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 is at rest 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 (of course, 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.
Wednesday, July 4, 2018
Very well the differences between set and multiset are described in Wikipedia. We look.
As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.
Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of 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 desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.
First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will frantically recall physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms for each coin is unique ...
And now I have the most interesting question: where is the boundary beyond which 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 here is not even close.
Look here. We select football stadiums with the same field area. The area of the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace 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 will 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 they are shamans for that, 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 "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can 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 elementarily.
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's say we 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 to a number graphic symbol. This is not a mathematical operation.
2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic characters to numbers. This is not a mathematical operation.
4. Add up the resulting numbers. Now that's mathematics.
The sum of the digits of the number 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that is not all.
From the point of view of mathematics, it does not matter in which number system we write the 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 a large number of 12345, I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have 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 like finding the area of a rectangle in meters and centimeters would give you completely different results.
Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement of 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 action does not depend on the value of the number, the unit of measure used, and on who performs this action.
Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?
Female... A halo on top and an arrow down is male.
If you have such a work of design art flashing before your eyes several times a day,
Then it is not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.
1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.
TABLE OF VALUES OF TRIGONOMETRIC FUNCTIONS
The table of values of trigonometric functions is compiled for angles of 0, 30, 45, 60, 90, 180, 270 and 360 degrees and their corresponding angles in radians. Of the trigonometric functions, the table shows the sine, cosine, tangent, cotangent, secant and cosecant. For the convenience of solving school examples, the values \u200b\u200bof trigonometric functions in the table are written as a fraction with the preservation of the signs of extracting the square root from numbers, which very often helps to reduce complex mathematical expressions. For tangent and cotangent, the values of some angles cannot be determined. For the values of the tangent and cotangent of such angles, there is a dash in the table of values of trigonometric functions. It is generally accepted that the tangent and cotangent of such angles is equal to infinity. On a separate page are formulas for reducing trigonometric functions.
The table of values for the trigonometric function sine shows the values \u200b\u200bfor 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 degree measure, which corresponds to cos 0 pi, cos pi to 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 function tangent gives values \u200b\u200bfor 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 functions of the tangent are not defined tg 90, tg 270, tg pi/2, tg 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, tg 3 pi/2 in radian measure of angles. The following values of 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 simplify the reduction of fractions in school examples.
Three more monsters of trigonometry. The first is the tangent of 1.5 degrees and a half, 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 the values of the sine and cosine functions visually represents the signs of the sine and cosine depending on the magnitude of the angle. Especially for blondes, the cosine values are underlined with a green dash in order to be less confused. The conversion of degrees to radians is also very clearly presented, when radians are expressed through pi.
This trigonometric table presents the values of sine, cosine, tangent and cotangent for angles from 0 zero to 90 ninety degrees in one degree intervals. For the first forty-five degrees, the names of trigonometric functions must 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 trigonometric functions in the lower part of the trigonometric table are different from the names in the upper part 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. The sine has positive values from 0 to 180 degrees or from 0 to pi. The negative values of the sine are 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 from pi to 3/2 pi. Negative tangent and cotangent are 90 to 180 degrees and 270 to 360 degrees, or 1/2 pi to pi and 3/2 pi to 2 pi. When determining the signs of trigonometric functions for angles greater than 360 degrees or 2 pi, the periodicity properties of these functions should be used.
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. When multiplying and dividing trigonometric functions, you must follow the rules of signs.
The table of values for the trigonometric function sine shows the values \u200b\u200bfor the following angles
DocumentA separate page contains casting formulas trigonometricfunctions. AT tablevaluesfortrigonometricfunctionssinusgivenvaluesfornextcorners: sin 0, sin 30, sin 45 ...
The proposed mathematical apparatus is a complete analogue of the complex calculus for n-dimensional hypercomplex numbers with any number of degrees of freedom n and is intended for mathematical modeling of nonlinear
Document... functions equals functions Images. From this theorem should, what for finding the coordinates U, V, it is enough to calculate function... geometry; polynar functions(multidimensional analogues of two-dimensional trigonometricfunctions), their properties, tables and application; ...
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1. Trigonometric functions are elementary functions whose argument is injection. Trigonometric functions describe the relationships between sides and acute angles in a right triangle. The areas of application of trigonometric functions are extremely diverse. So, for example, any periodic processes can be represented as a sum of trigonometric functions (Fourier series). These functions often appear when solving differential and functional equations.
2. Trigonometric functions include the following 6 functions: sinus, cosine, tangent,cotangent, secant and cosecant. For each of these functions, there is an inverse trigonometric function.
3. It is convenient to introduce the geometric definition of trigonometric functions using unit circle. The figure below shows a circle with radius r=1. The point M(x,y) is marked on the circle. The angle between the radius vector OM and the positive direction of the Ox axis is α.
4. sinus the angle α is the ratio of the ordinate y of the point M(x,y) to the radius r:
sinα=y/r.
Since r=1, then the sine is equal to the ordinate of the point M(x,y).5. cosine the angle α is the ratio of the abscissa x of the point M(x,y) to the radius r:
cosα=x/r6. tangent the angle α is the ratio of the ordinate y of the point M(x,y) to its abscissa x:
tanα=y/x,x≠07. Cotangent the angle α is the ratio of the abscissa x of the point M(x,y) to its ordinate y:
cotα=x/y,y≠08. Secant angle α is the ratio of the radius r to the abscissa x of the point M(x,y):
secα=r/x=1/x,x≠09. Cosecant angle α is the ratio of the radius r to the ordinate y of the point M(x,y):
cscα=r/y=1/y,y≠010. In the unit circle of the projection x, y, the points M(x, y) and the radius r form a right triangle, in which x, y are the legs, and r is the hypotenuse. Therefore, the above definitions of trigonometric functions as applied to a right triangle are formulated as follows:
sinus angle α is the ratio of the opposite leg to the hypotenuse.
cosine angle α is the ratio of the adjacent leg to the hypotenuse.
tangent angle α is called the opposite leg to the adjacent one.
Cotangent angle α is called the adjacent leg to the opposite.
Secant angle α is the ratio of the hypotenuse to the adjacent leg.
Cosecant angle α is the ratio of the hypotenuse to the opposite leg.11. sine function graph
y=sinx, domain: x∈R, domain: −1≤sinx≤112. Graph of the cosine function
y=cosx, domain: x∈R, range: −1≤cosx≤113. tangent function graph
y=tanx, domain: x∈R,x≠(2k+1)π/2, domain: −∞
14. Graph of the cotangent function
y=cotx, domain: x∈R,x≠kπ, domain: −∞
15. Graph of the secant function
y=secx, domain: x∈R,x≠(2k+1)π/2, domain: secx∈(−∞,−1]∪∪)
