2.05.2011

Division by zero is maybe!

Division by zero is considered impossible. Why? Because so mathematicians consider and learn others to think in the same way. Why mathematicians so consider? And here it is already interesting question. Everything that you will read further, can show to the friends and girlfriends, but do not take in head to show to mathematicians. Mathematicians can consider that you it is cleverer than them and terribly and cruelly to you will revenge. And you after all perfectly know that such revenge. Here when become the mathematician then can argue on a theme of division into a zero. Now we will try to understand, why division by zero is impossible?

We open a page of Wikipedia "Zero" (in Russian language) and attentively we read that to us offers collective reason. Mathematicians take number a and divide it into 0 (zero). Include the mathematical logic Further and begin reasonings. Let as a result of division the number c has turned out. In this case at number multiplication c by zero we should receive number a. But at multiplication to a zero of any number the zero as a result turns out. From this it follows that the number c cannot be result of division of number an by zero.

Personally I think hardly. We will try to write down all it easier:

a : 0 = c


c · 0 = a

Yes, really, not beautifully it turns out, especially wrong second equality. So in the mathematician does not happen. After all all know that any number increased by a zero, equals to zero.

Here it is a high time to remember a small lesson of history. Once, very long time ago, ALL considered that the Earth is a flat island in the middle of an ocean chasm which keeps on backs of three whales. This island is the World Center round which, on heavenly spheres, different features rotate: the sun, the Moon, asterisks.

Division by zero maybe! Earth on three whale. Mathematics  for blondes.
Division by zero is maybe!
Those who thought differently, it was accepted to burn on fires. Now we became more civilised, on fires we burn nobody. But to put in you the two or fill up you at examination can very simply.

And the lesson of this history says: ALL can be mistaken. If ALL around will start to be hung up, I hope, you not begin to do the same? In other words, the opinion of one person or opinion of the majority of people CAN be ERRONEOUS. If we do not acquire this lesson of history, history to us it will give once again, and once again until we will not grow wiser. Apparently, the situation with division into a zero is a repetition of lesson already passed by us.

a · 0 = 0

Now we will execute transformations and we will receive:

0 : a = 0

0 : 0 = a

The first turned out line corresponds to a mathematical rule which says that if a zero to divide into any number, the zero as a result will turn out. Concerning the second line of mathematics speak, expression a zero to divide into a zero it is not meaningful, as cannot be defined. Hi-hi, and we have defined that the zero деленный on a zero equals to any number. To which number? And what number is necessary to you, such and write. By the way, one rich dad searched to itself in the firm of chief accountant. To all applicants, besides other, it asked a question: "How many will be twice two?". For work it has accepted the person who has answered this question with the question: "And how many it is necessary to you?".

And so, mathematicians want to see concrete number, instead of the general phrase as a result of division by zero. For this reason they consider division by zero impossible. Here it is a high time to remember the second lesson of history.

You about integrals heard? If is not present, it is not terrible. It looks so: the uncertain integral something equals to something plus a constant. What is "constant"? This any number which we do not know. Yes-yes, just the same unknown number, as well as result of division by zero. So, the integral is not meaningful? Has not only sense, but also it is often applied in engineering calculations. Focus consists that the uncertain integral is only the theory. In practical calculations certain integrals are applied. When business reaches the certain integrals, all problems with a constant, that is unknown number, disappear. As a result of integration quite concrete numbers turn out.

Why it is possible to find integral and to divide into a zero it is impossible? I think that in due time engineers have come to mathematicians on a throat and the decision of a concrete problem have demanded. Mathematicians have thought up integrals and ways of their finding. It has saved to mathematicians life.

Why engineers so will not make with division by zero? To the decision of practical problems with division by zero to our engineers still as to babies to pension. Engineers simply have not grown to this level of knowledge and technics. And to mathematicians division by zero and so will descend. What difference that you to learn? Nobel Prizes do not give to them so what for once again in vain to strain.

Considering the second lesson of history, it is possible to declare with all responsibility that arguments of mathematicians concerning impossibility of division by zero are not accepted, as in a similar case with uncertain integrals were them, mathematicians, are denied.

