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.


How to memorize trigonometric functions?

How to memorize trigonometric functions? This question is set to itself very many at the study of trigonometric functions. On own experience I know that this business is not simple. Through many years I succeeded to carry two moments only, that behave to the trigonometric functions. First, that well burned into my memory, this determination of trigonometric functions generally: a trigonometric function is this relation something to something. Thus in memory at once there is a shot, at that I do not remember that stands in a numerator forgot that was in a denominator. From all trigonometric relations I remember only a tangent: a tangent is a sine on a cosine. A sine costs on the first place, he means in the numerator of shot. A cosine stands on the second place, he means in the denominator of shot. Sines with cosines I generally on life always mixed up. To the last time...

I succeeded to find a that failing element that so was not enough in the days of my educating. Today even I can faultlessly dab with a finger in a cosine to tell that on what in him divided.

The portrait of tangent appeared this magic stick. Yes, yes, not surprised - exactly portrait, at a look to that faultlessly know a tangent. This portrait of tangent is only work by oil of genius artist of beginning twenty the first century, that on auction of Sothebys in 2137 was sold for one hundred million dollars. Oddly enough, but a picture is written was on reverse side of canvas. Where do I it know from? I am a mathematician and I am able to see the future. By the way, yesterday I spoke with this artist, he asserts that such picture did not yet write, but an idea very pleased him. Specially for you I publish reproduction of this masterpiece here.

Masterpiece. Tangent. Mathematics For Blondes

Now we have all for creation of our own trigonometric Bible that will need to be learned by heart. Do not be frightened, in our Bible there will be only one line:

tangent - is sine by cosine

One-only picture from this Bible I already showed you. Now, as the most real preacher, I will teach you all of it it is correct to read. I think, the best method of educating will be a gape-seed of trigonometric comics. Let in the near time by it and we will engage.


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.


How to calculate the area of surface of direct three-cornered prism

How to calculate the area of surface of direct three-cornered prism? For this purpose the best of all to imagine a three-cornered prism in all her beauty. How does a three-cornered prism look? Look at a picture

Three-cornered prism. Area of surface of three-cornered prism. Mathematics for blondes.
Now next question: do you have a tube of lipstick in form three-cornered prism? If you answered this question (and or no) though as, means now you already know about what three-cornered prism and as she looks. Here those two triangles, from above and from below, are named the grounds of prism. Three rectangles on each side are named the verges of prism. As a triangle lies in founding of prism, verges for us exactly three. If there will be a pentagon in founding, verges it will be five. And if is there 1234-cornered in founding of prism? Correctly, verges there will be 1234 things. With the construction of direct prism we understood, after it it is possible to take up the mathematical calculations of area of surface of three-cornered prism.

As a prism is a geometrical body, her structure can be investigated on the example of body of blonde. That is included in the area of surface of prism? If you will come a heel on something sharp or knocked by the обо-что top of the blond head, it will be badly you. That is why both heels and top of head enter in the complement of surface of your body. Just overhead and lower triangles are included in the area of surface of three-cornered prism. If to dab with you a finger in a side, you will say "Ouch"! because a side belongs to your body. Here those three rectangles, that are from the sides of three-cornered prism, make the area of her side.

By the result of our scientific researches of bodies of blonde and three-cornered prism, we came to the conclusion, that the area of surface of three-cornered prism consists of areas overhead and lower grounds (they are equal) and area of side. The area of triangle needs to be found on one of formulas. I remind that the area of triangle found on a formula at the calculation of area of surface of prism needs to be taken two times, id est to increase her on two.

The area of side of prism is determined as a sum of areas of rectangular verges. It is needed to increase length of every side of triangle (that triangle that is in founding) on the height of prism and lay down three got areas together. It is possible to act simpler: to increase the perimeter of triangle (a perimeter of triangle is a sum of lengths of all his parties) on the height of prism. But adding up not to avoid you in any case: either to the increase (we fold long parties) or after an increase (we fold the areas of rectangular verges). Another trouble that you can lie in wait on a way to the area of surface of three-cornered prism is this absence of values of all lengths of parties. It is not mortal for the calculation of area of triangle. And for the area of side of prism you will have preliminary to find lengths of all three parties of triangle, applying the mathematical knowledge.


About symmetry of mathematical actions

About symmetry of mathematical actions - is my first official publication. To all appearances, my flaming speech under the name "Mathematics forever!" remained unnoticed. It is clear. Reading a like, I would say that a next idiot rushed about on all Internet with the ridiculous idea. But... All, that is here written, I write exceptionally for you and publish here in an only copy, unlike other authors of raving ideas. With my article about symmetry of mathematical actions you are first can become familiar right here and now. I am herein anything interesting or no - decide. In brackets I will give some comments (specially for you) that in the printed variant of the article are absent.

