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.


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.


Four-valued mathematical tables for blondes

Four-valued mathematical tables of Bradis - this was the basic mathematical reference book of soviet schoolchildren, students, engineers to appearance of calculators.

A trigonometric table for blondes is done by me to the navigation more informatively saturated in a plan. It what you did not lose way and did not entangle trigonometric functions.

Trigonometric table of sines and cosines
- from 0 to 90 degrees punctually to the minute corner.

Trigonometric table tangent cotangent in degrees - from 0 to 90 degrees punctually to the minute corner.

Trigonometric table in radians - sin, cos, tan.

We will do justice to work of Bradis and we will remember a bit history. His book "Tables of the four-valued logarithms and natural trigonometric sizes" went out in 1921. This book was repeatedly reprinted, but already under more simple name "The Four-valued mathematical tables". This bestseller looked approximately so.

Four-valued mathematical tables of Bradis for blondes. The basic mathematical reference book of soviet schoolchildren, students, engineers
It is possible to say without a false modesty, that on these tables all Soviet Union was built, a man started to fly in space, a soviet nuclear club was created et cetera. Schoolchildren, engineers, scientists - all used the tables of Bradis. We will remember those distant times - the Internet is not present, mobile telephones are not present, computers and calculators are not present. Even televisions were not then! There were only books. In many books formulas were written for a calculation, and necessary for calculations numerical values were taken from the tables of Bradis. And what did numbers multiply by then? Not on accounts... Well and time was! As it was then possible normally to live??? But lived somehow.

An interesting question arises up. What mathematical tables did Americans build famous sky-scrapers on and created the nuclear club? In fact did not they steal for us tables of Bradis? Omniscient Wikipedia is quiet on this occasion, and information about the table of Bradis in Wikipedia in English language I did not find. There is there a mathematical reference book of Abramowitz, Milton and Irene A. Stegun with tables, but he is published was only in 1964. And that did Americans have to these tables?

I used the table of Bradis once. This whole art, to search the value of trigonometric functions punctually to the minute corner. Fortunately, today we have calculators.

Unfortunately, work of great soviet writer of mathematical tables Bradis me does not interest. Therefore the four-valued mathematical table into language of blondes transferred will not be. In the simplified kind the table of Bradis will be presented in the tables of sine, cosine, tangent and cotangent.


Trigonometric circle - comments 2

Trigonometric circle - comments 2. This continuation of publication of comments to the page the "Trigonometric circle", left by visitors in the Russian version of blog "Mathematics For Blondes".

Comment - I am very thankful you, I skipped these themes and there is nothing could understand, and now though that I understand)))

Comment - I in general understood nothing)

Nikolay Khyzhnjak - Nothing, it is not needed to be disordered. All do not can everything it is identically good to understand. A trigonometric circle someone understands well, someone can not understand in general.

Comment - Class site))))))

Nikolay Khyzhnjak - Concerning a site not quite true: here almost is not that it must be are mathematicians. Simple, clear, evident. I will try to reform)))

Comment - Your table very pleased me all is simple, evidently and clear. Thank you!!!

Nikolay Khyzhnjak - It is my little strokes to stranger work. The trigonometric circle I will draw together with you straight in this blog. I hope, soon this we will take up)))

Comment - I thank you. I do not present although, how all of it I will teach... But for a crib it very well))

Nikolay Khyzhnjak - And really do the salespeople of estimations require cramming yet?))) Personally I prefer to understand mathematics, but not to learn by rote ridiculous rules)))

Comment - cos pi/12 as possible to present?

Nikolay Khyzhnjak - cos pi/12 = cos (180/12) degrees = cos 15 degrees = 0,9659 Finding this value is possible on the table of cosines.

Comment - cos corner in intervals from pi to 3pi/2 does take on negative values??

Nikolay Khyzhnjak - Yes, in the third fourth at a cosine negative values, as well as in fourth, from 3pi/2 to 2pi.

Comment - how to find the cosine of pi on 12

Nikolay Khyzhnjak - It is hardly higher written in comments, that on the table of cosines it is needed to find a cosine 15 degrees.

Comment - you are dimwits

Comment - Thank you for the name "Mathematics for blondes". I 30 years engage in that I translate school mathematics into the language of "mathematics for blondes". Actuality grew with introduction of single state examination. I add to the line of tangents and cotangents (students understand it).