Division by zero is maybe. We are not able to divide into a zero yet. Anything shameful in it is not present. The third lesson of history. Once people in general were not able to divide. Even number on number. Anything, have learnt both to divide, and to integrate. We will learn and on a zero to divide. Certainly, if for it us will not burn on fires. Or to arrange round us wild dances of shamans under a tambourine, inspiring us the sacred spells "Division by zero it is impossible!", "Division by zero is not meaningful!".

Enter number, how many there will be a factorial of three

"Enter number, how many there will be a factorial of three" Ogo! Here it is an ambush for blondes! Judging by that is written "enter number" is a site of any pervert from mathematics asks to enter into a window number. I would be frightened of such request! Same it is necessary to be such sadist that the visitors by means of a torture factorial to arrange! And it would be desirable to ask: "And you, the clever man, can count a factorial of four?! Or without the calculator poorly?!" Is not present, with this disgrace it is necessary to struggle. Anybody from Internet users is not obliged to know factorials - the Internet exists for all. It is obvious discrimination of blondes!

With a view of struggle against similar mathematical rudeness, we will create the small table of factorials for the aid to poor blondes and not only to it.

0! = 1
1! = 1
2! = 1 х 2 = 2
3! = 1 х 2 х 3 = 6
4! = 1 х 2 х 3 х 4 = 24
5! = 1 х 2 х 3 х 4 х 5 = 120


And so, the zero factorial is equal to unit, unit factorial is equal to unit, the factorial of two is equal to two, the factorial of three is equal six, the factorial of four is equal twenty four, the factorial of five is equal hundred twenty. I think, it will be quite enough for personal self-defence on the Internet.

By the way, as chance offers can ask those, who considers itself strongly literate, whether they know, how many figures at a factorial, well... For example, two hundred fifty five? In this number of five hundred five figures!!!

255! = 3350850684932979117652665123754814942022584063591740702576779884286208799035732771005626138126763314259280802118502282445926550135522251856727692533193070412811083330325659322041700029792166250734253390513754466045711240338462701034020262992581378423147276636643647155396305352541105541439434840109915068285430675068591638581980604162940383356586739198268782104924614076605793562865241982176207428620969776803149467431386807972438247689158656000000000000000000000000000000000000000000000000000000000000000

This number is five times longer, than the well-known number googol, in which only one unit and hundred zeroes. And at us five hundred five it is not simple zeroes, and different numbers)))

I hope, to anybody from you will not offer: "Enter number, how many there will be a factorial of two hundred fifty five")))

2.03.2011

Bermuda triangle and eternal youth

The Bermuda triangle is, we already spoke about it, when considered a triangle. With the Bermuda triangle it is connected a lot of mysterious and terrible stories. That in these stories truth, and that a usual advertising gimmick for advertising of favourite, - is difficult to understand. The Bermudas triangle on a card looks so:

Bermuda triangle and eternal youth. Mathematics for blondes.
Bermuda triangle
We will not remember all stories connected with the Bermuda triangle, and to assort them on stones. On the Internet it is information full. We will consider only one, curious enough, a case connected with the Bermuda triangle.

The passenger plane "the Boeing 727" with passengers onboard from Atlantic ocean flies up to the airport of Miami. According to rules, the crew spends verification of hours with land services. In 20 minutes the plane disappears from screens of a radar. Radio contact to plane has interrupted. In 10 minutes after disappearance, the plane again appears on screens of a radar. The plane has safely landed. A whole hours passengers and the plane lagged behind for 10 minutes. Anybody from crew or passengers unusual has not noticed anything.

Truth it or fiction? Our habit always and all to say lies, and also the mania of government officials to code all, does not give us possibility precisely to answer this question. If it is fiction - it is a pity. If it is truth - before us interesting enough possibilities open.

Present that you have come to school. Today at you control on the mathematician. The lesson begins, the teacher gives out all the task. After that you imperceptibly get a small charm from the beautician, press a button - the teacher disappears. You easy solve all examples and problems. By the end of a lesson the teacher appears again. With astonishment looks at the watch in a class - shout, ор, but the lesson already comes to an end and it is necessary to collect examinations. Naturally, control all have written this perfectly well!

