January 20, 2015

Right triangle trigonometry

I will not tell you the rules and definitions of trigonometric functions on a right triangle. Mathematics is happy to do without me. I'll just show you a picture, which shows a right triangle trigonometry.

Right triangle trigonometry. Mathematics For Blondes.
Right triangle trigonometry

The top row shows who is who in the trigonometric zoo. Sine and cosine of the angle alpha - a relationship of legs to the hypotenuse. Tangent and cotangent - a relationship of legs. With hypotenuse usually no problem, and it is located opposite one of the right angle. But the legs as much as two and they are different. One is located in front of the angle alpha is called opposed (the picture side a). Another gently pressed against the corner called alpha and an adjacent (on the picture side b). Now, looking at the picture, you can easily formulate the definition of trigonometric functions on a right triangle.

The bottom row of images shows how to find the sides of a right triangle, if we know one side and the angle alpha. Known side is green. Using this side and trigonometric functions, you can easily find the other two sides of a right triangle.

Spin this picture as you like, turn face down and look up to the light - trigonometric dependence in a right triangle from it are not changed.

Right triangle trigonometry. Spin this picture. Mathematics For Blondes.
Spin this picture

This picture you can come in handy in the future, the study of physics, theoretical mechanics, in engineering calculations. By the time you've forgotten how to define and use trigonometric functions in a right triangle.

And the saddest thing in the end. If you teach mathematics to use mathematics, these pictures you drew to themselves for a few minutes, without any textbooks. After all this is done simply elementary.

January 17, 2015

Hilbert's sixth problem

Here I have in the comments appeared one modest request:

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

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

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

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

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

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

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

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

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

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

- Kingdom of God - a complex space;

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

- Holy Cross - Cartesian coordinate system.

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

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

January 7, 2015

Happy New Year!

Happy New Year
Happy New Year!
Happy New Year to you all! Do not look at the numbers in 2015, they may look much prettier. It all depends on what we want to see them.

2015 year in the binary system
2015 year
The number 2015 in the binary system looks much nicer and has an almost perfect symmetry. But it will look like a calendar for 2015 in the binary system.

Calendar for 2015 in the binary system
Calendar for 2015
And New Year's letter to Santa Claus. To the Russian Santa Claus (Ded Moroz) from Ukrainian girl whose father was killed in the Russian-Ukrainian war. No comment.
New Year's letter to Santa Claus from Ukrainian girl
New Year's letter to Santa Claus from Ukrainian girl

 "Good afternoon, Santa Claus!
I have two big dreams.
My first dream is to end the war and did not die, our Pope!
Second, I want the laptop to communicate with children who also lost their dads, and who also want peace.
Thank you for understanding.
Please help me in my dreams.
From Levchuk Marija"

Here's how Putin congratulates Ukrainian people on the New Year.

Happy New Year from Putin
Happy New Year from Putin
And here is how the humanitarian assistance from Putin in Ukraine.

Humanitarian assistance from Putin in Ukraine
Humanitarian assistance from Putin in Ukraine

December 20, 2014

Arctangent of the calculator

Sorry to trouble you.
My wife is studying trigonometry, she learns in the legal (!) Institute (small computer science course - to "work with the calculator"). I studied Algebra 10 years ago, but with sines and cosines,  I somehow figured out.
In the impasse puts me "elementary" question ...
We have tg3x = 4, it is necessary to calculate the angle "x" ...
I do not know how to approach this issue ...
Explain it to me please ... Thank you.

Tame this beast through the inverse trigonometric functions. In this case it is necessary to use the arctangent. It looks approximately like this:


Then quite simply, I explain to the legal institute. Arctangent of tangent is just a corner, in this case 3x. It's like that steal and put back. The literal translation of mathematical language to household will sound like this:

"The Tangent stole of the angle 3x" - tg3x;
"Tangent stole and put in place a corner 3x" - arctg (tg3x)

Now it is a children's question: "What lies at the place?". That's right, the angle 3x. On the left side we have understood.

Now right-hand side. The police discovered that the fence number 4. Police know that traded stolen fence corner on the number at the rate of a tangent. The question arises: "What angle was stolen, if a fence to get him the number 4?". In this case, we can use a table of values of the trigonometric tangent function as a price list on the exchange of angles among the fence.

