Each trigonometric function for a given angle (or number) α corresponds to a certain meaning this function. From the definitions of sine, cosine, tangent and cotangent, it is clear that the value of the sine of the angle α is the ordinate of the point to which the initial point of the unit circle passes after it has rotated through the angle α, the value of the cosine is the abscissa of this point, the value of the tangent is the ratio of the ordinate to the abscissa, and the value of the cotangent is the ratio of the abscissa to the ordinate.
Quite often, when solving problems, it becomes necessary to find the values of the sines, cosines, tangents and cotangents of the indicated angles. For some angles, for example, at 0, 30, 45, 60, 90, ... degrees, it is possible to find the exact values of trigonometric functions, for other angles, finding the exact values is problematic and one has to be content with approximate values.
In this article, we will figure out what principles should be followed when calculating the value of the sine, cosine, tangent or cotangent. Let's list them in order.
- The approximate value of the specified trigonometric function can be found by definition. And for angles 0, ±90, ±180, etc. degrees definition of trigonometric functions allows you to specify the exact values of the sine, cosine, tangent and cotangent.
- The ratios between the sides and angles in a right triangle allow you to find the values of sine, cosine, tangent and cotangent for the "basic" angles 30, 45, 60 degrees.
- If the angle is outside the range from 0 to 90 degrees, then you should first use the reduction formulas, which will allow you to proceed to the calculation of the value of trigonometric functions with an argument from 0 to 90 degrees.
- If the value of one of the trigonometric functions for a given angle α is known, then we can always calculate the value of any other trigonometric function of the same angle. This allows us to do basic trigonometric identities.
- It is sometimes possible to calculate the value of a given trigonometric function for a given angle, starting from the values of the functions for the principal angles and using the appropriate trigonometry formulas. For example, given the known value of the sine of 30 degrees and the half angle formula for the sine, you can find the value of the sine of 15 degrees.
- Finally, you can always find the approximate value of a given trigonometric function for a given angle by referring to the one you need from the tables of sines, cosines, tangents and cotangents.
Now let's consider each of the listed principles for calculating the values of sines, cosines, tangents and cotangents in detail.
Page navigation.
Finding the values of sine, cosine, tangent and cotangent by definition
Based on the definition of sine and cosine, you can find the values of the sine and cosine of a given angle α. To do this, you need to take a unit circle, rotate the starting point A (1, 0) by an angle α, after which it will go to point A 1. Then the coordinates of the point A 1 will give, respectively, the cosine and sine of the given angle α. The tangent and cotangent of the angle α can then be calculated by calculating the ratios of the ordinate to the abscissa and the abscissa to the ordinate, respectively.
By definition, we can calculate the exact values of the sine, cosine, tangent, and cotangent of angles 0, ±90, ±180, ±270, ±360, … degrees ( 0, ±π/2, ±π, ±3π/2, ±2π, … radians). Let's divide these angles into four groups: 360 z degrees (2π z rad), 90+360 z degrees (π/2+2π z rad), 180+360 z degrees (π+2π z rad), and 270+360 z degrees (3π/2+2π z rad), where z is any . Let's depict in the figures where the point A 1 will be located, resulting from the rotation of the starting point A by these angles (if necessary, study the material of the article the angle of rotation).
For each of these groups of angles, we find the values of the sine, cosine, tangent and cotangent using the definitions.
As for the other angles other than 0, ±90, ±180, ±270, ±360, … degrees, then by definition we can only find approximate values of sine, cosine, tangent and cotangent. For example, let's find the sine, cosine, tangent and cotangent of the angle −52 degrees.
Let's build.
According to the drawing, we find that the abscissa of point A 1 is approximately 0.62, and the ordinate is approximately −0.78. In this way, 
and 
. It remains to calculate the values of the tangent and cotangent, we have 
and 
.
It is clear that the more accurately the constructions are performed, the more accurately the approximate values of the sine, cosine, tangent and cotangent of a given angle will be found. It is also clear that finding the values of trigonometric functions, by definition, is not convenient in practice, since it is inconvenient to carry out the described constructions.
Lines of sines, cosines, tangents and cotangents
Briefly, it is worth dwelling on the so-called lines of sines, cosines, tangents and cotangents. Lines of sines, cosines, tangents and cotangents are called lines depicted together with a unit circle, having a reference point and a unit of measurement equal to one in the introduced rectangular coordinate system, they clearly represent all possible values sines, cosines, tangents and cotangents. We depict them in the drawing below.
