Degree measure of angle. Radian measure of angle. Converting degrees to radians and vice versa.

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

In the previous lesson we learned how to measure angles on a trigonometric circle. Learned how to count positive and negative angles. We learned how to draw an angle greater than 360 degrees. It's time to figure out how to measure angles. Especially with the number "Pi", which strives to confuse us in tricky tasks, yes...

Standard problems in trigonometry with the number "Pi" are solved well. Visual memory helps. But any deviation from the template is a disaster! To avoid falling - understand necessary. Which is what we will do now with success. I mean, we’ll understand everything!

So, what do angles count? IN school course trigonometry uses two measures: degree measure of angle And radian angle measure. Let's look at these measures. Without this, there is nowhere in trigonometry.

Degree measure of angle.

We somehow got used to degrees. At the very least we passed geometry... And in life we ​​often come across the phrase “turned 180 degrees,” for example. A degree, in short, is a simple thing...

Yes? Answer me then what is a degree? What, it doesn’t work out right away? That's it...

Degrees were invented in Ancient Babylon. It was a long time ago... 40 centuries ago... And they came up with a simple idea. We took and divided the circle into 360 equal parts. 1 degree is 1/360 of a circle. That's all. They could have broken it into 100 pieces. Or 1000. But they divided it into 360. By the way, why exactly 360? How is 360 better than 100? 100 seems to be somehow smoother... Try to answer this question. Or weakly against Ancient Babylon?

Somewhere at the same time, in Ancient Egypt were tormented by another question. How many times is the length of a circle greater than the length of its diameter? And they measured it this way, and this way... Everything worked out a little more than three. But somehow it turned out shaggy, uneven... But they, the Egyptians, are not to blame. After them, they suffered for another 35 centuries. Until they finally proved that no matter how finely you cut a circle into equal pieces, from such pieces you can make smooth the length of the diameter is impossible... In principle, it is impossible. Well, how many times the circumference is greater than the diameter was established, of course. Approximately. 3.1415926... times.

This is the number "Pi". So shaggy, so shaggy. After the decimal point there is an infinite number of numbers without any order... Such numbers are called irrational. This, by the way, means that from equal pieces of a circle the diameter smooth don't fold. Never.

For practical application It is customary to remember only two digits after the decimal point. Remember:

Since we understand that the circumference of a circle is greater than its diameter by “Pi” times, it makes sense to remember the formula for the circumference of a circle:

Where L- circumference, and d- its diameter.

Useful in geometry.

For general education I will add that the number “Pi” is not only found in geometry... In various branches of mathematics, and especially in probability theory, this number appears constantly! By itself. Beyond our desires. Like this.

But let's return to degrees. Have you figured out why in Ancient Babylon the circle was divided into 360 equal parts? And not by 100, for example? No? OK. I'll give you a version. You can’t ask the ancient Babylonians... For construction, or, say, astronomy, it is convenient to divide the circle into equal parts. Now figure out what numbers it is divisible by completely 100, and which ones - 360? And in what version of these divisors completely- more? This division is very convenient for people. But...

As it turned out much later than Ancient Babylon, not everyone likes degrees. Higher mathematics doesn't like them... Higher mathematics- a serious lady, organized according to the laws of nature. And this lady declares: “Today you broke the circle into 360 parts, tomorrow you will break it into 100, the day after tomorrow into 245... And what should I do? No, really...” I had to listen. You can't fool nature...

We had to introduce a measure of angle that did not depend on human inventions. Meet - radian!

Radian measure of angle.

What is a radian? The definition of a radian is still based on a circle. An angle of 1 radian is an angle that cuts an arc from a circle whose length is ( L) is equal to the length of the radius ( R). Let's look at the pictures.

Such a small angle, it’s almost non-existent... We move the cursor over the picture (or touch the picture on the tablet) and we see about one radian. L = R

Do you feel the difference?

One radian is much more than one degree. How many times?

Let's look at the next picture. On which I drew a semicircle. The unfolded angle is, naturally, 180°.

Now I'll cut this semicircle into radians! We hover the cursor over the picture and see that 180° fits 3 and a half radians.

Who can guess what this tail is equal to!?

Yes! This tail is 0.1415926.... Hello, number "Pi", we haven't forgotten you yet!

Indeed, 180° degrees contains 3.1415926... radians. As you yourself understand, writing 3.1415926 all the time... is inconvenient. Therefore, instead of this infinite number, they always write simply:

But on the Internet the number

It’s inconvenient to write... That’s why I write his name in the text - “Pi”. Don't get confused, okay?...

Now we can write down an approximate equality in a completely meaningful way:

Or exact equality:

Let's determine how many degrees are in one radian. How? Easily! If there are 180° degrees in 3.14 radians, then there are 3.14 times less in 1 radian! That is, we divide the first equation (the formula is also an equation!) by 3.14:

This ratio is useful to remember. One radian is approximately 60°. In trigonometry, you often have to estimate and assess the situation. This is where this knowledge helps a lot.

But the main skill of this topic is converting degrees to radians and vice versa.

If the angle is given in radians with the number "Pi", everything is very simple. We know that "Pi" radians = 180°. So we substitute radians for “Pi” - 180°. We get the angle in degrees. We reduce what is reduced, and the answer is ready. For example, we need to find out how many degrees in angle "Pi"/2 radian? So we write:

Or, a more exotic expression:

Easy, right?

The reverse translation is a little more complicated. But not much. If the angle is given in degrees, we must figure out what one degree is equal to in radians and multiply that number by the number of degrees. What is 1° equal to in radians?

We look at the formula and realize that if 180° = “Pi” radians, then 1° is 180 times smaller. Or, in other words, we divide the equation (a formula is also an equation!) by 180. There is no need to represent “Pi” as 3.14; it is always written with a letter anyway. We find that one degree is equal to:

That's all. We multiply the number of degrees by this value and get the angle in radians. For example:

Or, similarly:

As you can see, in a leisurely conversation with lyrical digressions, it turned out that radians are very simple. And the translation is no problem... And “Pi” is a completely tolerable thing... So where does the confusion come from!?

I'll reveal the secret. The fact is that in trigonometric functions the degrees symbol is written. Always. For example, sin35°. This is sine 35 degrees . And the radian icon ( glad) - not written! It's implied. Either mathematicians were overwhelmed by laziness, or something else... But they decided not to write. If there are no symbols inside the sine-cotangent, then the angle is in radians ! For example, cos3 is the cosine of three radians .

This leads to confusion... A person sees “Pi” and believes that it is 180°. Anytime and anywhere. By the way, this works. For the time being, the examples are standard. But "Pi" is a number! The number is 3.14, but not degrees! This is "Pi" radians = 180°!

Once again: “Pi” is a number! 3.14. Irrational, but a number. Same as 5 or 8. You can, for example, do about "Pi" steps. Three steps and a little more. Or buy "Pi" kilograms of candy. If an educated seller comes across...

