like this:

- Sacred texts in religion - axioms and definitions in mathematics;

- All that is in this world created by God - in mathematics, the phrase "Let us given ..." by default assumes that everything God gives us;

- The story of Noah's ark and the "pair of every creature" - set theory;

- Man is composed of body and soul - complex numbers consist of real and imaginary parts;

- Kingdom of God - a complex space;

- God and the devil, good and evil - the positive and negative numbers;

- Holy Cross - Cartesian coordinate system.

If desired, you can thoroughly examine the sacred texts of religion and the sacred texts of Mathematicians (axioms and definitions) in search of other matches. In the language of religion, math is pretty good presents.

In my opinion, the problems in modern mathematics and believers alike - autism. They live in an imaginary world and not pay attention to their surroundings. Teaching of mathematics is very similar to the missionary preaching - we need to learn and do what they say preachers. All attempts to draw attention to the preachers of the surrounding reality ends are being sent to the sacred texts: "Read the Bible", "Read the definition."



Scientists say that to the complete extinction of dinosaurs led to the fall of the Earth asteroid the size of 10 kilometers. For the complete extinction of mankind only one idiot.

If you doubt that, let's look at one recent event through the lens of mathematics. The fact that we use mathematics nobody ever taught, except for those tasks that we face pose. In the analysis of the events I am using mathematical tricks such as symmetry, equality, like ... We strongly do not recommend using such tricks when dealing with others.

People with sick ego and nervous to read is strictly prohibited!

October 1, 2013 the U.S. government system was in a state of coma. The U.S. Congress has not passed the budget. About 800 000 (eight hundred thousand) for public officials sent on leave without pay. And it happened in a country that claims to be the world leader.

 I like to compare people with trained monkeys. I have to apologize to them - some monkeys may find the comparison offensive. The situation in the American system of government is much more similar to the behavior of cockroaches in the kitchen when you turn on the light. Here there are some very interesting questions that are usually no accepted.

Why not passed the U.S. budget? U.S. lawmakers failed to resolve the question of who would steal from the state budget. Budget of any state - is a feeding trough for locusts. No matter how much money is allocated, the state locusts will devour everything and say that it is not enough. The state budget deficit - a situation in which the appetite of locusts exceed the capabilities of state taxpayers. 

Why does the U.S. government employees stop working? Such legislation "the most democratic country." Very often, law-abiding reduced to absurdity: fools laws are written, others are fools them perform. This criminals can not obey the law, law-abiding rebel against legislative idiocy can crush a cockroach.

Who is responsible for the bombing of the world economy (remember, the United States - is the market leader)? Nobody. After all, legally no crime there.

Now try to compare the effects of legislative absurdity October 1, 2013 with the consequences of terrorist attacks September 11, 2001.

Terrorist attack and attack Congress
Terrorist attack and attack Congress
As a result of the terrorist attacks killed nearly 3,000 people. As a result of the struggle of Congressmen no one was killed. It is not lost.

The terrorists destroyed the World Trade Center and damaged the Pentagon building, destroyed planes and so on. Which will result in the absence of destruction of public servants in their jobs in public institutions?

After the terrorist attacks of U.S. Department of Defense, NASA, government agencies continue their work. After the attack, the congressmen have stopped their work, many U.S. government agencies. NASA has stopped working, even the site is closed. Two U.S. astronauts were in the position of Fools, which launched into orbit and abandoned to their fate.

NASA's not working
NASA's not working
  How many helicopters with military would send the U.S. president in the Capitol building for the " triumph of justice "? To kill a foreigner in his own bedroom without permission of the government - is the American president called justice. I'm not defending terrorists. I just want to pay special attention to the fact that, with respect to terrorists , the U.S. government did not comply with the laws as they did not comply with the terrorists against U.S. citizens. As they say, they fought for it and ran . And in relation to other criminals to apply the same principle - it is weak? There is a very interesting question : to kill the murderer - is a crime or punishment? It is clear that all criminals will require compliance with the law. But why laws must comply only we, the potential victims, and they are not the criminals?

Correctly or incorrectly similar comparison - decide for yourself. We all do not like it when our eyes are telling the truth.

Is it possible to avoid a similar fiscal situation in the future? Elementary. It suffices to apply the most basic mathematical operations. For example, division.

Let all civil servants continue their work, but the funding needed to carry out their work from their own pockets congressmen. The required amount of expenses divided by the number of members of Congress. The longer Congress will adopt the budget, the more money they lose. Congress will take a similar law? Never in my life. After all, it is we are fools, and the congressmen themselves feel smart.