Whether it is possible to use such charm what to keep the youth? Yes. It is necessary simply most to disappear for some time. All around live, grow old - and you here are not present. It is possible to disappear for the period of vacation. And it is possible to disappear and for all academic year - how many the time you can save! Present, your girlfriends ugly old cows, and you - young and beautiful! Here the main thing - not to overdo. Passing employment, you risk to turn in обезьянку from a zoo. With you all will be photographed, feed you with bananas, and you nothing will not think from this that round you occurs.

And if it is serious, the similar charm will help to keep lives to people in extreme situations. For example, if urgent medical aid is necessary to the person, and this help it is necessary to wait. That the person has not died, you do so that it has disappeared and return it already after occurrence of doctors which can rescue this person.

Fantasy it or real possibility? From the point of view of mathematics here all speaks simply enough. If you have a desire in it to understand - write in comments. We can plunge into this mathematical jungle. But consider that mathematics specific enough science. Look how simply to become the millionaire from the point of view of mathematics: scrape out from a purse all money which at you are, and add to them one million dollars. All - you the proprietress of treasured one million. Where and as you this one million dollars take - mathematics at all does not excite. But the mathematics can prompt variants. It is possible to find one rich man which will give you one million. And it is possible to find one million not rich man which will give to you on one dollar, the same one million dollars as a result will turn out.

You ask, and how eternal youth? Elementary. Disappear from this world young and beautiful. Also do not come back any more never. Than this way differs from death? From our point of view - practically than. So for preservation of eternal youth it is necessary to look for other recipes.

2.02.2011

Types of triangles

Types of triangles differ on clothes. Here when you look at the person, you, first of all, estimate it clothes and at once much becomes clear to you - what dirty trick from this person can be expected. Precisely as well at triangles. However, triangles have only two clothes which can be estimated simultaneously are corners and the parties. On a picture of a triangle it looks here so:

Types of triangles. Rectangular  triangle, right triangle, obtuse and acute triangle. Mathematics for Blondes.
Types of triangles.
The very first triangle on a picture - a right triangle. It is called so because this triangle has one right angle. Two right angles at a triangle does not happen - remember the triangle theorem? Correctly, if two corners take away to themselves on 90 degrees, to the third corner nothing does not remain. And what can be a triangle with two corners? There are no two-coal triangles. Everything, the theorem of a right angle in a right triangle for blondes is proved!

Types of triangles. Scalene, equilateral and isosceles triangle. Mathematics for Blondes.
Types of triangles.
It is possible to name a right triangle a triangle of Pythagor, on it Pythagor proved the well-known Pythagorean theorem. Let for Pythagor a right triangle also remains. Further on a picture at us the acute triangle is drawn. In it all three corners sharp, as female uvulas. Means, this triangle female. The top part of portrait gallery of triangles comes to an end with an obtuse triangle. This triangle has one big corner, and this corner stupid. It is a man's triangle. No, not because men stupid! No! Think. If there is a female triangle, means should be and man's. And all other triangles are already disassembled! There was only this... Here!

In the bottom part of a cloth of a brush of the unknown artist of the XXI-st century we see triangles which differ with lengths of the parties. To the first in the bottom number there is an equilateral triangle - a democracy symbol. Because all three parties at an equilateral triangle of equal length so - they are equal among themselves.

Further the isosceles triangle is drawn. A symbol of grace and beauty of women. At this triangle two parties have identical length. You imagine the woman with hips of different length? Here, and I do not represent. Why an isosceles triangle named so? In honour of blondes of whom mathematicians dreamt, drawing a pencil a triangle that under a hand has got. After all the majority of mathematicians there were no Leonardo's da Vinci's. Instead of fine Mona Liza's at them them isosceles triangles turned out.

The latest triangle - a scalene triangle - a symbol of geometrical tortures. And you thought, it is connected with erudition of this triangle? As though not so! At this triangle the teacher wants to know all three parties of different length and length of each of the parties of a triangle! After tortures by such triangle even Pythagor has not sustained and has told the theorem. In general, the scalene triangle is the most terrible nightmare of pupils on an extent already more than two thousand years.

Triangle names and pictures - it here. Types of triangles and their name.

