Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
Very often terms trigonometric circle, unit circle, number circle poorly understood by students. And completely in vain. These concepts are a powerful and universal assistant in all areas of trigonometry. In fact, this is a legal cheat sheet! Drew trigonometric circle– and immediately saw the answers! Tempting? So let's learn, it would be a sin not to use such a thing. Moreover, it is not at all difficult.
To successfully work with the trigonometric circle, you need to know only three things.
First. You need to know what sine, cosine, tangent and cotangent are as applied to a right triangle. Follow the link if you haven't been yet. Then everything will be clear here too.
Second. Need to know what it is trigonometric circle, unit circle, number circle. I will tell you this right here and now.
Third. You need to know how to measure angles on a trigonometric circle, and what degree and radian measures of angles are. This will be in the next lessons.
All. Having dealt with these three whales, we get reliable, trouble-free and completely legal cheat sheet for all trigonometry at once.
And then in school textbooks with this same trigonometric circle it’s somehow not very good...
Let's start, little by little.
In the previous lesson, you learned that sine, cosine, tangent and cotangent (i.e. trigonometric functions) depend only on the angle. And they do not depend on the lengths of the sides in a right triangle. This raises an interesting question. Let us have this angle. Let's call its angle β. The letter is beautiful.)
Since there is an angle, it must have trigonometric functions! Sine, say, or cotangent... Where can I get them? There is no hypotenuse, no legs...
How to Determine Trigonometric Angle Functions without right triangle? Problem... We'll have to delve into the treasury of world knowledge again. To medieval people. They knew everything...
First of all, let's take the coordinate plane. These are the most common coordinate axes, OX - horizontally, OY - vertically. And... let's nail one side of the angle to the positive semi-axis OX. The top of the corner, naturally, is at point O. We’ll nail it firmly so as not to tear it off! Let's leave the other side movable so that the angle can be changed. We will have a sliding corner. We denote the end of the unattached side of the corner with a dot A. We get this picture:
So, the corner was added. Where is its sine and cosine? Calmly! Everything will be now.
Let's mark the coordinates of the point A on the axes. Hover your mouse over the picture and you will see everything. On OX it will be a period IN, on ОY - point WITH. It is clear that IN And WITH - These are some numbers. Point coordinates A.
So, number B will be the cosine of the angle β, and number C- his sinus!
Why did it happen? Ancient people taught us that sine and cosine are ratios of sides! Which do not depend on the lengths of the sides. And here we came up with the coordinates of the point... But! Look at the triangle OAV. Rectangular, by the way... According to the ancient definition, the cosine of the angle β is equal to the ratio of the adjacent leg to the hypotenuse. Those. OB/OA. Okay, we don't mind. Moreover, cosine and sine do not depend on the lengths of the sides. And this is absolutely great! This means that the lengths of the sides can be whatever you want. We have every right to take the length OA for a unit! It doesn't matter what. Even a meter, even a kilometer, the sine still does not change. And in this case
Like this. If we carry out the same reasoning for the sine, we find that the sine of the angle β is equal to AB. But AB = OS. Hence,
It can be said quite simply. The sine of angle β will be game coordinate of point A, and cosine – x. The words are non-standard, but so much the better. It is remembered more reliably! And you need to remember this. It's important to remember. Cosine - according to X, sine - according to Y.
No, the medieval people did not offend the ancients! Preserved the heritage! And the relationship between the parties was preserved, and the possibilities were expanded enormously!
However, where trigonometric circle!? Where unit circle!? There was not a word about circles!
Right. But there's nothing left. Take the moving side OA and turn it around point O a full turn. What kind of figure do you think point A will draw? Absolutely right! Circle! Here she is.
This is what will happen trigonometric circle.
Like this. Why is the circle trigonometric? Circle and circle... A reasonable question. Let me explain. Each point on the circle corresponds to two numbers. The X coordinate of this point and the Y coordinate of this point. What are our coordinates? Hover your cursor over the picture. Our coordinates are points B and C. That is. cosine and sine angle β. Those. trigonometric functions. That's why the circle is called trigonometric.
Remembering that OA= 1, a OA– radius, let’s figure out what this is – and unit circle Same.
And since sine and cosine are just some numbers- this trigonometric circle will also be number circle.
Three terms in one bottle.)