"Pi" is a number! What, did I annoy you with this phrase? Have you already understood everything long ago? OK. Let's check. Tell me, which number is greater?

Or what is less?

This is one of a series of slightly non-standard questions that can drive you into a stupor...

If you, too, have fallen into a stupor, remember the spell: “Pi” is a number! 3.14. In the very first sine it is clearly stated that the angle is in degrees! Therefore, it is impossible to replace “Pi” by 180°! "Pi" degrees is approximately 3.14°. Therefore, we can write:

There are no notations in the second sine. So, there - radians! This is where replacing “Pi” by 180° will work just fine. Converting radians to degrees, as written above, we get:

It remains to compare these two sines. What. forgot how? Using a trigonometric circle, of course! Draw a circle, draw approximate angles of 60° and 1.05°. Let's see what sines these angles have. In short, everything is described as at the end of the topic about the trigonometric circle. On a circle (even the crooked one!) it will be clearly visible that sin60° significantly more than sin1.05°.

We will do exactly the same thing with cosines. On the circle, draw angles of approximately 4 degrees and 4 radian(Have you forgotten what 1 radian is approximately equal to?). The circle will say everything! Of course, cos4 is less than cos4°.

Let's practice using angle measures.

Convert these angles from degrees to radians:

360°; 30°; 90°; 270°; 45°; 0°; 180°; 60°

You should get these values ​​in radians (in a different order!)

0

By the way, I specifically highlighted the answers in two lines. Well, let's figure out what the corners are in the first line? At least in degrees, at least in radians?

Yes! These are the axes of the coordinate system! If you look at the trigonometric circle, then the moving side of the angle with these values fits exactly on the axes. These values ​​​​need to be known. And I noted the angle of 0 degrees (0 radians) for good reason. And then some people just can’t find this angle on a circle... And, accordingly, they get confused in the trigonometric functions of zero... Another thing is that the position of the moving side at zero degrees coincides with the position at 360°, so there are always coincidences on the circle near.

In the second line there are also special angles... These are 30°, 45° and 60°. And what's so special about them? Nothing special. The only difference between these angles and all the others is that you should know about these angles All. And where they are located, and what trigonometric functions these angles have. Let's say the value sin100° you don't have to know. A sin45°- please be so kind! This is mandatory knowledge, without which there is nothing to do in trigonometry... But more about this in the next lesson.

In the meantime, let's continue training. Convert these angles from radian to degree:

You should get results like this (in disarray):

210°; 150°; 135°; 120°; 330°; 315°; 300°; 240°; 225°.

Happened? Then we can assume that converting degrees to radians and back- no longer your problem.) But translating angles is the first step to understanding trigonometry. There you also need to work with sines and cosines. And with tangents and cotangents too...

The second powerful step is the ability to determine the position of any angle on a trigonometric circle. Both in degrees and radians. I will give you boring hints about this very skill throughout trigonometry, yes...) If you know everything (or think you know everything) about the trigonometric circle, and the measurement of angles on the trigonometric circle, you can check it out. Solve these simple tasks:

1. Which quarter do the angles fall into:

45°, 175°, 355°, 91°, 355° ?

Easily? Let's continue:

2. Which quarter do the corners fall into:

402°, 535°, 3000°, -45°, -325°, -3000°?

No problem too? Well, look...)

3. You can place the corners in quarters:

Could you? Well, you give..)

4. Which axes will the corner fall on:

and corner:

Is it easy too? Hm...)

5. Which quarter do the corners fall into:

And it worked!? Well, then I really don’t know...)

6. Determine which quarter the corners fall into:

1, 2, 3 and 20 radians.

I will give an answer only to the last question (it’s a little tricky) of the last task. An angle of 20 radians will fall in the first quarter.

I won’t give the rest of the answers, not out of greed.) Simply, if you haven't decided something you doubt it as a result, or spent on task No. 4 more than 10 seconds, you are poorly oriented in a circle. This will be your problem in all of trigonometry. It’s better to get rid of it (the problem, not trigonometry!) immediately. This can be done in the topic: Practical work with the trigonometric circle in section 555.

It tells you how to solve such tasks simply and correctly. Well, these tasks have been solved, of course. And the fourth task was solved in 10 seconds. Yes, it’s been decided that anyone can do it!

If you are absolutely confident in your answers and you are not interested in simple and trouble-free ways of working with radians, you don’t have to visit 555. I don’t insist.)

A good understanding is a good enough reason to move on!)

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

Table of values ​​of trigonometric functions compiled for angles of 0, 30, 45, 60, 90, 180, 270 and 360 degrees and the corresponding angle values ​​in radians. From trigonometric functions the table shows sine, cosine, tangent, cotangent, secant And cosecant. For the convenience of solving school examples of meaning trigonometric functions in the table are written in the form of a fraction while preserving the signs of extracting the square root of numbers, which very often helps to reduce complex mathematical expressions. For tangent And cotangent Some angles cannot be determined. For values tangent And cotangent There is a dash in the table of values ​​of trigonometric functions for such angles. It is generally accepted that tangent And cotangent of such angles equals infinity. On a separate page there are formulas for reducing trigonometric functions.

In the table of values ​​for trigonometric function sine values ​​are given for the following angles: sin 0, sin 30, sin 45, sin 60, sin 90, sin 180, sin 270, sin 360 in degree measure, which corresponds to sin 0 pi, sin pi/6, sin pi/4, sin pi/3, sin pi/2, sin pi, sin 3 pi/2, sin 2 pi in radian measure of angles. School table of sines.

For the trigonometric cosine function, the table shows the values ​​for the following angles: cos 0, cos 30, cos 45, cos 60, cos 90, cos 180, cos 270, cos 360 in degrees, which corresponds to cos 0 pi, cos pi by 6, cos pi by 4, cos pi by 3, cos pi by 2, cos pi, cos 3 pi by 2, cos 2 pi in radian measure of angles. School table of cosines.

The trigonometric table for the trigonometric tangent function gives values ​​for the following angles: tg 0, tg 30, tg 45, tg 60, tg 180, tg 360 in degree measure, which corresponds to tg 0 pi, tg pi/6, tg pi/4, tg pi/3, tg pi, tg 2 pi in radian measure of angles. The following values ​​of the trigonometric tangent functions are not defined tan 90, tan 270, tan pi/2, tan 3 pi/2 and are considered equal to infinity.

For the trigonometric function cotangent in the trigonometric table the values ​​of the following angles are given: ctg 30, ctg 45, ctg 60, ctg 90, ctg 270 in degree measure, which corresponds to ctg pi/6, ctg pi/4, ctg pi/3, tg pi/ 2, tan 3 pi/2 in radian measure of angles. The following values ​​of the trigonometric cotangent functions are not defined ctg 0, ctg 180, ctg 360, ctg 0 pi, ctg pi, ctg 2 pi and are considered equal to infinity.