Values of sines, cosines, tangents and cotangents of angles of 30, 45 and 60 degrees
For angles of 30, 45 and 60 degrees, the exact values of sine, cosine, tangent and cotangent are known. They can be obtained from the definitions of the sine, cosine, tangent and cotangent of an acute angle in a right triangle using pythagorean theorems.
To obtain the values of trigonometric functions for angles of 30 and 60 degrees, consider a right triangle with these angles, and take it such that the length of the hypotenuse is equal to one. It is known that the leg opposite the angle of 30 degrees is half the hypotenuse, therefore, its length is 1/2. We find the length of the other leg using the Pythagorean theorem: 
.
Since the sine of an angle is the ratio of the opposite leg to the hypotenuse, then 
and 
. In turn, the cosine is the ratio of the adjacent leg to the hypotenuse, then 
and 
. The tangent is the ratio of the opposite leg to the adjacent leg, and the cotangent is the ratio of the adjacent leg to the opposite leg, therefore, 
and 
, as well as 
and 
.
It remains to get the values of sine, cosine, tangent and cotangent for an angle of 45 degrees. Let's turn to a right triangle with angles of 45 degrees (it will be isosceles) and a hypotenuse equal to one. Then, by the Pythagorean theorem, it is easy to check that the lengths of the legs are equal. Now we can calculate the values of sine, cosine, tangent and cotangent as the ratio of the lengths of the corresponding sides of the considered right triangle. We have and 
.
The obtained values of the sine, cosine, tangent and cotangent of the angles 30, 45 and 60 degrees will be very often used in solving various geometric and trigonometric problems, so we recommend that you remember them. For convenience, we will list them in the table of basic values of sine, cosine, tangent and cotangent.
In conclusion of this paragraph, we will give an illustration of the values of the sine, cosine, tangent and cotangent of the angles 30, 45 and 60 using the unit circle and the lines of sine, cosine, tangent and cotangent.
Flattening to an angle from 0 to 90 degrees
We note right away that it is convenient to find the values of trigonometric functions when the angle is in the range from 0 to 90 degrees (from zero to pi in half rad). If the argument of the trigonometric function, the value of which we need to find, goes beyond the limits from 0 to 9 0 degrees, then we can always use the reduction formulas to find the value of the trigonometric function, the argument of which will be within the specified limits.
For example, let's find the value of the sine of 210 degrees. By representing 210 as 180+30 or as 270−60, the corresponding reduction formulas reduce our problem from finding the sine of 210 degrees to finding the value of the sine of 30 degrees, or the cosine of 60 degrees.
Let's agree for the future when finding the values of trigonometric functions, always using the reduction formulas, go to angles from the interval from 0 to 90 degrees, unless, of course, the angle is already within these limits.
It is enough to know the value of one of the trigonometric functions
Basic trigonometric identities establish relationships between the sine, cosine, tangent and cotangent of the same angle. Thus, with their help, we can use the known value of one of the trigonometric functions to find the value of any other function of the same angle.
Let's consider an example solution.
Example.
Determine what equals sine angle pi by eight, if 
.
Solution.
First, find what the cotangent of this angle is: 
Now using the formula 
, we can calculate what the square of the sine of the angle pi by eight is equal to, and therefore the desired value of the sine. We have 
It remains only to find the value of the sine. Since the angle pi by eight is the angle of the first coordinate quarter, then the sine of this angle is positive (if necessary, see the section on the theory of the signs of sine, cosine, tangent and cotangent by quarters). In this way, 
.
Answer:
.
Finding values using trigonometric formulas
In the two previous paragraphs, we have already begun to cover the issue of finding the values of sine, cosine, tangent and cotangent using trigonometry formulas. Here we only want to say that it is sometimes possible to calculate the required value of a trigonometric function using trigonometric formulas and known values \u200b\u200bof sine, cosine, tangent and cotangent (for example, for angles of 30, 45 and 60 degrees).
For example, using trigonometric formulas, we calculate the value of the tangent of the angle pi by eight, which we used in the previous paragraph to find the value of the sine.
11 degrees? The question is very difficult.