In this topic these concepts are: trigonometric circle, unit circle and number circle- same. More broadly, unit circle is any circle with a radius equal to one. Trigonometric circle- a practical term, just for working with the unit circle in trigonometry. That's what we're doing now. Working with the trigonometric circle.
We have already completed the first half of the work. We drew a trigonometric circle using an angle (sounds cool, right?).
Now let's do the second half of the work. Let's do the same thing, only in reverse. Let's go from the trigonometric circle to the corner.
Let us be given a unit circle. Those. simply a circle drawn on the coordinate plane with a radius of one. Let's take an arbitrary point A on the circle. Let's mark its coordinates with points B and C on the axes. As we remember, its coordinates are cosβ(by X) and sinβ(according to the game). And let's note the sine and cosine. We get this picture:
All clear? Attention, question!
Where is β!? Where is the angle β, without which there are no sine and cosine!?
We move the cursor over the picture, and... here it is, here it is angle β! It is its sine and cosine that are the coordinates of point A.
By the way, the nailed side of the corner is not drawn here. It is not needed in the previous drawings, just for understanding... Angle Always measured from the positive direction of the OX axis. From the direction of the arrow.
What if point A is taken in a different place? A circle is round... Yes please! Anywhere! Let's place, for example, point A in the second quarter, mark its coordinates, sine, cosine, as expected. Like this:
The most observant will notice that the sine of angle β is positive (point WITH- on the positive semi-axis OY), but the cosine - negative! Dot IN lies on the negative semi-axis OX.
We move the cursor over the picture and see the angle β. Angle β here is obtuse. Which, by the way, absolutely does not happen in a right triangle. Was it in vain that we expanded our capabilities?
Got the point trigonometric circle? If you take a point anywhere on a circle, its coordinates will be the cosine and sine of the angle. The angle is measured from the positive direction of the OX axis and to the straight line connecting the coordinate center with this very point on the circle.
That's all. I would like it to be simpler, but there is nowhere. By the way, my advice to you. When working with the trigonometric circle, draw not only points on the circle, but also the corner itself! Like in these pictures. It will be clearer.
You will constantly have to draw this circle in trigonometry. This is not mandatory, this is the legal cheat sheet that smart people use. Doubts? Then call me by memory signs of such expressions, for example: sin130 0, cos150 0, sin250 0, cos330 0? I’m not asking about cos1050 0 or sin(-145 0)... Such angles are written about in the next lesson.
And you won’t find a hint anywhere. Only on the trigonometric circle. Let's draw exemplary the angle is in the correct quarter and we immediately see where its sine and cosine fall. On positive semi-axes, or negative ones. By the way, determining the signs of trigonometric functions is constantly required in a variety of tasks...
Or else, purely for example... Do you need, for example, to find out what is greater, sin130 0, or sin155 0? Just try and figure it out...
And we are smart, we will draw a trigonometric circle. And draw an angle on it approximately 130 degrees. Based on only from that that it is more than 90 and less than 180 degrees. Let's focus on the angle, not the circle! Where the moving side of the angle intersects the circle, it will intersect there. We mark the game coordinate of the intersection point. This will be sin130 0 . Like this picture:
And then, right here, we'll draw an angle of 155 degrees. Let's draw it approximately, knowing that it is more than 130 degrees. And less than 180. Let's also note its sine. Hover your cursor over the picture and you will see everything. So what, which sine is greater? It’s really hard to make a mistake here! Of course sin130 0 is greater than sin155 0!
For a long time? Yah?! Nobody requires you to carefully draw the picture and provide animation! You will work with this site, and for this task you will draw a picture like this in 10 seconds:
Another person won’t even understand what kind of scribbles this is, yes... But you will calmly and confidently give the correct answer! Although, accuracy doesn’t hurt... Otherwise, you can draw such a “circle” that the answer will be the opposite...
This problem is just one example of the wide possibilities of the trigonometric circle. It is quite possible to master these opportunities. That's what we'll do next.
Most often you will have to deal with trigonometric functions in regular, algebraic notation. Like sin45 0, tg(-3), cos(x+y) and so on. Without any pictures or trigonometric circles! You have to draw this very circle yourself. With your hands. If, of course, you want to solve trigonometry problems easily and correctly. Including the most advanced ones. But don't worry too much. On this site, in trigonometry, I will provide you with drawing circles! And you will master this extremely useful technique. Definitely.