You can adopt the budget on individual articles. Those budget items that have already been made, taking into law. Regardless of the decision remaining articles. Even if there are contentious issues in the work of the state apparatus it will not change.

You can automatically count the budget adopted last year. Let legislators then adjust individual articles if were not able to agree on a budget under consideration. 

You can allocate a budget for a number of articles that guarantee the smooth functioning of government, and that do not require annual approval. They can only be adjusted annually.

I am not an expert in the law of different countries, but these recipes will help any country to avoid U.S. budget insanity. 

Finally, the most interesting question: U.S. taxpayers with October 1, 2013 as well together do not pay taxes, as public servants do not go to work? If you continue to pay, then I do not understand anything in this madhouse called "The United States of America." By the way, I personally doubt very much that in my country the legal situation is better than the American.


Phlogiston theory and complex numbers

At the beginning of XVIII century the theory of phlogiston appeared in chemistry and explained the process of burning the presence of "a fiery substance" to combustibles. It was believed that when the material is burned, it disappears from the phlogiston. For its time it was the most advanced theory, the first theory in the history of chemistry, through which chemicals become a science.

In the development of this theory, there was such a paradox: in chemical experiments on calcination have found that scale weighs more than the metal itself. Instead, that would decrease the mass scale increases when the metal is released phlogiston. Here come to the aid of physics mathematics with its negative numbers. The logic was clear - if there are positive and negative numbers, so there is a positive and negative mass. The theory of phlogiston recorded that phlogiston has negative mass, which is confirmed by experiments - negative weight of phlogiston leaves the metal and enters the air, resulting in a positive mass scale increases.

Thanks to the work of Antoine Lavoisier was digging the role of oxygen in the combustion process and replaced the phlogiston theory came the oxygen theory of combustion. In the new theory, the increase in mass scale was explained in the positive numbers: the weight of the metal added weight of oxygen from the air, resulting in a mass scale increases. That's so simple and effective negative numbers were expelled from chemistry. Because of this, today we enjoy the benefits of chemistry in the form in which they exist.

But physics lurking different fate. Where did the complex numbers? From the same place there was a weight of phlogiston - of negative numbers. This is the common parent of the two theories are even conduct DNA analysis is not necessary. Why is this? To understand this, one must look at the history of mathematics.


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.


Why are sines and cosines needed?

Why are sines and cosines needed? Really, interesting question. In comments to the trigonometric circle of sines and cosines such question appeared:

where will sin and cos be useful in life?
p.s why are they needed sines cosines?

Let us will call a spade a spade. To swingeing majority from you they will be never useful. Unless, when will your children go out into school and will begin to study trigonometric functions, they too will put question you "Why are sines and cosines needed?" and, in addition, will ask to explain, what is it.

Money we use every day already not alone thousand years and perfectly we do without every sines, cosines and other elegant mathematical pieces. I assure you, and through millions of years in the count of money nothing will change. Not because we are such dull, and because such are mathematical properties of money : it is impossible to increase roubles on roubles and with money in the second degree to hurry in a motor show to buy "Lamborghini".

On a kitchen, in culinary recipes, you will meet neither sines nor cosines too. If to give a glance soberly on our everyday life, then all our everyday mathematics remains somewhere at the level of knowledge of Ancient Greece. We are enough with a head.

So why are sines and cosines needed? As compared to Ancient Greece, we have very much different pieces about that ancient Greeks could not dream even today. Even their Gods did not ride on machines, did not use mobile communication, did not communicate on the Internet. But we have all of it and we use this constantly. Did all this extraordinary riches undertake from where? He was created by us. At first scientists did the scientific opening. Then engineers, on the basis of done by the scientists of opening, created every useful things. We use these things today, not having not the least concept about that is into these things and what scientific laws are fixed in basis of their work. So, if there were not sines and cosines, there would not be all these useful things.

Sines and cosines are used most effectively scientists and engineers. I will not say that they continuously trigonometric functions are used only. No, they use them rarely, but well-aimed. Sines and cosines often are in the formulas of different calculations an engineer or scientific.

Often with sines and cosines it is necessary to clash to the geodesists. They have the special instruments for an exact goniometry. Through sines and cosines corners can be converted into lengths or coordinates of points on an earth surface.