The values ​​of the trigonometric functions secant and cosecant are given for the same angles in degrees and radians as sine, cosine, tangent, cotangent.

The table of values ​​of trigonometric functions of non-standard angles shows the values ​​of sine, cosine, tangent and cotangent for angles in degrees 15, 18, 22.5, 36, 54, 67.5 72 degrees and in radians pi/12, pi/10, pi/ 8, pi/5, 3pi/8, 2pi/5 radians. The values ​​of trigonometric functions are expressed in terms of fractions and square roots to make it easier to reduce fractions in school examples.

Three more trigonometry monsters. The first is the tangent of 1.5 one and a half degrees or pi divided by 120. The second is the cosine of pi divided by 240, pi/240. The longest is the cosine of pi divided by 17, pi/17.

The trigonometric circle of values ​​of the functions sine and cosine visually represents the signs of sine and cosine depending on the magnitude of the angle. Especially for blondes, the cosine values ​​are underlined with a green dash to reduce confusion. The conversion of degrees to radians is also very clearly presented when radians are expressed in terms of pi.

This trigonometric table presents the values ​​of sine, cosine, tangent, and cotangent for angles from 0 zero to 90 ninety degrees at one-degree intervals. For the first forty-five degrees, the names of trigonometric functions should be looked at at the top of the table. The first column contains degrees, the values ​​of sines, cosines, tangents and cotangents are written in the next four columns.

For angles from forty-five degrees to ninety degrees, the names of the trigonometric functions are written at the bottom of the table. The last column contains degrees; the values ​​of cosines, sines, cotangents and tangents are written in the previous four columns. You should be careful because the names of the trigonometric functions at the bottom of the trigonometric table are different from the names at the top of the table. Sines and cosines are interchanged, just like tangent and cotangent. This is due to the symmetry of the values ​​of trigonometric functions.

The signs of trigonometric functions are shown in the figure above. Sine has positive values ​​from 0 to 180 degrees, or 0 to pi. Sine has negative values ​​from 180 to 360 degrees or from pi to 2 pi. Cosine values ​​are positive from 0 to 90 and 270 to 360 degrees, or 0 to 1/2 pi and 3/2 to 2 pi. Tangent and cotangent have positive values ​​from 0 to 90 degrees and from 180 to 270 degrees, corresponding to values ​​from 0 to 1/2 pi and pi to 3/2 pi. Negative values ​​of tangent and cotangent are from 90 to 180 degrees and from 270 to 360 degrees, or from 1/2 pi to pi and from 3/2 pi to 2 pi. When determining the signs of trigonometric functions for angles greater than 360 degrees or 2 pi, you should use the periodicity properties of these functions.

The trigonometric functions sine, tangent and cotangent are odd functions. The values ​​of these functions for negative angles will be negative. Cosine is an even trigonometric function—the cosine value for a negative angle will be positive. Sign rules must be followed when multiplying and dividing trigonometric functions.

Root 2/2 is how much pi?— It happens in different ways (see picture). You need to know which trigonometric function is equal to root two divided by two.

If you liked the post and want to know more, I have more in the works.

cos pi divided by 2

Home > Directory > Mathematical formulas.

Mathematical formulas.

Convert radians to degrees.
A d = A r * 180 / pi

Converting degrees to radians.
A r = A d * pi / 180
Where A d is the angle in degrees, A r is the angle in radians.

Circumference.
L = 2 * pi * R

Length of the arc of a circle.
L=A*R

Area of ​​a triangle.

p=(a+b+c)/2 - semi-perimeter.

Area of ​​a circle.
S = pi * R 2

Sector area.
S = L d * R/2 = (A * R 2)/2

Surface area of ​​the ball.
S = 4 * pi * R 2

S = 2 * pi * R * H

Where S is the area of ​​the lateral surface of the cylinder, R is the radius of the base of the cylinder, H is the height of the cylinder.

S = pi * R * L

S = pi * R * L + pi * R 2

Volume of the ball.
V = 4 / 3 * pi * R 3

Cylinder volume.
V = pi * R 2 * H

Cone volume.

Posted: 01/15/13
Updated: 11/15/14
Total views: 10754
today: 1

Home > Directory > Mathematical formulas.

Egor

Good evening! You asked a very interesting question, I hope we can help you.

How to solve C1. Lesson 2. Unified State Exam in Mathematics 2014

You and I need to solve the following problem: find cos pi divided by 2.
Most often, to solve such problems you need to determine the cosine or sine exponents. For angles from 0 to 360 degrees, almost any value of cos or sin can be easily found in the corresponding plates that exist and are widespread, such as these:

But you and I do not have a sine (sin), but a cosine. Let's first understand what cosine is. Cos (cosine) is one of the trigonometric functions. In order to calculate the cosine of the acute right triangle You will need to know the ratio of the side of the adjacent angle to the hypotenuse. The cosine pi divided by 2 can be easily calculated using trigonometric formula, which refers to standard trigonometry formulas. But if we are talking about the value of the cosine pi divided by 2, then for this we will use the table that we have already mentioned more than once:

Good luck to you in future solutions to similar tasks!
Answer:

Home > Directory > Mathematical formulas.

Mathematical formulas.

Convert radians to degrees.
A d = A r * 180 / pi

Converting degrees to radians.
A r = A d * pi / 180
Where A d is the angle in degrees, A r is the angle in radians.

Circumference.
L = 2 * pi * R
Where L is the circumference, R is the radius of the circle.

Length of the arc of a circle.
L=A*R
Where L is the length of the circular arc, R is the radius of the circle, A is the central angle, expressed in radians
For a circle A = 2*pi (360 degrees), we get L = 2*pi*R.

Area of ​​a triangle.
S = (p * (p-a) * (p-b) * (p-c)) 1/2
Where S is the area of ​​the triangle, a, b, c are the lengths of the sides,
p=(a+b+c)/2 - semi-perimeter.

Area of ​​a circle.
S = pi * R 2
Where S is the area of ​​the circle, R is the radius of the circle.

Sector area.
S = L d * R/2 = (A * R 2)/2
Where S is the area of ​​the sector, R is the radius of the circle, L d is the length of the arc.

Surface area of ​​the ball.
S = 4 * pi * R 2
Where S is the surface area of ​​the ball, R is the radius of the ball.

The lateral surface area of ​​the cylinder.
S = 2 * pi * R * H
Where S is the area of ​​the lateral surface of the cylinder, R is the radius of the base of the cylinder, H is the height of the cylinder.

Square full surface cylinder.
S = 2 * pi * R * H + 2 * pi * R 2
Where S is the area of ​​the lateral surface of the cylinder, R is the radius of the base of the cylinder, H is the height of the cylinder.

The area of ​​the lateral surface of the cone.
S = pi * R * L
Where S is the area of ​​the lateral surface of the cone, R is the radius of the base of the cone, L is the length of the generatrix of the cone.