However, the exact values of trigonometric functions in practice are often not so necessary. Approximate values with some required degree of accuracy are usually sufficient. There are tables of values of trigonometric functions, from where we can always find the approximate value of the sine, cosine, tangent or cotangent of a given angle that we need. Examples of such tables are the tables of sines, cosines, tangents and cotangents of V. M. Bradis. These tables contain the values of trigonometric functions with an accuracy of four decimal places.
Bibliography.
- Algebra: Proc. for 9 cells. avg. school / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M.: Enlightenment, 1990.- 272 p.: Ill.- ISBN 5-09-002727-7
- Algebra and the beginning of the analysis: Proc. for 10-11 cells. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorova.- 14th ed.- M.: Enlightenment, 2004.- 384 p.: ill.- ISBN 5-09-013651-3.
- Bashmakov M.I. Algebra and the beginning of analysis: Proc. for 10-11 cells. avg. school - 3rd ed. - M.: Enlightenment, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
Table of values of trigonometric functions
Note. This table of values for trigonometric functions uses the √ sign to denote the square root. To denote a fraction - the symbol "/".
see also useful materials:
For determining the value of a trigonometric function, find it at the intersection of the line indicating the trigonometric function. For example, a sine of 30 degrees - we are looking for a column with the heading sin (sine) and we find the intersection of this column of the table with the line "30 degrees", at their intersection we read the result - one second. Similarly, we find cosine 60 degrees, sine 60 degrees (once again, at the intersection of the sin (sine) column and the 60 degree row, we find the value sin 60 = √3/2), etc. In the same way, the values of sines, cosines and tangents of other "popular" angles are found.
Sine of pi, cosine of pi, tangent of pi and other angles in radians
The table of cosines, sines and tangents below is also suitable for finding the value of trigonometric functions whose argument is given in radians. To do this, use the second column of angle values. Thanks to this, you can convert the value of popular angles from degrees to radians. For example, let's find the 60 degree angle in the first line and read its value in radians under it. 60 degrees is equal to π/3 radians.
The number pi uniquely expresses the dependence of the circumference of a circle on the degree measure of the angle. So pi radians equals 180 degrees.
Any number expressed in terms of pi (radian) can be easily converted to degrees by replacing the number pi (π) with 180.
Examples:
1. sine pi.
sin π = sin 180 = 0
thus, the sine of pi is the same as the sine of 180 degrees and is equal to zero.
2. cosine pi.
cos π = cos 180 = -1
thus, the cosine of pi is the same as the cosine of 180 degrees and is equal to minus one.
3. Tangent pi
tg π = tg 180 = 0
thus, the tangent of pi is the same as the tangent of 180 degrees and is equal to zero.
Table of sine, cosine, tangent values for angles 0 - 360 degrees (frequent values)
|
angle α (degrees) |
angle α (via pi) |
sin (sinus) |
cos (cosine) |
tg (tangent) |
ctg (cotangent) |
sec (secant) |
cause (cosecant) |
| 0 | 0 | 0 | 1 | 0 | - | 1 | - |
| 15 | π/12 | 2 - √3 | 2 + √3 | ||||
| 30 | π/6 | 1/2 | √3/2 | 1/√3 | √3 | 2/√3 | 2 |
| 45 | π/4 | √2/2 | √2/2 | 1 | 1 | √2 | √2 |
| 60 | π/3 | √3/2 | 1/2 | √3 | 1/√3 | 2 | 2/√3 |
| 75 | 5π/12 | 2 + √3 | 2 - √3 | ||||
| 90 | π/2 | 1 | 0 | - | 0 | - | 1 |
| 105 | 7π/12 |
- |
- 2 - √3 | √3 - 2 | |||
| 120 | 2π/3 | √3/2 | -1/2 | -√3 | -√3/3 | ||
| 135 | 3π/4 | √2/2 | -√2/2 | -1 | -1 | -√2 | √2 |
| 150 | 5π/6 | 1/2 | -√3/2 | -√3/3 | -√3 | ||
| 180 | π | 0 | -1 | 0 | - | -1 | - |
| 210 | 7π/6 | -1/2 | -√3/2 | √3/3 | √3 | ||
| 240 | 4π/3 | -√3/2 | -1/2 | √3 | √3/3 | ||
| 270 | 3π/2 | -1 | 0 | - | 0 | - | -1 |
| 360 | 2π | 0 | 1 | 0 | - | 1 | - |
If in the table of values of trigonometric functions, instead of the value of the function, a dash is indicated (tangent (tg) 90 degrees, cotangent (ctg) 180 degrees), then for a given value of the degree measure of the angle, the function does not have a definite value. If there is no dash, the cell is empty, so we have not yet entered the desired value. We are interested in what requests users come to us for and supplement the table with new values, despite the fact that the current data on the values of cosines, sines and tangents of the most common angle values is enough to solve most problems.
