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.

11.24.2010

Zero is equal to unit

A zero is equal to unit - beautiful mathematical equality turns out. One variant of origin of similar equality did I consider in a note "Why is the factorial of zero equal to unit"? from the cycle of "Legend about mathematics". But this was just a joke on a theme a zero. Some time about this mathematical masterpiece will tell anecdotes. Statement that the factorial of zero is equal to unit it is possible bravely to attribute to the digit of mathematical funny things. But... let attentively we will look closely to equality a zero is equal to unit:

0 = 1


Looks beautifully. But there is a little problem in legitimacy of existence of similar equality - it we. Did you hear the fable of Ivan Krylov "Marmoset and glasses"?

Ivan Krylov. Marmoset and glasses. Mathematics for blondes. Nikolay Khyzhnjak.
Ivan Krylov. Marmoset and glasses.
This fable about us and about this equality. What will we do with this equality? Correctly, we will begin to stick this equality in all possible mathematical holes. Here simple example:

5 = 5 - undeniable equality, all marmosets approve amicably;

5 = 5 + 0 - was there obvious curiosity on the snouts of marmosets - now there will be something interesting, otherwise why to add a zero to the number?;

Presentiment of marmosets did not deceive - we use equality a "zero is equal to unit" and we put instead of zero unit:

5 = 5 + 1 - marmosets titter;

5 = 6 - marmosets amicably laugh at us, we in bewilderment.

"Any number is equal to any number" is the most terrible nightmare of mathematicians, which we so easily and simply got. Superstitious horror before similar equality is very well described in the story of Ted Chiang "Divizion by Zero". In this fantastic story woman-mathematician managed mathematical methods to divide a number into a zero. She got equality:

any number = any number

Finale of story "Divizion by Zero" tragic enough. But, here is a very interesting question: "Are there glasses by a that universal object which ideally goes Near ALL parts of body of marmoset?". We are not marmosets. We know a right answer - glasses are put exceptionally on a nose for the improvement of sight.

I would not begin to write about equality a "zero equal to unit", if this was the fruit of leisure fiction. But, strangely enough, it is the future of mathematics. That in what we today refuse to believe, tomorrow will be one of basic equalizations of mathematics. True, mathematics then will be other, not for monkeys with glasses.

Here little fact in support of my words. One of leading physicists-theorists, Andrei Linde, in the lecture "Many-sided Universe" said: "The sum of energy of matter and gravity energy in the scales of Universe is equal to a zero".

A zero can be equal to something, different from a zero! you will turn the special attention, that the question in this equality is about SUM, but not difference of two physical sizes. If physicists subtracted from one physical size other is be much simpler. As early as school we taught that if from something to subtract such is exact something, then as a result we will get a zero. But, physicists assure that we need to something to add something and as a result we will get a zero! The law of maintenance of energies in the scale of Universe equals a zero. Our equality a "zero is equal to unit" presents this law in a general view for any pieces, taken in the scales of universe. Strange, it turns out, what the amount of blondes in the scales of Universe must be equal to the zero... And it yet needs to be proved! Where do they, at that rate, go?!

Think, now to our marmosets not to the laughter. Does "this law of maintenance spread to all marmoset or only on her part? If on part, then on which one? If on all marmoset, then does this law operate on other marmosets? If does operate, then on all marmosets without an exception or only on select? If only on select, then who, at that rate, them does elect there? Write down and me!!! I want to be select!")))

As see, every new step in science generates mass of new questions. And for that, what not to look a marmoset with glasses, something needs to be understood. Though slightly.

How do you behave to mathematics?

"How do you behave to mathematics?" - such questioning I conducted in spring of 2010. In questioning 70 persons took part from a number the visitors of the Russian blog of "Mathematician for blondes". Here I bring the results of questioning and comments to them.

Most appeared those, who loves mathematics - 21 percent. It makes happy. But I do not want, what all of them grew into ordinary mathematical robots. I think, here physicists prevail on storage of thought in the hands of good teachers. Here can be lyric poets in the hands of talented teachers.