The total surface area of ​​a cone.
S = pi * R * L + pi * R 2
Where S is the total surface area of ​​the cone, R is the radius of the base of the cone, L is the length of the generatrix of the cone.

Volume of the ball.
V = 4 / 3 * pi * R 3
Where V is the volume of the ball, R is the radius of the ball.

Cylinder volume.
V = pi * R 2 * H
Where V is the volume of the cylinder, R is the radius of the base of the cylinder, H is the height of the cylinder.

Cone volume.
V = pi * R * L = pi * R * H/cos (A/2) = pi * R * R/sin (A/2)
Where V is the volume of the cone, R is the radius of the base of the cone, L is the length of the generatrix of the cone, A is the angle at the apex of the cone.

Posted: 01/15/13
Updated: 11/15/14
Total views: 10742
today: 1

Home > Directory > Mathematical formulas.

Egor
You can secure the wire to the terminals of the Crohn battery with a tube cut from the cap of a medical needle.

Simply put, these are vegetables cooked in water according to a special recipe. I will consider two initial components (vegetable salad and water) and the finished result - borscht. Geometrically, it can be thought of as a rectangle, with one side representing lettuce and the other side representing water. The sum of these two sides will indicate borscht. The diagonal and area of ​​such a “borscht” rectangle are purely mathematical concepts and are never used in borscht recipes.

How do lettuce and water turn into borscht from a mathematical point of view? How can the sum of two line segments become trigonometry? To understand this, we need linear angular functions.

You won't find anything about linear angular functions in math textbooks. But without them there can be no mathematics. The laws of mathematics, like the laws of nature, work regardless of whether we know about their existence or not.

Linear angular functions are addition laws. See how algebra turns into geometry and geometry turns into trigonometry.

Is it possible to do without linear angular functions? It’s possible, because mathematicians still manage without them. The trick of mathematicians is that they always tell us only about those problems that they themselves know how to solve, and never tell us about those problems that they cannot solve. Look. If we know the result of addition and one term, we use subtraction to find the other term. All. We don’t know other problems and we don’t know how to solve them. What should we do if we only know the result of the addition and do not know both terms? In this case, the result of the addition must be decomposed into two terms using linear angular functions. Next, we ourselves choose what one term can be, and linear angular functions show what the second term should be so that the result of the addition is exactly what we need. There can be such pairs of terms infinite set. IN Everyday life We can do just fine without decomposing the sum; subtraction is enough for us. But when scientific research laws of nature, decomposing a sum into its components can be very useful.

Another law of addition that mathematicians don't like to talk about (another of their tricks) requires that the terms have the same units of measurement. For salad, water, and borscht, these could be units of weight, volume, value, or unit of measurement.

The figure shows two levels of difference for mathematical . The first level is the differences in the field of numbers, which are indicated a, b, c. This is what mathematicians do. The second level is the differences in the field of units of measurement, which are shown in square brackets and indicated by the letter U. This is what physicists do. We can understand the third level - differences in the area of ​​​​the objects being described. Different objects can have the same number of identical units of measurement. How important this is, we can see in the example of borscht trigonometry. If we add subscripts to the same unit designation for different objects, we can say exactly what mathematical quantity describes a particular object and how it changes over time or due to our actions. Letter W I will designate water with a letter S I'll designate the salad with a letter B- borsch. This is what linear angular functions for borscht will look like.

If we take some part of the water and some part of the salad, together they will turn into one portion of borscht. Here I suggest you take a little break from borscht and remember your distant childhood. Remember how we were taught to put bunnies and ducks together? It was necessary to find how many animals there would be. What were we taught to do then? We were taught to separate units of measurement from numbers and add numbers. Yes, any one number can be added to any other number. This is a direct path to the autism of modern mathematics - we do it incomprehensibly what, incomprehensibly why, and very poorly understand how this relates to reality, because of the three levels of difference, mathematicians operate with only one. It would be more correct to learn how to move from one unit of measurement to another.

Bunnies, ducks, and little animals can be counted in pieces. One common unit of measurement for different objects allows us to add them together. This children's version tasks. Let's look at a similar task for adults. What do you get when you add bunnies and money? There are two possible solutions here.

First option. We determine the market value of the bunnies and add it to the available amount of money. We got the total value of our wealth in monetary terms.

Second option. You can add the number of bunnies to the number of banknotes we have. We will receive the amount of movable property in pieces.

As you can see, the same addition law allows you to get different results. It all depends on what exactly we want to know.

But let's get back to our borscht. Now we can see what will happen when different meanings angle of linear angular functions.

The angle is zero. We have salad, but no water. We can't cook borscht. The amount of borscht is also zero. This does not mean at all that zero borscht is equal to zero water. There can be zero borscht with zero salad (right angle).

For me personally, this is the main mathematical proof of the fact that . Zero does not change the number when added. This happens because addition itself is impossible if there is only one term and the second term is missing. You can feel about this as you like, but remember - all mathematical operations with zero were invented by mathematicians themselves, so throw away your logic and stupidly cram the definitions invented by mathematicians: “division by zero is impossible”, “any number multiplied by zero equals zero” , “beyond the puncture point zero” and other nonsense. It is enough to remember once that zero is not a number, and you will never again have a question whether zero is a natural number or not, because such a question loses all meaning: how can something that is not a number be considered a number? It's like asking what color an invisible color should be classified as. Adding a zero to a number is the same as painting with paint that is not there. We waved a dry brush and told everyone that “we painted.” But I digress a little.

The angle is greater than zero but less than forty-five degrees. We have a lot of lettuce, but not enough water. As a result, we will get thick borscht.

The angle is forty-five degrees. We have equal quantities of water and salad. This is the perfect borscht (forgive me, chefs, it's just math).

The angle is greater than forty-five degrees, but less than ninety degrees. We have a lot of water and little salad. You will get liquid borscht.

Right angle. We have water. All that remains of the salad are memories, as we continue to measure the angle from the line that once marked the salad. We can't cook borscht. The amount of borscht is zero. In this case, hold on and drink water while you have it)))

Here. Something like this. I can tell other stories here that would be more than appropriate here.

Two friends had their shares in a common business. After killing one of them, everything went to the other.

The emergence of mathematics on our planet.

All these stories are told in the language of mathematics using linear angular functions. Some other time I will show you the real place of these functions in the structure of mathematics. In the meantime, let's return to borscht trigonometry and consider projections.

Saturday, October 26, 2019

I watched an interesting video about Grundy series One minus one plus one minus one - Numberphile. Mathematicians lie. They did not perform an equality check during their reasoning.

This echoes my thoughts about .

Let's take a closer look at the signs that mathematicians are deceiving us. At the very beginning of the argument, mathematicians say that the sum of a sequence DEPENDS on whether it has an even number of elements or not. This is an OBJECTIVELY ESTABLISHED FACT. What happens next?