Annotation: Rules of symmetry of mathematical actions allow to apply a commutative law to all mathematical actions: to addition, deduction, multiplication and division. (An annotation is this obligatory condition for the publication of the article. Such are rules of the bureaucratic playing science)

Changes in the surrounding world are expressed by mathematical actions. Quantitative changes are expressed by addition and deduction. Quality changes are expressed by an increase and division. No quantitative changes can cause the change of quality.

Quantitative changes reflect the change of amount of the separately taken unit. Addition and deduction are symmetric mathematical actions reflecting the quantitative changes of any unit. Addition and deduction are mirror symmetric relatively neutral element are points zero.

An increase and division similarly are symmetric mathematical actions reflecting the quality changes of units. An increase and division are back symmetric relatively neutral element are points one.

Rules of symmetry of mathematical actions:

1. Any mathematical action is begun with a neutral element.

2. A sign of mathematical action is the inalienable attribute of number before that he stands.
(This fragment is distinguished by me by fat text specially for you)

Application of these rules allows to apply a commutative law to all mathematical actions reflecting quality or quantitative changes.

0 + 3 + 7 + 4 = 0 + 7 + 3 + 4 = 14

0 – 3 – 7 – 4 = 0 – 7 – 3 – 4 = –14

0 + 3 – 7 – 4 = 0 – 7 + 3 – 4 = –8

1 х 3 х 7 х 4 = 1 х 7 х 3 х 4 = 84

1 : 3 : 7 : 4 = 1 : 7 : 3 : 4 = 1/84

1 х 3 : 7 : 4 = 1 : 7 х 3 : 4 = 3/28

A commutative law can not be used in the cases of the mixed implementation of mathematical actions reflecting quality and quantitative changes in one mathematical expression.

The change of the mathematical operating on symmetric gives a symmetric result, here the point of symmetry is a neutral element. Application of commutative law does not influence on a result.

0 – 3 – 7 – 4 = 0 – 7 – 3 – 4 = –14

0 + 3 + 7 + 4 = 0 + 7 + 3 + 4 = 14

0 – 3 + 7 + 4 = 0 + 7 – 3 + 4 = 8

1 : 3 : 7 : 4 = 1 : 7 : 3 : 4 = 1/84

1 х 3 х 7 х 4 = 1 х 7 х 3 х 4 = 84

1 : 3 х 7 х 4 = 1 х 7 : 3 х 4 = 28/3

Running the numbers in the mathematical operating on symmetric relatively neutral element of number gives a symmetric result.

0 + (–3) + (–7) + (–4) = 0 + (–7) + (–3) + (–4) = –14

0 – (–3) – (–7) – (–4) = 0 – (–7) – (–3) – (–4) = 14

0 + (–3) – (–7) – (–4) = 0 – (–7) + (–3) – (–4) = 8

1 х 1/3 х 1/7 х 1/4 = 1 х 1/7 х 1/3 х 1/4 = 1/84

1 : 1/3 : 1/7 : 1/4 = 1 : 1/7 : 1/3 : 1/4 = 84

1 х 1/3 : 1/7 : 4 = 1 : 1/7 х 1/3 : 1/4 = 28/3

Simultaneous change of the mathematical operating on symmetric and running the numbers on symmetric relatively neutral element of number abandons a result without changes.

0 – (–3) – (–7) – (–4) = 0 – (–7) – (–3) – (–4) = 14

0 + (–3) + (–7) + (–4) = 0 + (–7) + (–3) + (–4) = –14

0 – (–3) + (–7) + (–4) = 0 + (–7) – (–3) + (–4) = –8

1 : 1/3 : 1/7 : 1/4 = 1 : 1/7 : 1/3 : 1/4 = 84

1 х 1/3 х 1/7 х 1/4 = 1 х 1/7 х 1/3 х 1/4 = 1/84

1 : 1/3 х 1/7 х 4 = 1 х 1/7 : 1/3 х 1/4 = 3/28

The neutral elements of mathematical actions it is not accepted to write at the decision of mathematical problems and examples, as they do not influence on a result. Before application of commutative law introduction of neutral elements allows correctly to apply a commutative law.

All of it is written, certainly, not for blondes, and for mathematicians. In the future we yet not once will call to this article. And while... you know any more mathematician about symmetry of mathematical actions.