Mathematics of an inequality

In the mathematician equalities and inequalities are very extended. Whether the inequality can be useful practically to us in an everyday life? Perhaps if you learn to make inequalities of the most simple situations and to analyze them.

Here the most usual situation. You have got acquainted with the guy who was presented to Vasej. Вася has told that it the friend Bill Gates. Here we also will use mathematical principles of construction and the inequality analysis.

First of all, we will remember one mathematical rule - an addition rule. This rule says that from shift composed the sum does not change. Other mathematical rule asserts that at shift of factors product does not change. Here we apply this principle of shift to words of Vasi and we look that at us it has turned out:

(Vasja has told that it the friend Bill Gates) is equal (Bill Gates has told that Vasja its friend)

Now we will analyse both received phrases. You personally are not familiar neither with Vasej, nor with Bill Gates. But Bill Gates - the celebrated personality, therefore you will trust its words much more, than to words of known Vasi any to nobody. The probability of that Vasja REALLY is the friend Bill Gates, is smallest. Hence, we have received an inequality which looks so:

(Vasja has told that it the friend Bill Gates) is not equal (Bill Gates has told that Vasja its friend)

From the phrase analysis it is possible to draw a conclusion that from you something is necessary for this Vasja. If simply to make upon you impression is one. If speech comes about money is another.

Similar reception - to be called as someone concerning the celebrated personality - any swindlers very often use. Especially, when speech comes about money: "the pupil such", "the right hand of that" and so on, and so forth.

If you learn easily and quickly to define situation high lights, mentally to change a situation on opposite and to analyze the received results, it will be much easier to you to make correct decisions.

1.30.2011

Triangle

The triangle is the elementary polygon. Such definition of a triangle is given by Wikipedia. We will not argue with collective reason, we will try to judge by own strength. The triangle is a geometrical figure which consists of three tops and three parties. As appears from the name "triangle", this geometrical figure has three corners. How the triangle looks? Here a triangle photo in all its beauty.

Triangle, corners, parties of triangle. Mathematics  for blondes.
Triangle

Despite such simple-minded appearance of a triangle, I very much doubt correctness of the statement of Wikipedia that the triangle is the elementary polygon. It is too much at a triangle of any gadgets. Has specially looked at the mathematical directory and here acknowledgement of my words: to a triangle it is devoted 6 (six!) the pages, to all quadrangles together taken, only 5 (five) pages. External simplicity of a triangle at all does not mean simplicity mathematical.

And so, we will start to assort a triangle on stones. Three points which are not lying on one straight line, form a triangle. These points are called as triangle tops. The request not to confuse to mountain tops is absolutely another. The triangle has three tops which are designated by the big Latin letters A, B, C (it is a surname at tops such). You ask, than these Latin letters differ from Russian letters And, In, With? A family tree at these letters different, and consequently also an order of an arrangement of these letters in the alphabet.

Between triangle tops there are triangle parties. These are such equal paths on which it is possible to run across from one top of a triangle to other top. The triangle parties also are inviolable frontiers of a triangle. In these borders there is a triangle area. Everything that is outside behind these borders, into the triangle area does not enter. Along these borders frontier guards with dogs go and watch, that another's area has not got into the triangle area, and the triangle area has not run away abroad from such good life in a triangle. The triangle parties by small Latin letters a, b, c are designated.

Just the same small Latin letters a, b, c the length of these parties of a triangle is designated. The strict boundary heads after all need to know, how many kilometres poor boundary dogs have run? How many kilometres passed unfortunate frontier guards are, for some reason, chiefs never interests. When speech comes about length of the parties of a triangle, probably other designation - two big Latin letters with two vertical sticks: |AB |, |BC |, |AC |. In this case the designation of length of the party of a triangle undertakes on a surname of tops of a triangle between which there is this party (look a photo triangle). In full conformity with rules of mathematical bureaucracy it is possible to write down:

a = |BC|

b = |AC|

c = |AB|

From here the first law of a triangle for blondes is very easily deduced: the length of the party of a triangle is designated by two big Latin буковками or one small буковкой, that which is not present among big буковок.