Nikolay Khyzhnjak - Accept my sincere condolences))) Unlike you, I have a right to be quiet. As a teacher of mathematics, you such right do not have. you are under an obligation to explain ALL themes, I can simply not talk about separate moments.
Yes, lines of tangents and cotangents simply are under an obligation to be on a trigonometric circle. Although I and lazy, but I will try in the near time to draw such picture.
very interesting me, how do you explain negative and complex numbers in language of blondes?

Comment - creator, for you alike at with a head badly, if you consider blondes fools.

Nikolay Khyzhnjak - I think that my articles "Mathematics - the blonde searches x", "http://www.mathforblondes.com/2010/11/wunderkindes-and-cretin-with-blondes.html" testify to reverse.

Comment - thank you that who did everything I am a not blonde and not girl but all is even

Nikolay Khyzhnjak - I write for all, who wants to read it)))

Comment - good site, to the author respect and respect, though I and not blonde, I badly understand in mathematics, but all is normally written with a clear language, thank you

Nikolay Khyzhnjak - Thank you for a comment. Not everybody can understand mathematics - it firstly. Secondly, someone must and mathematicians to feed, dress, write their portraits...

Comment - seems, you similarly are blond

Nikolay Khyzhnjak - When I became a mathematician, I, seems, began to understand logic of thinking of blondes. Paradox, but for blondes, alike, mathematical character of thinking.

Trigonometric circle - comments

Math blonde. A Paris Hilton on a picture reads a book Discrete Mathematics. Mathematics For Blondes.
The Russian specialists of Photoshop compelled Paris Hilton to read a book "Discrete mathematics".

Comment - and to me reached!)

Nikolay Khyzhnjak - it is stranger work, this picture of trigonometric circle I only adorned))) it will be Needed to draw the trigonometric circle, by sight clear at first blush)))

Comment - you are a fool bald, explain normally! I am a hard blonde, I understand nothing!

Nikolay Khyzhnjak - If you are a hard fool, then medicine is here powerless. Mathematics - moreover. Now try once again, only already politely, to answer a question: that you are not clear exactly?

Comment - what if at being of value alpha equal 4pi, for me 720 degrees, how farther to find a sine and cosine alpha??

Comment - however I hate geometry, though I am a not blonde=)

Nikolay Khyzhnjak - concerning a hatred. It is your problems((( to Mathematics on a drum, blonde you or no, you love geometry or hate, in mathematics twice two however will equal four, and the laws of geometry will continue to work, as well as before. It is needed simply to reconcile oneself to with this fact. All people can not identically masterly own geometry. But it is desirable to know about basic concepts. In fact you are a reasonable creature, but not animal. It not needed the animal of knowledge, for life by him there are fully enough instincts.
Concerning 720 degrees now I will write on a separate page.

Comment - on my you engage in unavailing enough business. To teach blondes, it too same, that to try to drive away a tram... Although, good luck)

Comment - better will explain on an example - how to decide cos(x - pi/2)

Nikolay Khyzhnjak - Concerning educating of blondes. Blondes are a simply advertising trick))) Although in some places I understand their logic perfectly)))
In business-literature a phrase pleased me: "If not able to do business, teach other people, as needed to do business. If not able to teach people to business, teach teachers, as needed to teach people to business"))) As I in trigonometry small that I understand, I teach other and study. Blondes - it my level just)))
About the decision of examples. Examples without решебника I I am not able to decide. I am simpler to set about in a fuzz and ashes all trigonometric circle, what to decide a предложнный example in due form educational-bureaucratic art))) I will Find a decision - I will publish)))

Comment - is pi divided by 12 exists? If yes, then what is it equal to?

Nikolay Khyzhnjak - Yes, pi divided on 12 is a not mirage, it exists. For that, what finding, what it is equal to, is needed to divide 180 degrees on 12. 15 degrees ensue.
pi/12 = 180/12 = 15 degrees
next equality is Here used for translation of radian measure of corners in a degree:
pi radian = 180 degrees

Comment - Thank you, then yet: (Pi/6+2Pi)/2=13Pi/6. If is it calculated right, then how to bring this expression over? and whether it is possible to simplify him not taking away Pi.

Nikolay Khyzhnjak - It is calculated not correctly. After equal sign the result of calculation of expression is writtenin in brackets, and he needs to be yet divided in half. In the total it must turn out 13pi/12. With the value of corner already nothing will do, unless to write down as 1 and 1/12 pi. And trigonometric functions with the corner of such size it is possible to lead on the special formulas. Every trigonometric function, has the formulas of coersion. All trigonometric functions of any corners can be driven away in a stall measuring from 0 to 90 degrees. The formulas of bringing stray sheep over we will consider in the nearest time)))


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.