Consider mathematics interesting science are 11 percents. Mathematical slavery does not threaten this category of students. When them it will be compelled intensively to decide different family tasks and examples - they quickly will say to mathematics "no" and will go away from mathematics a little rather. I think, lyric poets which drove mathematicians with teachers prevail here.

Indifferently behave to mathematics are 14 percents. It is victims of ineffectual teachers. Teachers indifferently expound educational material and require the same indifferent answers in reply. For it the teachers of mathematics need to put large two.

Consider mathematics bad science are 15 percents. It is victims of terror from the side of ineffectual teachers. Ineffectual teachers consider that mathematics all must know. Lyric poets on such violence answer a hatred. Personally exact sciences were much easier given me. Languages and literature I hated. From all long-term school course of languages I memorized only one: a rule of grammar is this large black spot in a textbook, which needs to be learned. From a look to these rules became me depressed.

Those, whoever knows about what mathematics is 9 percents. As I suppose, these are blondes. To them I belong. I also try to understand, what such the mathematics. An existent set governed and determinations personally me does not arrange already. Too much I am questions, on which mathematicians simply unable to give an answer. Simplest example. Increasing length of two perpendicular parties of rectangle, we get meters square and we get an area. Increasing length of two parallel parties, we get meters square but we do not get the area of rectangle. Why? An answer "so can not be done" me does not arrange. An answer "so can not be done" me does not arrange. It is so "possible", and it is so "impossible" is already not mathematics, it is a spiritualistic session.

11.13.2010

Wunderkindes and cretin with blondes

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

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

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

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

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

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


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

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

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

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

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

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

Why is the factorial of zero equal to unit?

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

5! = 1 х 2 х 3 х 4 х 5 = 120

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

0 х 1 = 0

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

0 х 1 = 1

Whew mathematics grows into a marasmus.

11.12.2010

Why is trigonometry needed?

If to judge on that trigonometry which me unsuccessfully tried to teach at school and in the subsequent years of my studies, then trigonometry is thought of exceptionally in an order to complicate life to us. Sometimes, here and there, very rarely in life it was necessary to run into trigonometry, and that only because at school I was taught to decide some tasks through trigonometric functions.

Today I a few othergates I look at all things in general, and on trigonometry in particular. I became a blonde on character of thought and I see everything in characters, but not I learn by heart bluntly someone the thought of rules and determinations. So, from the height of flight of blond idea, I can bravely declare, whatever mathematicians understand, what trigonometry. They thought of many determinations, swept everything, that can be swept, in one heap, drew many pictures, made many tables glad, as children - present all of it to us. And we rush about with all this trigonometry, as a crackpot.

Why is trigonometry needed? Twice two equal four? Mathematics For Blondes.
Why is trigonometry needed?
And meantime, trigonometry - it one of the most important things in the surrounding us world. It is needed only to dissociate trigonometry from other mathematical concepts which in trigonometry are ordinary garbage. you did culinary recipes read? We will "take a soupspoon that, we will add полстакана of it, will season here by this nasty thing and all is careful we will mix - a dish is ready, I ask to the table". But it is possible to prepare and on other principle: "take all edible, that will manage to find in a house, pour it in a large tub, carefully mix and gorge on a health, enjoy your meal". School trigonometry is prepared exactly on such recipe. Once I was handsomely tangled at the decision of geodesic task with the use of mathematical reference book. But it is other history.

I will make an effort, besides standard school letups, to show the component elements of different mathematical dishes to you. Skilful cooks use the comparatively limited set of foods, but can create all variety of national kitchens of the world. It is just possible to act and in mathematics. You will be very surprised, knowing, from what a few of component elements it is possible to create the most different mathematical masterpieces. The determination of natural numbers, integers and the brandname recipe of receipt of the greatest in the world number I already gave you. Let us a bit understand with school trigonometry, and then I will show you, as through trigonometry it is possible to measure one of most important for us things - is love.