But we have set the theme of "work with a calculator." The calculator "Windows" enter the number 4, then click the button "Inv". At the same time closely following button "tan". This button should appear degree minus one. Four should remain unchanged. Now press the button tangent to the minus first degree and obtain the value of the angle whose tangent is 4. If we have included in the calculator button "Degrees", then we get 75.963756532073521417107679840837 degrees when the button is turned "Radian", we'll get 1, 3258176636680324650592392104285 radians.

Now we can restore the picture to commit a crime (in degrees and radians oil paintings look different). We write from the beginning in degrees, rounded to three characters:

х=25,321 (degrees)

The same, but now in radians:

х=0,442 (radians)

If you're holding a calculator any other design, you will need an educated bet calculator dig out the desired result.

Unexpected response:

A huge thank you for a mathematical investigation. Your explanation is very inspired us and my wife and I decided all the "problems with a calculator." Thanks again!
P.S. I read your profile in Google. As a citizen of Russia say that I share your views on terrorist organizations of Donetsk and Lugansk. I wish you and the Ukrainian people defeat the terrorists and to regain the Crimea. We believe in Ukraine without the Kremlin! Good luck to you!

December 18, 2014

How to find the number of

Hello. Please help me to solve the problem.
Subject: decimals.
The sum of three numbers is 10 whole 4/10. The second number is 3 times smaller than the first, and at one whole 1/10 more than the third. Find these numbers.

Carefully read the conditions and form the equation. "The second number is 3 times smaller than the first" - this means that the first number is three times langer than the second. If the second number is denoted by X, then the first number is three X.

"The second number is (((3 times smaller than the first, and))) at one whole 1/10 more than the third" - means the third number is equal to X minus one whole 1/10.

Now write the sum of three numbers equate it to 10.4 and obtain the equation.

3x + x + x - 1.1 = 10.4
5x = 10.4 + 1.1
5x = 11.5
x = 11.5 / 5
x = 2.3

The second number is equal to 2.3
The first number is equal to 2.3 * 3 = 6.9
The third number is 2.3-1.1 = 1.2

Check the amount of received numbers

6.9 + 2.3 + 1.2 = 10.4

October 11, 2013


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

April 11, 2013

Division by zero in physics

Switch. Dividing by a zero in physics. Example of application of mathematics. Mathematics for blondes. Mathforblondes.
All laws can be divided into two groups - invented by us and the laws of mathematics. Laws invented by us may not work in spite of the fact that we invented them. The laws of mathematics, which are displaying the laws of nature, always work, regardless of whether we know them or do not know. That is the case with the law of multiplication and division by zero.

There is an old student's joke that the device that performs mathematical operations of multiplication and division by zero is an ordinary switch. Personally, I more trust the not fresh views of students, than the "scientific" opus of different "scientists". Usually, the first impression is correct.

All mathematical letups, that were written here, I deleted at first, because I believed that the harm from them will be more than good. Comments I cleaned similarly. But then changed my mind. If I will not tell about mathematical principles of work of electric switch, then others will not soon do it. And so, to begin a comment (to the page in Russian language):

"Idiot, Ohm's law correctly write down. Why is current is not equal to zero at zero voltage? And why does a resistance of burned out bulb is zero? You're either blind or can not to read. All is excellent divided by zero when you know the function of dependence. In most cases, the result is infinity. And about the real uncertainties you, probably, did not hear in general. I'm not going to talk about all the options, where it will not work. Even if not to pay attention to the serious errors in the examples. So I conclude you don't have a brain."

This is a typical reaction of a person to coach at zero, as a trained dog on a command "Attack!". It is thus needed to remember that the math is considering abstract concepts that aren't to need understood. In the result, we all turn into trained animals who think exactly as they were taught. I know from own experience how hard it is to get rid from the generally accepted stereotypes. So I explain that throughout the following discussion I will talk not about numerical values of physical quantities, and about the units, which are usually not considered in general mathematics.

Each of us in our daily lives every day many times uses multiplication and division by zero. Engineers, as well as we, suspecting nothing, created a special device that allows to multiply and divide by zero. And all of this is so firmly established in our lives, that without these devices is impossible to imagine our life around. But let's begin one after another, from the math.