Let's summarize the lesson.
In this topic, we smoothly moved from trigonometric functions of an angle in a right triangle to trigonometric functions any corner. To do this, we needed to master the concepts "trigonometric circle, unit circle, number circle." It is very useful.)
Here I talked about the trigonometric circle as applied to sine and cosine. But tangent and cotangent are also possible see on the circle! One movement of the pen, and you can easily and correctly determine the sign of the tangent - the cotangent of any angle, solve trigonometric inequalities and generally amaze those around you with your trigonometric abilities.)
If you are interested in such perspectives, you can visit the lesson “Tangent and cotangent on the trigonometric circle” in Special Section 555.
What do 1000 degree angles look like? What do negative angles look like? What is this mysterious number “Pi” that you inevitably come across in any section of trigonometry? And which way is this “Pi” attached to the corners? All this is in the following lessons.
On the trigonometric circle, in addition to angles in degrees, we observe .
More information about radians:
A radian is defined as the angular value of an arc whose length is equal to its radius. Accordingly, since the circumference is equal to 
, then it is obvious that radians fit into the circle, that is
1 rad ≈ 57.295779513° ≈ 57°17′44.806″ ≈ 206265″.
Everyone knows that a radian is
So, for example, , and . That's how we learned to convert radians to angles.
Now it's the other way around let's convert degrees to radians.
Let's say we need to convert to radians. It will help us. We proceed as follows:
Since, radians, let’s fill out the table:
We are training to find the values of sine and cosine in a circle
Let's clarify the following.
Well, okay, if we are asked to calculate, say, - there is usually no confusion here - everyone first starts looking on the circle.
And if you are asked to calculate, for example,... Many people suddenly begin to not understand where to look for this zero... They often look for it at the origin. Why?
1) Let's agree once and for all! What comes after or is the argument = angle, and our corners are located on the circle, don't look for them on the axes!(It’s just that individual points fall on both the circle and the axis...) And we look for the values of sines and cosines themselves on the axes!
2) And one more thing! If we depart from the “start” point counterclock-wise(the main direction of traversing the trigonometric circle), then we postpone the positive values of the angles, the angle values increase when moving in this direction.
If we depart from the “start” point clockwise, then we plot negative angle values.
Example 1.
Find the value.
Solution:
We find it on a circle. We project the point onto the sine axis (that is, we draw a perpendicular from the point to the sine axis (oy)).
We arrive at 0. So, .
Example 2.
Find the value.
Solution:
We find it on the circle (we go counterclockwise and again). We project the point onto the sine axis (and it already lies on the axis of sines).
We get to -1 along the sine axis.
Note that behind the point there are “hidden” points such as (we could go to the point marked as , clockwise, which means a minus sign appears), and infinitely many others.
We can give the following analogy:
Let's imagine a trigonometric circle as a stadium running track.
You may find yourself at the “Flag” point, starting from the start counterclockwise, having run, say, 300 m. Or having run, say, 100 m clockwise (we assume the length of the track is 400 m).
You can also end up at the Flag point (after the start) by running, say, 700m, 1100m, 1500m, etc. counter-clockwise. You can end up at the Flag point by running 500m or 900m etc. clockwise from the start.
Mentally turn the stadium treadmill into a number line. Imagine where on this line the values 300, 700, 1100, 1500, etc. will be, for example. We will see points on the number line that are equally spaced from each other. Let's turn back into a circle. The points “stick together” into one.
So it is with the trigonometric circle. Behind each point there are infinitely many others hidden.
Let's say angles , , , etc. are represented by one dot. And the values of sine and cosine in them, of course, coincide. (Did you notice that we added/subtracted or ? This is the period for the sine and cosine function.)
Example 3.
Find the value.
Solution:
Let's convert to degrees for simplicity.
(later, when you get used to the trigonometric circle, you won't need to convert radians to degrees):
We will move clockwise from the point We will go half a circle () and another
We understand that the value of the sine coincides with the value of the sine and is equal to
Note that if we took, for example, or, etc., then we would get the same sine value.
Example 4.
Find the value.
Solution:
However, we will not convert radians to degrees, as in the previous example.
That is, we need to go counterclockwise half a circle and another quarter half a circle and project the resulting point onto the cosine axis (horizontal axis).
Example 5.
Find the value.