It is very logical to assume that triangle corners also have the designations. Each corner of a triangle has cosy settled down in triangle top between two parties. Designate triangle corners small letters of the Greek alphabet α (alpha), β (beta), γ (gamma). The sum of all corners of a triangle is equal 180 degrees. Such to itself democracy of a triangle: if you a corner also want to be more, select at other corner and use. All as in life. Therefore corners in a triangle meet the different: stupid and pompous (you such perfectly know) near to thin and graceful (blondes). Democracy of a triangle in the mathematician is called "the triangle Theorem" and it sounds so: the sum of all corners of a triangle equals hundred eighty degrees. In mathematical symbols the triangle theorem looks so:

α + β + γ = 180°

Depending on a kind of the corners which have formed Open Company "Triangle Ltd.", triangles differ on appearance. It is such geometrical dress-code for triangles. But about it we will talk next time.

Here you will find answers for such questions: the sum of all corners on a triangle.

1.29.2011

The triangle area

The triangle area concerns that number of school problems which very often should be solved and in the further life. Now we speak not about the area of a triangle Bermudas which at desire can be calculated (by the way, it makes more than one million square kilometres), and not about the area of a triangle love which basically it is impossible to calculate. We speak about the area of that triangle which is a geometrical figure. By the way, from a figure of the blonde of mathematics something took for the favourite triangles. But about it another time.

Today we will look one eye at formulas of a triangle for an area finding. The triangle area in Wikipedia is given in the form of the whole heap of formulas. I стырил them therefrom also have made in the most unscrupulous image for you a crib on geometry with formulas of the area of a triangle and a portrait of the hero of the festivities - a triangle. This crib on favourite our mathematics can be downloaded free of charge - the right button of a mouse (fiiii!) "to keep drawing as". This crib remains at you in the computer.

Triangle, area of triangle. Area  of rectangular triangle. Mathematics  for blondes. Nikolay Khyzhnjak.
The triangle area
Let's begin with a triangle picture. In different textbooks there can be different designations of tops, the parties and triangle corners. Therefore, before it is stupid to apply formulas from a crib to the decision of the problems, compare all designations of the parties, corners and tops of triangles. Quite probably that some letters in formulas to you should be changed, what on a crib and in your textbook there were identical designations.

Now about formulas. Under figure 1 (one) costs the most widespread formula for a finding of the area of a triangle in length of the party of a triangle and the height of a triangle lowered on this party. The area of a triangle to half of product of the party of a triangle on height equals.

The second formula allows to find the triangle area on length of two parties and a corner between them. At the formula there is a sine of the angle the scale for which value it is necessary to search under the trigonometrical table or to calculate on the calculator.

The third formula allows to find the triangle area on three parties and radius of the entered circle. Equal-signs in this formula divide different variants of this formula in which the triangle semiperimeter, radius of the extraentered circle, concerning is applied by one of the parties. With these features we somehow will make an effort understand. For now below our crib it is possible to find that designation who is who in our triangular zoo. And formulas for calculation of height and triangle semiperimeter.

On lengths of three parties and radius of the described circle it is possible to find the area by means of the fourth formula.

My most favourite formula - the formula of Geron - at number five. This formula allows to find the triangle area on three parties. Than so this Heron's formula is good? On work I with its help calculated the area of almost any geometrical figure. For example, if the room had the form of a wrong polygon was to measure lengths of walls and distance between room corners (lengths of the parties and distance between polygon tops enough). Then under the formula of Gerona the area of triangles into which it is possible to break any polygon was calculated. The sum of the areas of triangles gave the polygon area, that is the room area. In a crib the Heron's formula is presented in two variants - a finding of the area of a triangle through semiperimeter and in lengths of the parties.

Further we can find the triangle area on one party and three corners (the formula 6), on radius of the described circle and three corners (the formula 7), on co-ordinates of tops of a triangle (the formula 8). In last formula vertical sticks in numerator designate the number module, after all the area cannot be negative are mathematicians know even.

The area of a rectangular triangle (the formula 9) can be found as half of product of legs of a triangle or through radiuses of the entered and described circle.

In conclusion of our excursion on Placco De the Triangle we can find the triangle area on the party and two corners, using cotangent (the formula 10) or sine (the formula 11) these corners.