Next, mathematicians subtract the sequence from unity. What does this lead to? This leads to a change in the number of elements of the sequence - an even number changes to an odd number, an odd number changes to an even number. After all, we added one element equal to one to the sequence. Despite all the external similarity, the sequence before the transformation is not equal to the sequence after the transformation. Even if we are talking about an infinite sequence, we must remember that an infinite sequence with an odd number of elements is not equal to an infinite sequence with an even number of elements.

By putting an equal sign between two sequences with different numbers of elements, mathematicians claim that the sum of the sequence DOES NOT DEPEND on the number of elements in the sequence, which contradicts an OBJECTIVELY ESTABLISHED FACT. Further reasoning about the sum of an infinite sequence is false, since it is based on a false equality.

If you see that mathematicians, in the course of proofs, place brackets, rearrange elements of a mathematical expression, add or remove something, be very careful, most likely they are trying to deceive you. Like card magicians, mathematicians use various manipulations of expression to distract your attention in order to ultimately give you a false result. If you cannot repeat a card trick without knowing the secret of deception, then in mathematics everything is much simpler: you don’t even suspect anything about deception, but repeating all the manipulations with a mathematical expression allows you to convince others of the correctness of the result obtained, just like when -they convinced you.

Question from the audience: Is infinity (as the number of elements in the sequence S) even or odd? How can you change the parity of something that has no parity?

Infinity is for mathematicians, like the Kingdom of Heaven is for priests - no one has ever been there, but everyone knows exactly how everything works there))) I agree, after death you will be absolutely indifferent whether you lived an even or odd number of days, but... Adding just one day into the beginning of your life, we will get a completely different person: his last name, first name and patronymic are exactly the same, only the date of birth is completely different - he was born one day before you.

Now let’s get to the point))) Let’s say that a finite sequence that has parity loses this parity when going to infinity. Then any finite segment of an infinite sequence must lose parity. We don't see this. The fact that we cannot say for sure whether an infinite sequence has an even or odd number of elements does not mean that parity has disappeared. Parity, if it exists, cannot disappear without a trace into infinity, like in a sharpie’s sleeve. There is a very good analogy for this case.

Have you ever asked the cuckoo sitting in the clock in which direction the clock hand rotates? For her, the arrow rotates in the opposite direction to what we call “clockwise”. As paradoxical as it may sound, the direction of rotation depends solely on which side we observe the rotation from. And so, we have one wheel that rotates. We cannot say in which direction the rotation occurs, since we can observe it both from one side of the plane of rotation and from the other. We can only testify to the fact that there is rotation. Complete analogy with the parity of an infinite sequence S.

Now let's add a second rotating wheel, the plane of rotation of which is parallel to the plane of rotation of the first rotating wheel. We still can't say for sure in which direction these wheels rotate, but we can absolutely tell whether both wheels rotate in the same direction or in the opposite direction. Comparing two infinite sequences S And 1-S, I showed with the help of mathematics that these sequences have different parities and putting an equal sign between them is a mistake. Personally, I trust mathematics, I don’t trust mathematicians))) By the way, to fully understand the geometry of transformations of infinite sequences, it is necessary to introduce the concept "simultaneity". This will need to be drawn.

Wednesday, August 7, 2019

Concluding the conversation about, we need to consider an infinite set. The point is that the concept of “infinity” affects mathematicians like a boa constrictor affects a rabbit. The trembling horror of infinity deprives mathematicians of common sense. Here's an example:

The original source is located. Alpha stands for real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take the infinite set as an example natural numbers, then the considered examples can be presented as follows:

To clearly prove that they were right, mathematicians came up with many different methods. Personally, I look at all these methods as shamans dancing with tambourines. Essentially, they all boil down to the fact that either some of the rooms are unoccupied and new guests are moving in, or that some of the visitors are thrown out into the corridor to make room for guests (very humanly). I presented my view on such decisions in the form of a fantasy story about the Blonde. What is my reasoning based on? Relocating an infinite number of visitors takes an infinite amount of time. After we have vacated the first room for a guest, one of the visitors will always walk along the corridor from his room to the next one until the end of time. Of course, the time factor can be stupidly ignored, but this will be in the category of “no law is written for fools.” It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.

What is an “endless hotel”? An infinite hotel is a hotel that always has any number of empty beds, regardless of how many rooms are occupied. If all the rooms in the endless "visitor" corridor are occupied, there is another endless corridor with "guest" rooms. There will be an infinite number of such corridors. Moreover, the “infinite hotel” has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians are not able to distance themselves from banal everyday problems: there is always only one God-Allah-Buddha, there is only one hotel, there is only one corridor. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to “shove in the impossible.”

I will demonstrate the logic of my reasoning to you using the example of an infinite set of natural numbers. First you need to answer a very simple question: how many sets of natural numbers are there - one or many? There is no correct answer to this question, since we invented numbers ourselves; numbers do not exist in Nature. Yes, Nature is great at counting, but for this she uses other mathematical tools that are not familiar to us. I’ll tell you what Nature thinks another time. Since we invented numbers, we ourselves will decide how many sets of natural numbers there are. Let's consider both options, as befits real scientists.

Option one. “Let us be given” one single set of natural numbers, which lies serenely on the shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and nowhere to take them. We cannot add one to this set, since we already have it. What if you really want to? No problem. We can take one from the set we have already taken and return it to the shelf. After that, we can take one from the shelf and add it to what we have left. As a result, we will again get an infinite set of natural numbers. You can write down all our manipulations like this:

I wrote down the actions in algebraic notation and in set theory notation, with a detailed listing of the elements of the set. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one is subtracted from it and the same unit is added.

Option two. We have many different infinite sets of natural numbers on our shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. Let's take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. This is what we get:

The subscripts "one" and "two" indicate that these elements belonged to different sets. Yes, if you add one to an infinite set, the result will also be an infinite set, but it will not be the same as the original set. If you add another infinite set to one infinite set, the result is a new infinite set consisting of the elements of the first two sets.

The set of natural numbers is used for counting in the same way as a ruler is for measuring. Now imagine that you added one centimeter to the ruler. This will be a different line, not equal to the original one.

You can accept or not accept my reasoning - it is your own business. But if someday you come across math problems, think about whether you are following the path of false reasoning trodden by generations of mathematicians. After all, studying mathematics, first of all, forms a stable stereotype of thinking in us, and only then adds to our mental abilities (or, conversely, deprives us of free-thinking).

pozg.ru

Sunday, August 4, 2019

I was finishing a postscript to an article about and saw this wonderful text on Wikipedia:

We read: "... rich theoretical basis The mathematics of Babylon did not have a holistic character and was reduced to a set of disparate techniques, devoid of common system and evidence base."

Wow! How smart we are and how well we can see the shortcomings of others. Is it difficult for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, I personally got the following:

The rich theoretical basis of modern mathematics is not holistic in nature and is reduced to a set of disparate sections, devoid of a common system and evidence base.