Table of values of trigonometric functions sin, cos, tg for the most popular angles
0, 15, 30, 45, 60, 90 ... 360 degrees
(numerical values "as per Bradis tables")
| angle value α (degrees) | value of angle α in radians | sin (sine) | cos (cosine) | tg (tangent) | ctg (cotangent) |
|---|---|---|---|---|---|
| 0 | 0 | ||||
| 15 |
0,2588 |
0,9659
|
0,2679 |
||
| 30 |
0,5000 |
0,5774 |
|||
| 45 |
0,7071 |
||||
|
0,7660 |
|||||
| 60 |
0,8660 |
0,5000
|
1,7321 |
||
|
7π/18 |
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, this can be represented as a rectangle in which one side denotes lettuce, the other side denotes water. The sum of these two sides will denote 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 in terms of mathematics? How can the sum of two segments turn into trigonometry? To understand this, we need linear angle functions.
You won't find anything about linear angle functions in math textbooks. But without them there can be no mathematics. The laws of mathematics, like the laws of nature, work whether we know they exist or not.
Linear angular functions are the laws of addition. See how algebra turns into geometry and geometry turns into trigonometry.
Is it possible to do without linear angular functions? You can, because mathematicians still manage without them. The trick of mathematicians lies in the fact that they always tell us only about those problems that they themselves can solve, and never tell us about those problems that they cannot solve. See. If we know the result of the addition and one term, we use subtraction to find the other term. Everything. We do not know other problems and we are not able to solve them. What to do if we know only the result of the addition and do not know both terms? In this case, the result of addition must be decomposed into two terms using linear angular functions. Further, we ourselves choose what one term can be, and the linear angular functions show what the second term should be in order for the result of the addition to be exactly what we need. There can be an infinite number of such pairs of terms. In everyday life, we do very well without decomposing the sum; subtraction is enough for us. But in scientific studies of the laws of nature, the expansion of the sum into terms can be very useful.
Another law of addition that mathematicians don't like to talk about (another trick of theirs) requires the terms to have the same unit of measure. For lettuce, water, and borscht, these may be units of weight, volume, cost, or unit of measure.
The figure shows two levels of difference for math. 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 area of units of measurement, which are shown in square brackets and are indicated by the letter U. This is what physicists do. We can understand the third level - the differences in the scope of the described objects. Different objects can have the same number of the same units of measure. How important this is, we can see on the example of borscht trigonometry. If we add subscripts to the same notation for the units of measurement of different objects, we can say exactly what mathematical quantity describes a particular object and how it changes over time or in connection with our actions. letter W I will mark the water with the letter S I will mark the salad with the letter B- borsch. Here's what the linear angle functions for borscht would look like.
If we take some part of the water and some part of the salad, together they will turn into one serving 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 will turn out. What then were we taught to do? We were taught to separate units from numbers and add numbers. Yes, any number can be added to any other number. This is a direct path to the autism of modern mathematics - we do not understand what, it is not clear why, and we understand very poorly how this relates to reality, because of the three levels of difference, mathematicians operate on only one. It will be more correct to learn how to move from one unit of measurement to another.
And bunnies, and 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 problem 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 cash. We got the total value of our wealth in terms of money.
Second option. You can add the number of bunnies to the number of banknotes we have. We will get 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 back to our borscht. Now we can see what happens 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. Zero borsch can also be at 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 is because addition itself is impossible if there is only one term and the second term is missing. You can treat it as you like, but remember - everything mathematical operations mathematicians themselves came up with zero, so discard your logic and stupidly cramming the definitions invented by mathematicians: “division by zero is impossible”, “any number multiplied by zero equals zero”, “behind the point zero” and other nonsense. It is enough to remember once that zero is not a number, and you will never have a question whether zero is a natural number or not, because such a question generally loses all meaning: how can one consider a number that which is not a number. It's like asking what color to attribute an invisible color to. Adding zero to a number is like painting with paint that doesn't exist. They waved a dry brush and tell everyone that "we have 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 little water. As a result, we get a thick borscht.