The teachers of mathematics on the sort of the duties constantly deal with trigonometric functions. This year they told about sines and cosines to you, the next year the teachers of mathematics will tell the same to other students. Such for them work - to teach.

Schoolchildren and students study trigonometric functions on the lessons of mathematics. Personally I got through tortures sines and cosines at school, техникуме, institute.

Adults sometimes engage in sines and cosines then, when their toschoolchildren need a help at preparation of homeworks.

All! Sines and cosines do not need other generally! In everyday life most people they are not used hardly ever. If I wrong, remedy me.

So why then generally to teach these sines and cosines? Well, firstly, such is the school program. Secondly, if you in life may need apply a sine or cosine, you know already, what is it and where it is needed to search information about them. The knowledge gained at school will fully have you, what is independent in everything to understand.

So what such the sines, cosines and other trigonometric functions? It is a mathematical instrument it is needed that to be able to use. That we this instrument we do not use hardly ever, talks not that studying them is not necessary, and that efficiency of application of the knowledge gained by us is practically equal to the zero. But it is quite another theme already.


Mathematics forever!

Mathematics forever!

You can congratulate me, my first scientific publication went out in light. There are "The papers of independent authors" in a magazine, producing № 18, to the page 110 the little article is printed under the modest name "About symmetry of mathematical actions".

At the last time (in a blog in Russian language) I promised you to show that needs to be done in order that impossible in mathematics became possible. In this little article, only on two pages of text, it is shown, as a commutative law works at deduction and division. If someone wants to look, can pass on this page (in Russian language), there is reference for a flush-off. Nothing difficult in this article is present - the half of text is occupied by examples on addition, deduction, increase and division. All at the level of middle classes of school. Such mathematics any blonde will understand.

Here so, unnoticed, we with you began to live in a completely another epoch - Epoch of Great Mathematical Opening. Doing these mathematical opening coming to you, I can only broadly speaking explain to you, what mathematics, where in mathematics, opening is hidden and as they need to be searched. By the way, my article is the first step on a way to dividing by a zero. Mathematical actions are symmetric: if in mathematics there is multiplying by a zero, means there is under an obligation to be dividing by a zero. If dividing by a zero is impossible, means multiplying is impossible by a zero. The third variant (that is known by us all) can not be. When I will show you, where and as there is dividing by a zero, you will understand that mathematics is this not паханное field on that we engage in the most primitive collector. To throw open and take the crop on the mathematical field, at a desire, any of you can.


Mathematics of blondes and not only...

Today for blondes we will conduct a lesson of biological mathematics. Surprisingly, but the fact:

If to count up quantity of hair on a head of the person (and it is an abstract mathematics without any prejudices) results of calculation will appear such: at their blondes nearby 150,000 (hundred fifty thousand), chestnut hair on a head we will count about 110,000 (hundred ten thousand), black hair it will appear about 100,000 (hundred thousand), red - 90,000 (ninety thousand). Blondes! You possess the thickest hair-do among all people! And, it not a leisure invention of the bald author of a blog "Mathematics for blondes", and the fact confirmed with mathematics.

Mathematics for blondes. Curious  facts. Humour. Curious facts  about numbers. A few  curious numbers are from  life of pilose people.
The number of the top eyelashes makes from 150 to 200, bottom - from 50 to 100 (horror, but even at blondes number of the bottom eyelashes in 2 - 3 times are less, than number of the top eyelashes - here even MaхFactor is powerless);

It is long the eyelashes, measured by the most ordinary ruler for measurement are long eyelashes, for the top eyelashes yields result 8 - 12 millimetres, for bottom equals 6 - 8 millimetres (here already is where to clear up not only Maksfaktoru, but also to false eyelashes);

The size of day growth of hair on a head makes 0.5 - 0.7 millimetres (specially for blondes I will write with letters these terrible numbers - the five tenth and seven tenth millimetre) is less than the tiniest division into your school ruler almost twice, and here day growth of hair of a beard already on the one tenth millimetre is more - from 0.6 to 0.8 for a day (it does not threaten blondes the same as to me - growth of hair on a head).

And some more the sad facts from life of the hairy. A day hair fall on a head - from 50 to 120 pieces (it is interesting, why I bald???!!! Where the mathematics looks!). Life expectancy of hair makes from 2 till 4 years. Life expectancy of eyelashes makes only 150 (hundred fifty!) days. You represent?! Already in half a year on your eyes there will be no familiar eyelash - all new! Horror!!!

Photo from Ledi Truth.