I won’t go far to confirm my words - it has a language and conventions that are different from the language and symbols many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole series of publications to the most obvious mistakes of modern mathematics. See you soon.

Saturday, August 3, 2019

How to divide a set into subsets? To do this, you need to enter a new unit of measurement that is present in some of the elements of the selected set. Let's look at an example.

May we have plenty A consisting of four people. This set is formed on the basis of “people.” Let us denote the elements of this set by the letter A, the subscript with a number will indicate the serial number of each person in this set. Let's introduce a new unit of measurement "gender" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set A based on gender b. Notice that our set of “people” has now become a set of “people with gender characteristics.” After this we can divide the sexual characteristics into male bm and women's bw sexual characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, no matter which one - male or female. If a person has it, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we use the usual school math. Look what happened.

After multiplication, reduction and rearrangement, we ended up with two subsets: the subset of men Bm and a subset of women Bw. Mathematicians reason in approximately the same way when they apply set theory in practice. But they don’t tell us the details, but give us the finished result - “a lot of people consist of a subset of men and a subset of women.” Naturally, you may have a question: how correctly has the mathematics been applied in the transformations outlined above? I dare to assure you that, in essence, the transformations were done correctly; it is enough to know the mathematical basis of arithmetic, Boolean algebra and other branches of mathematics. What it is? Some other time I will tell you about this.

As for supersets, you can combine two sets into one superset by selecting the unit of measurement present in the elements of these two sets.

As you can see, units of measurement and ordinary mathematics make set theory a relic of the past. A sign that all is not well with set theory is that for set theory mathematicians invented own language and own notations. Mathematicians acted as shamans once did. Only shamans know how to “correctly” apply their “knowledge.” They teach us this “knowledge”.

In conclusion, I want to show you how mathematicians manipulate
Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ...discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes...were involved in the study of the issue mathematical analysis, set theory, new physical and philosophical approaches; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.

From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.

If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must not be sought endlessly large numbers, but in units of measurement.

Another interesting aporia of Zeno tells about a flying arrow:

A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to the car, you need two photographs taken from different points space at one point in time, but it is impossible to determine the fact of movement from them (naturally, additional data is still needed for calculations, trigonometry will help you). What I want to point out Special attention, is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
I'll show you the process with an example. We select the “red solid in a pimple” - this is our “whole”. At the same time, we see that these things are with a bow, and there are without a bow. After that, we select part of the “whole” and form a set “with a bow”. This is how shamans get their food by tying their set theory to reality.

Now let's do a little trick. Let’s take “solid with a pimple with a bow” and combine these “wholes” according to color, selecting the red elements. We got a lot of "red". Now the final question: are the resulting sets “with a bow” and “red” the same set or two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so it will be.

This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We formed a set of "red solid with a pimple and a bow." The formation took place in four different units of measurement: color (red), strength (solid), roughness (pimply), decoration (with a bow). Only a set of units of measurement allows us to adequately describe real objects in the language of mathematics. This is what it looks like.

The letter "a" with different indices indicates different units of measurement. The units of measurement by which the “whole” is distinguished at the preliminary stage are highlighted in brackets. The unit of measurement by which the set is formed is taken out of brackets. The last line shows the final result - an element of the set. As you can see, if we use units of measurement to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not the dancing of shamans with tambourines. Shamans can “intuitively” come to the same result, arguing that it is “obvious,” because units of measurement are not part of their “scientific” arsenal.

Using units of measurement, it is very easy to split one set or combine several sets into one superset. Let's take a closer look at the algebra of this process.

Simply put, these are vegetables cooked in water according to a special recipe. I will consider two initial components (vegetable salad and water) and the finished result - borscht. Geometrically, it can be thought of as a rectangle, with one side representing lettuce and the other side representing water. The sum of these two sides will indicate borscht. The diagonal and area of ​​such a “borscht” rectangle are purely mathematical concepts and are never used in borscht recipes.

How do lettuce and water turn into borscht from a mathematical point of view? How can the sum of two line segments become trigonometry? To understand this, we need linear angular functions.

You won't find anything about linear angular functions in math textbooks. But without them there can be no mathematics. The laws of mathematics, like the laws of nature, work regardless of whether we know about their existence or not.

Linear angular functions are addition laws. See how algebra turns into geometry and geometry turns into trigonometry.

Is it possible to do without linear angular functions? It’s possible, because mathematicians still manage without them. The trick of mathematicians is that they always tell us only about those problems that they themselves know how to solve, and never tell us about those problems that they cannot solve. Look. If we know the result of addition and one term, we use subtraction to find the other term. All. We don’t know other problems and we don’t know how to solve them. What should we do if we only know the result of the addition and do not know both terms? In this case, the result of the addition must be decomposed into two terms using linear angular functions. Next, we ourselves choose what one term can be, and linear angular functions show what the second term should be so that the result of the addition is exactly what we need. There can be an infinite number of such pairs of terms. In everyday life, we get along just fine without decomposing the sum; subtraction is enough for us. But in scientific research into the laws of nature, decomposing a sum into its components can be very useful.

Another law of addition that mathematicians don't like to talk about (another of their tricks) requires that the terms have the same units of measurement. For salad, water, and borscht, these could be units of weight, volume, value, or unit of measurement.

The figure shows two levels of difference for mathematical . The first level is the differences in the field of numbers, which are indicated a, b, c. This is what mathematicians do. The second level is the differences in the field of units of measurement, which are shown in square brackets and indicated by the letter U. This is what physicists do. We can understand the third level - differences in the area of ​​​​the objects being described. Different objects can have the same number of identical units of measurement. How important this is, we can see in the example of borscht trigonometry. If we add subscripts to the same unit designation for different objects, we can say exactly what mathematical quantity describes a particular object and how it changes over time or due to our actions. Letter W I will designate water with a letter S I'll designate the salad with a letter B- borsch. This is what linear angular functions for borscht will look like.

If we take some part of the water and some part of the salad, together they will turn into one portion of borscht. Here I suggest you take a little break from borscht and remember your distant childhood. Remember how we were taught to put bunnies and ducks together? It was necessary to find how many animals there would be. What were we taught to do then? We were taught to separate units of measurement from numbers and add numbers. Yes, any one number can be added to any other number. This is a direct path to the autism of modern mathematics - we do it incomprehensibly what, incomprehensibly why, and very poorly understand how this relates to reality, because of the three levels of difference, mathematicians operate with only one. It would be more correct to learn how to move from one unit of measurement to another.

Bunnies, ducks, and little animals can be counted in pieces. One common unit of measurement for different objects allows us to add them together. This is a children's version of the problem. Let's look at a similar task for adults. What do you get when you add bunnies and money? There are two possible solutions here.

First option. We determine the market value of the bunnies and add it to the available amount of money. We got the total value of our wealth in monetary terms.

Second option. You can add the number of bunnies to the number of banknotes we have. We will receive the amount of movable property in pieces.