All of you know the mathematical law of multiplication:

a*b = c

On the pages of this web-site I told about the mathematical rules of multiplication and division by zero. Take from this page these formulas, which we will use:

ab*0 = 0 and a or b

a/0 = ab

We will rewrite these formulas in a that kind, in what we will use them in our concrete example:

a*b*0 = (a*0)*b = a*(b*0) = c*0

c*0 = {c=0; a=0; b≠0} = {c=0; a≠0; b=0}

a/0 = b/0 = a*b = c

Now we'll check how these algebraic letups correspond to reality. For this, we represent the most ordinary domestic situation: you're sitting in the evening in the livingroom and suddenly light shuts off. What is your first thought? Correctly, either electricity disappeared or a bulb burned out. Or if trying is possible to think of two others variants: either you suddenly became blind or you suddenly died. Since the latter two options are more to biology, we do not consider they. But as far as exactly the first two variants can be described by our algebraic formulas, let's look.

Luminescence of bulb in physics is described by the Ohm's law that looks so:

I*R = U

In this kind the Ohm's law fully coincides the law of increase presented by us in an algebraic kind:

a*b = c

According to this, in the further reasoning, we can replace the algebraic elements of formulas by physical quantities:

a = I - it is a current that flows on wires, it is measured in Amperes;

b = R - it is a resistance to the electric current is in the spiral of bulb, it is measured in Ohms;

c = U - it is a voltage in an electrical circuit, that compels a bulb to shine, it is measured in Volts.

The first variant of the apocalyptic gloom involves to shutdown of knife-switch by some evil man (for to save energy but without our consent), as a result an electric current stops to enter on wires. Or broken wire in an accident on electrical networks.

U*0 = {U=0; I=0; R≠0}

As we see, this mathematical result reports us, that a bulb really left off to burn, as an electric current disappeared in wires, but with our bulb all is normal and she is ready again to begin to shine, as soon as a current will appear.

Now we will look at the second variant of total eclipse, when a bulb simply burned out for us, and with a current in electric networks all is normal:

U*0 = {U=0; I≠0; R=0}

As see, unlike traditional "any number increased on a zero equals a zero", we got establishment of fact of extinct bulb not only U=0, but also two possible reasons of this annoying incident : {I=0; R≠0} and {I≠0; R=0}.

It is here needed to mark that in traditional mathematics multiplying by a zero what or element of equality taken to one of basic mathematical equalities:


Usually all mathematics closes thereon. In the variant of multiplying offered by me by a zero this situation means disappearance of primary equality and passing to two inequalities - tension is not equal to strength of current and tension is not equal to resistance of electric chain :

U ≠ I

U ≠ R

In a general view for algebraic expression a*b = c it looks so:

c ≠ a

c ≠ b

For renewal of primary equality it is necessary to execute the mathematical operation of dividing by a zero. In our example it is necessary either to recover an electric current in wires or replace a burneout bulb. Thus there is the following:

I/0 = [I*(R*0)]/0 = I*(R*0/0) = I*(R*1) = I*R = U

R/0 = [(I*0)*R]/0 = (I*0/0)*R = (I*1)*R = I*R = U

Equality is used in our case 0/0=1, where as unit units come forward electric that or electric resistance. Introduction to the formula of any other unit of measuring will not result in a primary result, as electric tension ensues exceptionally co-operation of strength of current and resistance. you can go about in circles, winding meters long and почесывая itself in the back of head. You can get a purse and throw about money. Burneout bulb from it will not begin to shine:

I/0 = [I*(L*0)]/0 = I*(L*0/0) = I*(L*1) = I*L ≠ U

I/0 = [I*($*0)]/0 = I*($*0/0) = I*($*1) = I*$ ≠ U

As see, application of dividing by a zero supposes the presence of reason, but not dull implementation of mathematical actions.

In conclusion I want to say that engineers did switches allowing to execute multiplying and dividing by a zero in electric chains already a long ago. This device is basic custom control by electric chains. Switches are equip practically all electric devices: bulbs, engines, televisions, mobile telephones and other.