The angle is forty-five degrees. We have equal amounts of water and lettuce. This is the perfect borscht (may the cooks forgive me, 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 lettuce. Get liquid borscht.
Right angle. We have water. Only memories remain of the lettuce, as we continue to measure the angle from the line that once marked the lettuce. We can't cook borscht. The amount of borscht is zero. In that case, hold on and drink water while it's available)))
Here. Something like this. I can tell other stories here that will be more than appropriate here.
The two friends had their shares in the common business. After the murder of 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 the trigonometry of borscht and consider projections.
Saturday, October 26, 2019
I watched an interesting video about Grandi's row One minus one plus one minus one - Numberphile. Mathematicians lie. They did not perform an equality test in their reasoning.
This resonates with my reasoning about .
Let's take a closer look at the signs that mathematicians are cheating us. At the very beginning of the reasoning, mathematicians say that the sum of the sequence DEPENDS on whether the number of elements in it is even 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 in the sequence - an even number changes to an odd number, an odd number changes to an even number. After all, we have 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.
Putting an equal sign between two sequences different in the number 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, because it is based on a false equality.
If you see that mathematicians place brackets in the course of proofs, rearrange the elements of a mathematical expression, add or remove something, be very careful, most likely they are trying to deceive you. Like card magicians, mathematicians distract your attention with various manipulations of expression in order to eventually slip you false result. If you can’t repeat the card trick without knowing the secret of cheating, then in mathematics everything is much simpler: you don’t even suspect anything about cheating, but repeating all the manipulations with a mathematical expression allows you to convince others of the correctness of the result, just like when have convinced you.
Question from the audience: And infinity (as the number of elements in the sequence S), is it even or odd? How can you change the parity of something that has no parity?
Infinity for mathematicians is like the Kingdom of Heaven 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 at 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.
And now to the point))) Suppose a finite sequence that has parity loses this parity when going to infinity. Then any finite segment of an infinite sequence must also lose parity. We do not observe this. The fact that we cannot say for sure whether the number of elements in an infinite sequence is even or odd does not mean at all that the parity has disappeared. Parity, if it exists, cannot disappear into infinity without a trace, as in the sleeve of a card sharper. There is a very good analogy for this case.
Have you ever asked a cuckoo sitting in a clock in which direction the clock hand rotates? For her, the arrow rotates in the opposite direction to what we call "clockwise". It may sound paradoxical, but 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 tell exactly which direction these wheels are spinning, but we can tell with absolute certainty whether both wheels are spinning in the same direction or in opposite directions. Comparing two infinite sequences S and 1-S, I showed with the help of mathematics that these sequences have different parity and putting an equal sign between them is a mistake. Personally, I believe in mathematics, I do not trust mathematicians))) By the way, in order 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. Gave in that the concept of "infinity" acts on mathematicians, like a boa constrictor on a rabbit. The quivering horror of infinity deprives mathematicians of common sense. Here is an example:
The original source is located. Alpha denotes a 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 an infinite set of natural numbers as an example, then the considered examples can be represented as follows:
To visually prove their case, mathematicians have come up with many different methods. Personally, I look at all these methods as the dances of shamans with tambourines. In essence, they all come down to the fact that either some of the rooms are not occupied and new guests are settled in them, or that some of the visitors are thrown out into the corridor to make room for the guests (very humanly). I presented my view on such decisions in the form of a fantastic story about the Blonde. What is my reasoning based on? Moving an infinite number of visitors takes an infinite amount of time. After we have vacated the first guest room, 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 already be from the category of "the law is not written for fools." It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.
What is an "infinite hotel"? The Infinite Hotel is a hotel that always has any number of free places, no matter how many rooms are occupied. If all the rooms in the endless hallway "for visitors" are occupied, there is another endless hallway with rooms for "guests". There will be an infinite number of such corridors. At the same time, 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 endless amount Gods. Mathematicians, on the other hand, are not able to move away from banal everyday problems: God-Allah-Buddha is always only one, the hotel is one, the corridor is only one. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to "shove the unpushed".