As you can see, the same addition law allows you to get different results. It all depends on what exactly we want to know.

But let's get back to our borscht. Now we can see what will happen for different angle values ​​of linear angular functions.

The angle is zero. We have salad, but no water. We can't cook borscht. The amount of borscht is also zero. This does not mean at all that zero borscht is equal to zero water. There can be zero borscht with zero salad (right angle).

For me personally, this is the main mathematical proof of the fact that . Zero does not change the number when added. This happens because addition itself is impossible if there is only one term and the second term is missing. You can feel about this as you like, but remember - all mathematical operations with zero were invented by mathematicians themselves, so throw away your logic and stupidly cram the definitions invented by mathematicians: “division by zero is impossible”, “any number multiplied by zero equals zero” , “beyond the puncture point zero” and other nonsense. It is enough to remember once that zero is not a number, and you will never again have a question whether zero is a natural number or not, because such a question loses all meaning: how can something that is not a number be considered a number? It's like asking what color an invisible color should be classified as. Adding a zero to a number is the same as painting with paint that is not there. We waved a dry brush and told everyone that “we painted.” But I digress a little.

The angle is greater than zero but less than forty-five degrees. We have a lot of lettuce, but not enough water. As a result, we will get thick borscht.

The angle is forty-five degrees. We have equal quantities of water and salad. This is the perfect borscht (forgive me, chefs, it's just math).

The angle is greater than forty-five degrees, but less than ninety degrees. We have a lot of water and little salad. You will get liquid borscht.

Right angle. We have water. All that remains of the salad are memories, as we continue to measure the angle from the line that once marked the salad. We can't cook borscht. The amount of borscht is zero. In this case, hold on and drink water while you have it)))

Here. Something like this. I can tell other stories here that would be more than appropriate here.

Two friends had their shares in a common business. After killing one of them, everything went to the other.

The emergence of mathematics on our planet.

All these stories are told in the language of mathematics using linear angular functions. Some other time I will show you the real place of these functions in the structure of mathematics. In the meantime, let's return to borscht trigonometry and consider projections.

Saturday, October 26, 2019

I watched an interesting video about Grundy series One minus one plus one minus one - Numberphile. Mathematicians lie. They did not perform an equality check during their reasoning.

This echoes my thoughts about .

Let's take a closer look at the signs that mathematicians are deceiving us. At the very beginning of the argument, mathematicians say that the sum of a sequence DEPENDS on whether it has an even number of elements or not. This is an OBJECTIVELY ESTABLISHED FACT. What happens next?

Next, mathematicians subtract the sequence from unity. What does this lead to? This leads to a change in the number of elements of the sequence - an even number changes to an odd number, an odd number changes to an even number. After all, we added one element equal to one to the sequence. Despite all the external similarity, the sequence before the transformation is not equal to the sequence after the transformation. Even if we are talking about an infinite sequence, we must remember that an infinite sequence with an odd number of elements is not equal to an infinite sequence with an even number of elements.

By putting an equal sign between two sequences with different numbers of elements, mathematicians claim that the sum of the sequence DOES NOT DEPEND on the number of elements in the sequence, which contradicts an OBJECTIVELY ESTABLISHED FACT. Further reasoning about the sum of an infinite sequence is false, since it is based on a false equality.

If you see that mathematicians, in the course of proofs, place brackets, rearrange elements of a mathematical expression, add or remove something, be very careful, most likely they are trying to deceive you. Like card magicians, mathematicians use various manipulations of expression to distract your attention in order to ultimately give you a false result. If you cannot repeat a card trick without knowing the secret of deception, then in mathematics everything is much simpler: you don’t even suspect anything about deception, but repeating all the manipulations with a mathematical expression allows you to convince others of the correctness of the result obtained, just like when -they convinced you.

Question from the audience: Is infinity (as the number of elements in the sequence S) even or odd? How can you change the parity of something that has no parity?

Infinity is for mathematicians, like the Kingdom of Heaven is for priests - no one has ever been there, but everyone knows exactly how everything works there))) I agree, after death you will be absolutely indifferent whether you lived an even or odd number of days, but... Adding just one day into the beginning of your life, we will get a completely different person: his last name, first name and patronymic are exactly the same, only the date of birth is completely different - he was born one day before you.

Now let’s get to the point))) Let’s say that a finite sequence that has parity loses this parity when going to infinity. Then any finite segment of an infinite sequence must lose parity. We don't see this. The fact that we cannot say for sure whether an infinite sequence has an even or odd number of elements does not mean that parity has disappeared. Parity, if it exists, cannot disappear without a trace into infinity, like in a sharpie’s sleeve. There is a very good analogy for this case.

Have you ever asked the cuckoo sitting in the clock in which direction the clock hand rotates? For her, the arrow rotates in the opposite direction to what we call “clockwise”. As paradoxical as it may sound, the direction of rotation depends solely on which side we observe the rotation from. And so, we have one wheel that rotates. We cannot say in which direction the rotation occurs, since we can observe it both from one side of the plane of rotation and from the other. We can only testify to the fact that there is rotation. Complete analogy with the parity of an infinite sequence S.

Now let's add a second rotating wheel, the plane of rotation of which is parallel to the plane of rotation of the first rotating wheel. We still can't say for sure in which direction these wheels rotate, but we can absolutely tell whether both wheels rotate in the same direction or in the opposite direction. Comparing two infinite sequences S And 1-S, I showed with the help of mathematics that these sequences have different parities and putting an equal sign between them is a mistake. Personally, I trust mathematics, I don’t trust mathematicians))) By the way, to fully understand the geometry of transformations of infinite sequences, it is necessary to introduce the concept "simultaneity". This will need to be drawn.

Wednesday, August 7, 2019

Concluding the conversation about, we need to consider an infinite set. The point is that the concept of “infinity” affects mathematicians like a boa constrictor affects a rabbit. The trembling horror of infinity deprives mathematicians of common sense. Here's an example:

The original source is located. Alpha stands for real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take the infinite set of natural numbers as an example, then the considered examples can be represented in this form:

To clearly prove that they were right, mathematicians came up with many different methods. Personally, I look at all these methods as shamans dancing with tambourines. Essentially, they all boil down to the fact that either some of the rooms are unoccupied and new guests are moving in, or that some of the visitors are thrown out into the corridor to make room for guests (very humanly). I presented my view on such decisions in the form of a fantasy story about the Blonde. What is my reasoning based on? Relocating an infinite number of visitors takes an infinite amount of time. After we have vacated the first room for a guest, one of the visitors will always walk along the corridor from his room to the next one until the end of time. Of course, the time factor can be stupidly ignored, but this will be in the category of “no law is written for fools.” It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.

