In a certain proportion with others. Make a proportion

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.

Sign on the door He opens the door and says:

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.

The ratio of the number x to the number y is the quotient of the numbers u, i.e. y/ x or x: y. The ratio shows how many times x is greater than y, or what fraction of y is x. Proportion is the equality of two ratios, i.e. a/b = x/y. The numbers a and y are called the extreme terms, and the numbers x and b are called the middle terms of the proportion.

Properties of proportion

(main): the product of the extreme terms of a proportion is equal to the product of its middle terms, i.e. if a / b = x / y, then ay = bx.

Conversely, the numbers a, b, x, y form the proportion a/b = x/y, if ay = bx.

If in a proportion we swap the extremes, middle terms, or both at the same time, we get the correct proportion.

To find the unknown middle (or extreme) term of the proportion, you need to divide the product of the extreme (middle) terms by the known middle (extreme) term of the proportion

Dividing a number into parts directly and inversely proportional to the given numbers

To divide a number in proportion to given numbers (divide in a given ratio), you need to divide this number by the sum of the given numbers and multiply the result by each of them.

To divide a number into parts inversely proportional to the given numbers, it is enough to divide this number into parts directly proportional to the numbers inversely proportional to the given numbers.

For example, let's divide 27 in inverse proportion to the numbers 4 and 5. The reciprocals of the data are related as (1/4) : (1/5) = 5: 4; then we get

A percentage is a hundredth part of a number. Percentage is indicated by the % sign.

If this number is taken as 1, then 1% is 0.01 of this number, 25% is 0.25 of the number (or 1/4 of the number), etc. Thus, in order to present the number of percents as a fraction, it is enough to divide the number of percents by 100. For example, 125% = 1.25; 2.3% = 0.023.

Basic problems on percentages

Finding the percentage of a given number.

To find a % of the number b, you need to express the percentages as a fraction: a /100 and multiply the number b by this fraction.

For example, 30% of 60 rubles. are 0.3 60 = 18 (rub.).

Finding a number by its percentage.

If it is known that a% of a number x is equal to b, then we find the number x using the formula. Those. You need to express the percentages as a fraction and divide the known number b by this fraction.

For example, if 3% of the cash deposit is 150 rubles, then the entire deposit is equal to 150/0.03 = 5000 (rubles).

Finding the percentage of numbers.

To find the percentage ratio of two numbers a and b, you need to multiply the ratio of these numbers by 100, i.e. calculate.

For example, if, with a planned target of 60 cars per day, the plant produced 90 cars, then it completed the target by

(90/60) 100% = 150%.

“The Father of Comedy” Aristophanes the Ancient Greek Several times in his comedies Aristophanes turns to such an unpleasant phenomenon as informers. For example, for Nikarch, the hero of “Akharnyan...

The theme of love in the Russian novel (Based on the novels “Oblomov”, “Fathers and Sons”, “War and Peace”)... The theme of love suffering is most closely associated with the main character of “Fathers and Sons” Evgeny Bazarov. His feeling is a heavy, all-consuming...

Solving most problems in high school mathematics requires knowledge of proportions. This simple skill will help you not only perform complex exercises from the textbook, but also delve into the very essence of mathematical science. How to make a proportion? Let's figure it out now.

The simplest example is a problem where three parameters are known, and the fourth needs to be found. The proportions are, of course, different, but often you need to find some number using percentages. For example, the boy had ten apples in total. He gave the fourth part to his mother. How many apples does the boy have left? This is the simplest example that will allow you to create a proportion. The main thing is to do this. Initially there were ten apples. Let it be 100%. We marked all his apples. He gave one-fourth. 1/4=25/100. This means he has left: 100% (it was originally) - 25% (he gave) = 75%. This figure shows the percentage of the amount of fruit remaining compared to the amount initially available. Now we have three numbers by which we can already solve the proportion. 10 apples - 100%, X apples - 75%, where x is the required amount of fruit. How to make a proportion? You need to understand what it is. Mathematically it looks like this. The equal sign is placed for your understanding.

10 apples = 100%;

x apples = 75%.

It turns out that 10/x = 100%/75. This is the main property of proportions. After all, the larger x, the greater the percentage of this number from the original. We solve this proportion and find that x = 7.5 apples. We do not know why the boy decided to give away an integer amount. Now you know how to make a proportion. The main thing is to find two relationships, one of which contains the unknown unknown.

Solving a proportion often comes down to simple multiplication and then division. Schools do not explain to children why this is so. Although it is important to understand that proportional relationships are mathematical classics, the very essence of science. To solve proportions, you need to be able to handle fractions. For example, you often need to convert percentages to fractions. That is, recording 95% will not work. And if you immediately write 95/100, then you can make significant reductions without starting the main calculation. It’s worth saying right away that if your proportion turns out to be with two unknowns, then it cannot be solved. No professor will help you here. And your task most likely has a more complex algorithm for correct actions.

Let's look at another example where there are no percentages. A motorist bought 5 liters of gasoline for 150 rubles. He thought about how much he would pay for 30 liters of fuel. To solve this problem, let's denote by x the required amount of money. You can solve this problem yourself and then check the answer. If you have not yet understood how to make a proportion, then take a look. 5 liters of gasoline is 150 rubles. As in the first example, we write down 5l - 150r. Now let's find the third number. Of course, this is 30 liters. Agree that a pair of 30 l - x rubles is appropriate in this situation. Let's move on to mathematical language.

5 liters - 150 rubles;

30 liters - x rubles;

Let's solve this proportion:

x = 900 rubles.

So we decided. In your task, do not forget to check the adequacy of the answer. It happens that with the wrong decision, cars reach unrealistic speeds of 5000 kilometers per hour and so on. Now you know how to make a proportion. You can also solve it. As you can see, there is nothing complicated about this.

Basic properties of proportions

Reversal of proportion. If a : b = c : d, That b : a = d : c
Multiplying the terms of a proportion crosswise. If a : b = c : d, That ad = bc.
Rearrangement of middle and extreme terms. If a : b = c : d, That

a : c = b : d(rearrangement of the middle terms of the proportion), d : b = c : a(rearrangement of the extreme terms of the proportion).

Increasing and decreasing proportions. If a : b = c : d, That

(a + b) : b = (c + d) : d (increase in proportion), (a – b) : b = (c – d) : d (decrease in proportion).

Making proportions by addition and subtraction. If a : b = c : d, That

(a + With) : (b + d) = a : b = c : d(composing proportions by addition), (a – With) : (b – d) = a : b = c : d(composing proportions by subtraction).