Solution:
How to plot on a trigonometric circle?
If we pass or, at least, we will still find ourselves at the point that we designated as “start”. Therefore, you can immediately go to a point on the circle
Example 6.
Find the value.
Solution:
We will end up at the point (it will still take us to point zero). We project the point of the circle onto the cosine axis (see trigonometric circle), we find ourselves in . That is .
The trigonometric circle is in your hands
You already understand that the main thing is to remember the values of the trigonometric functions of the first quarter. In the remaining quarters everything is similar, you just need to follow the signs. And I hope you won’t forget the “ladder chain” of values of trigonometric functions. 
How to find tangent and cotangent values main angles.
After which, having become familiar with the basic values of tangent and cotangent, you can pass
On a blank circle template. Train!
In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the “Achilles and the Tortoise” aporia. Here's what it sounds like: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.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary”. Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...
And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now this is mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different number systems the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With the large number 12345, I don’t want to fool my head, let’s consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.
As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
In this article we will analyze in great detail the definition of the number circle, find out its main property and arrange the numbers 1,2,3, etc. About how to mark other numbers on the circle (for example, \(\frac(π)(2), \frac(π)(3), \frac(7π)(4), 10π, -\frac(29π)( 6)\)) understands .
Number circle called a circle of unit radius whose points correspond , arranged according to the following rules:
1) The origin is at the extreme right point of the circle;
2) Counterclockwise - positive direction; clockwise – negative;
3) If we plot the distance \(t\) on the circle in the positive direction, then we will get to a point with the value \(t\);
4) If we plot the distance \(t\) on the circle in a negative direction, then we will get to a point with the value \(–t\).
Why is the circle called a number circle?
Because it has numbers on it. In this way, the circle is similar to the number axis - on the circle, like on the axis, there is a specific point for each number.
Why know what a number circle is?
Using the number circle, the values of sines, cosines, tangents and cotangents are determined. Therefore, to know trigonometry and pass the Unified State Exam with 60+ points, you must understand what a number circle is and how to place dots on it.
What do the words “...of unit radius...” mean in the definition?
This means that the radius of this circle is equal to \(1\). And if we construct such a circle with the center at the origin, then it will intersect with the axes at points \(1\) and \(-1\).
It doesn’t have to be drawn small; you can change the “size” of the divisions along the axes, then the picture will be larger (see below).
Why is the radius exactly one? This is more convenient, because in this case, when calculating the circumference using the formula \(l=2πR\), we get:
The length of the number circle is \(2π\) or approximately \(6.28\).
What does “...the points of which correspond to real numbers” mean?
As we said above, on the number circle for any real number there will definitely be its “place” - a point that corresponds to this number.
Why determine the origin and direction on the number circle?
The main purpose of the number circle is to uniquely determine its point for each number. But how can you determine where to put the point if you don’t know where to count from and where to move?
Here it is important not to confuse the origin on the coordinate line and on the number circle - these are two different reference systems! And also do not confuse \(1\) on the \(x\) axis and \(0\) on the circle - these are points on different objects.
Which points correspond to the numbers \(1\), \(2\), etc.?
Remember, we assumed that the number circle has a radius of \(1\)? This will be our unit segment (by analogy with the number axis), which we will plot on the circle.
To mark a point on the number circle corresponding to the number 1, you need to go from 0 to a distance equal to the radius in the positive direction.
To mark a point on the circle corresponding to the number \(2\), you need to travel a distance equal to two radii from the origin, so that \(3\) is a distance equal to three radii, etc.
When looking at this picture, you may have 2 questions:
1. What happens when the circle “ends” (i.e. we make a full revolution)?
Answer: let's go for the second round! And when the second one is over, we’ll go to the third one and so on. Therefore, an infinite number of numbers can be plotted on a circle.
2. Where will the negative numbers be?
Answer: right there! They can also be arranged, counting from zero the required number of radii, but now in a negative direction.
Unfortunately, it is difficult to denote integers on the number circle. This is due to the fact that the length of the number circle will not be equal to an integer: \(2π\). And at the most convenient places (at the points of intersection with the axes) there will also be fractions, not integers
In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the “Achilles and the Tortoise” aporia. Here's what it sounds like: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.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary”. Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...
And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now this is mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different number systems the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With the large number 12345, I don’t want to fool my head, let’s consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.
As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.