What is an “endless hotel”? An infinite hotel is a hotel that always has any number of empty beds, regardless of how many rooms are occupied. If all the rooms in the endless "visitor" corridor are occupied, there is another endless corridor with "guest" rooms. There will be an infinite number of such corridors. Moreover, the “infinite hotel” has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians are not able to distance themselves from banal everyday problems: there is always only one God-Allah-Buddha, there is only one hotel, there is only one corridor. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to “shove in the impossible.”

I will demonstrate the logic of my reasoning to you using the example of an infinite set of natural numbers. First you need to answer a very simple question: how many sets of natural numbers are there - one or many? There is no correct answer to this question, since we invented numbers ourselves; numbers do not exist in Nature. Yes, Nature is great at counting, but for this she uses other mathematical tools that are not familiar to us. I’ll tell you what Nature thinks another time. Since we invented numbers, we ourselves will decide how many sets of natural numbers there are. Let's consider both options, as befits real scientists.

Option one. “Let us be given” one single set of natural numbers, which lies serenely on the shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and nowhere to take them. We cannot add one to this set, since we already have it. What if you really want to? No problem. We can take one from the set we have already taken and return it to the shelf. After that, we can take one from the shelf and add it to what we have left. As a result, we will again get an infinite set of natural numbers. You can write down all our manipulations like this:

I wrote down the actions in algebraic notation and in set theory notation, with a detailed listing of the elements of the set. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one is subtracted from it and the same unit is added.

Option two. We have many different infinite sets of natural numbers on our shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. Let's take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. This is what we get:

The subscripts "one" and "two" indicate that these elements belonged to different sets. Yes, if you add one to an infinite set, the result will also be an infinite set, but it will not be the same as the original set. If you add another infinite set to one infinite set, the result is a new infinite set consisting of the elements of the first two sets.

The set of natural numbers is used for counting in the same way as a ruler is for measuring. Now imagine that you added one centimeter to the ruler. This will be a different line, not equal to the original one.

You can accept or not accept my reasoning - it is your own business. But if you ever encounter mathematical problems, think about whether you are following the path of false reasoning trodden by generations of mathematicians. After all, studying mathematics, first of all, forms a stable stereotype of thinking in us, and only then adds to our mental abilities (or, conversely, deprives us of free-thinking).

pozg.ru

Sunday, August 4, 2019

I was finishing a postscript to an article about and saw this wonderful text on Wikipedia:

We read: "... the rich theoretical basis of the mathematics of Babylon did not have a holistic character and was reduced to a set of disparate techniques, devoid of a common system and evidence base."

Wow! How smart we are and how well we can see the shortcomings of others. Is it difficult for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, I personally got the following:

The rich theoretical basis of modern mathematics is not holistic in nature and is reduced to a set of disparate sections, devoid of a common system and evidence base.

I won’t go far to confirm my words - it has a language and conventions that are different from the language and conventions of many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole series of publications to the most obvious mistakes of modern mathematics. See you soon.

Saturday, August 3, 2019

How to divide a set into subsets? To do this, you need to enter a new unit of measurement that is present in some of the elements of the selected set. Let's look at an example.

May we have plenty A consisting of four people. This set is formed on the basis of “people.” Let us denote the elements of this set by the letter A, the subscript with a number will indicate the serial number of each person in this set. Let's introduce a new unit of measurement "gender" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set A based on gender b. Notice that our set of “people” has now become a set of “people with gender characteristics.” After this we can divide the sexual characteristics into male bm and women's bw sexual characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, no matter which one - male or female. If a person has it, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we use regular school mathematics. Look what happened.

After multiplication, reduction and rearrangement, we ended up with two subsets: the subset of men Bm and a subset of women Bw. Mathematicians reason in approximately the same way when they apply set theory in practice. But they don’t tell us the details, but give us the finished result - “a lot of people consist of a subset of men and a subset of women.” Naturally, you may have a question: how correctly has the mathematics been applied in the transformations outlined above? I dare to assure you that, in essence, the transformations were done correctly; it is enough to know the mathematical basis of arithmetic, Boolean algebra and other branches of mathematics. What it is? Some other time I will tell you about this.

As for supersets, you can combine two sets into one superset by selecting the unit of measurement present in the elements of these two sets.

As you can see, units of measurement and ordinary mathematics make set theory a relic of the past. A sign that all is not well with set theory is that mathematicians have come up with their own language and notation for set theory. Mathematicians acted as shamans once did. Only shamans know how to “correctly” apply their “knowledge.” They teach us this “knowledge”.

In conclusion, I want to show you how mathematicians manipulate
Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.

From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.

If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia of Zeno tells about a flying arrow:

A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
I'll show you the process with an example. We select the “red solid in a pimple” - this is our “whole”. At the same time, we see that these things are with a bow, and there are without a bow. After that, we select part of the “whole” and form a set “with a bow”. This is how shamans get their food by tying their set theory to reality.

Now let's do a little trick. Let’s take “solid with a pimple with a bow” and combine these “wholes” according to color, selecting the red elements. We got a lot of "red". Now the final question: are the resulting sets “with a bow” and “red” the same set or two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so it will be.

This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We formed a set of "red solid with a pimple and a bow." The formation took place in four different units of measurement: color (red), strength (solid), roughness (pimply), decoration (with a bow). Only a set of units of measurement allows us to adequately describe real objects in the language of mathematics. This is what it looks like.

The letter "a" with different indices indicates different units of measurement. The units of measurement by which the “whole” is distinguished at the preliminary stage are highlighted in brackets. The unit of measurement by which the set is formed is taken out of brackets. The last line shows the final result - an element of the set. As you can see, if we use units of measurement to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not the dancing of shamans with tambourines. Shamans can “intuitively” come to the same result, arguing that it is “obvious,” because units of measurement are not part of their “scientific” arsenal.

Using units of measurement, it is very easy to split one set or combine several sets into one superset. Let's take a closer look at the algebra of this process.

There are several options for calculating the value of the expression cos (3 / 2 Pi).

First option. Usage
This option is the easiest and simplest and consists in the fact that you need to find the corresponding values ​​in the table.

There are many varieties of the table, some of which present arguments only in radians, others in degrees, and some that contain both radians and degrees.
Sometimes it is still useful to convert the angle value into degrees to make it easier to perceive the cosine value. But it is not prohibited to use a table with degrees and radians)).
From the table we determine the value of the cosine from 3 Pi / 2 - this is 0.
Mathematical notation:

Second option. .
A convenient option if a table of trigonometric functions is not available. Here the value of the trigonometric function can be determined using trigonometric circle.


On a trigonometric circle (or circle), the values ​​of the cosine function are located on the abscissa axis.
According to the assignment, the argument of the function is 3 Pi / 2. On the circle, this value is on the ordinate axis at the very bottom. To calculate the value of a given function, you need to lower the perpendicular to the Ox axis, after which we get the value 0. Thus, the cosine of 3 Pi / 2 is equal to 0.

Third option. Usage .
If there is no table, and it is difficult to navigate the trigonometric circle, then it is useful to use the cosine graph, from which you can also determine the value.