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 exist - one or many? There is no correct answer to this question, since we ourselves invented numbers, there are no numbers in Nature. Yes, Nature knows how to count perfectly, but for this she uses other mathematical tools that are not familiar to us. As Nature thinks, I will tell you another time. Since we invented the numbers, we ourselves will decide how many sets of natural numbers exist. Consider both options, as befits a real scientist.
Option one. "Let us be given" a single set of natural numbers, which lies serenely on a shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and there is 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 a unit from the set we have already taken and return it to the shelf. After that, we can take a unit from the shelf and add it to what we have left. As a result, we again get an infinite set of natural numbers. You can write all our manipulations like this:
I have written down the operations in algebraic notation and in set theory notation, listing the elements of the set in detail. 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 one is added.
Option two. We have many different infinite sets of natural numbers on the shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. We 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. Here's 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 another infinite set is added 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 for measurements. Now imagine that you have added one centimeter to the ruler. This will already be a different line, not equal to the original.
You can accept or not accept my reasoning - this is your own business. But if you ever run into mathematical problems, consider whether you are on the path of false reasoning, trodden by generations of mathematicians. After all, mathematics classes, first of all, form a stable stereotype of thinking in us, and only then they add mental abilities to us (or vice versa, they deprive us of free thinking).
pozg.ru
Sunday, August 4, 2019
I was writing a postscript to an article about and saw this wonderful text on Wikipedia:
We read: "... the rich theoretical basis of Babylonian mathematics 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 weak for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, personally I got the following:
The rich theoretical basis of modern mathematics does not have a holistic character 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 symbols 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 cycle of publications to the most obvious blunders of modern mathematics. See you soon.
Saturday, August 3, 2019
How to divide a set into subsets? To do this, you must enter a new unit of measure, which is present in some of the elements of the selected set. Consider an example.
May we have many BUT consisting of four people. This set is formed on the basis of "people" Let's designate the elements of this set through the letter a, the subscript with a digit will point to serial number each person in this set. Let's introduce a new unit of measurement "sexual characteristic" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set BUT on gender b. Notice that our "people" set has now become the "people with gender" set. After that, we can divide the sexual characteristics into male bm and women's bw gender characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, it does not matter which one is male or female. If it is present in a person, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we apply the usual school mathematics. See what happened.
After multiplication, reductions and rearrangements, we got two subsets: the male subset bm and a subset of women bw. Approximately the same way mathematicians reason when they apply set theory in practice. But they do not let us in on the details, but give us the finished result - "a lot of people consists of a subset of men and a subset of women." Naturally, you may have a question, how correctly applied mathematics in the above transformations? I dare to assure you that in fact the transformations are done correctly, it is enough to know the mathematical justification of arithmetic, Boolean algebra and other sections of mathematics. What it is? Some other time I will tell you about it.
As for supersets, it is possible to combine two sets into one superset by choosing a unit of measurement that is present in the elements of these two sets.
As you can see, units of measurement and common math make set theory a thing 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. The mathematicians did what the shamans once did. Only shamans know how to "correctly" apply their "knowledge". This "knowledge" they teach us.
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 paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about 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 universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.
From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.
If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its 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 infinitely quickly overtake the tortoise."
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise crawls 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 one hundred steps. Now Achilles is eight hundred paces 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 insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet 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 of a flying arrow:
A flying arrow is motionless, since at each 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 the flying arrow rests at different points in space, which, in fact, is movement. There is another point 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 the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (naturally, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.
I will show the process with an example. We select "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 a part of the "whole" and form a set "with a bow". This is how shamans feed themselves by tying their set theory to reality.
Now let's do a little trick. Let's take "solid in a pimple with a bow" and unite these "whole" by color, selecting red elements. We got a lot of "red". Now a tricky question: are the received 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 be it.
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 pimply with a bow". The formation took place according to four different units of measurement: color (red), strength (solid), roughness (in a bump), decorations (with a bow). Only a set of units of measurement makes it possible to adequately describe real objects in the language of mathematics. Here's what it looks like.
The letter "a" with different indices denotes different units measurements. In parentheses, units of measurement are highlighted, according to which the "whole" is allocated at the preliminary stage. The unit of measurement, according to 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 to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not the dances of shamans with tambourines. Shamans can “intuitively” come to the same result, arguing it with “obviousness”, because units of measurement are not included in their “scientific” arsenal.
With the help of units of measurement, it is very easy to break one or combine several sets into one superset. Let's take a closer look at the algebra of this process.