Sine and cosine 11 degrees and 32.7 minutes how to calculate?

When I went to school, I had to use the table of Bradisa for a finding of sine, cosine, tangents and cotangents. Has already forgotten, as this table to use. But today we live in a computer century, and what such the computer? Correctly, it is such big calculator. And in each big calculator there should be a calculator small. Here this calculator also needs to use. At me operating system Windows XP, on the screen the glory, below, is a button "Start-up". Press this button, then in the menu choose "All programs", from all programs choose "Standard". In standard programs the calculator of sine necessary to us and cosine hides.

The calculator usually has no sine and cosine. It is necessary to press a button "Kind" in top panel the calculator and to choose "Engineering". In the engineering calculator there are buttons necessary to us a sine "sin", cosine "cos" and a tangent "tg".

After that it is necessary to track that in the calculator decimal notation and degrees for corners would be included. For this purpose it is necessary to press "Dec" and "Degrees" as on a picture it is shown. Our tool for a finding of sine and cosine is ready. Now we will start directly process of extraction of useful trigonometrical minerals.

If it was not possible to you extract the calculator from the computer, do not despair! Specially for you I have placed in this blog "Mathematics for blondes" the calculator free of charge which you can use directly here and now!

web 2.0 scientific calculator

At first it is necessary to translate minutes in degrees. For this purpose 32.7 we divide on 60. It is As a result received 0.545 degrees. On 60 we divide because in one degree of 60 minutes. To received циферке it is added 11 degrees which at us already are, and it is received 11.545 degrees. Here from such corner on the calculator it is possible to take already sine and косинусы. For this purpose it is necessary to press simply a button "sin" or "cos".

All process press buttons looks so:

32.7 / 60 + 11 = sin

As a result in a calculator window there will be number 0.20013750391127021629780041181162

For math it registers so:

sin (11° 32.7') = sin 11.545° = 0.2001

For a cosine of the angle of 11 degrees of 32.7 minutes value is equal almost to unit and will register so:

cos (11° 32.7') = cos 11.545° = 0.9798

For a tangent all is carried out precisely also, only right at the end instead of a button "sin" the button "tg" is pressed. Here with cotangent, apparently, a problem. There is no such button in the calculator! But we clever, also remember that trigonometrical function cotangent is return trigonometrical function in relation to a tangent (so much clever words for once - already most terribly!). In practice it looks very simply: at first we find a tangent as it is described above. When numbers a tangent have appeared in a calculator window, we press a button "1/х". Numbers a tangent will exchange on numbers cotangent. And this additional magic button is called "number, the return entered". For the sake of a trick, enter number 2, press this magic button and you will have number 0.5 that equally 1/2.

For transfer of seconds in minutes, seconds as need to be divided on 60, as in each minute of 60 seconds. For transfer in the degrees, the received minutes it is necessary to divide once again on 60:

1" = 0.016667' = 0.00027778°

Here, apparently, all how to calculate a sine and косинус 11 degrees and 32,7 minutes. If someone still had questions, write to comments. If to someone laziness most to press buttons in the calculator, it is not necessary to write to comments! I understand, not imperial this business - to be picked the calculator. Then go on a site of the decision of problems, they will execute any your mathematical whim, naturally, for your money.

For the favourite blondes I can give some small helps. The sine of 6 degrees of 30 minutes needs to be typed on the calculator as 6,5 degrees, then to press a sine button.

Now example it is more difficult, with seconds: cosine 6 degrees of 7 minutes of 9 seconds. 9 seconds we divide on 60, we add 7 minutes, again we divide on 60, we add 6 degrees. Number 6.11916666 should turn out... Degrees. Now we press a button cosine "cos". An order press buttons the such:

9 / 60 + 7 / 60 + 6 = cos

For math recalculation of degrees, minutes and seconds in degrees for 6 degrees of 7 minutes of 9 seconds can be written down so:

(9 : 60 + 7) : 60 + 6 = 6.11916666...

In a general view for a corner in x degrees, y minutes, z seconds the transfer formula in degrees will look so:

(z : 60 + y) : 60 + x = degrees

I hope, this formula is useful to you.

How to find a value:

How do you find the sin of degrees and minutes on a calculator - here is a calculator it is written, how on him to calculate a sine