What is an integer. Whole numbers
Negative numbers were first used in ancient China and India; in Europe they were introduced into mathematical use by Nicolas Chuquet (1484) and Michael Stiefel (1544).
Algebraic properties
is not closed under division of two integers (for example, 1/2). The following table illustrates several basic properties of addition and multiplication for any integer a, b And c.
| addition | multiplication | |
| closedness: | a + b- whole | a × b- whole |
| associativity: | a + (b + c) = (a + b) + c | a × ( b × c) = (a × b) × c |
| commutativity: | a + b = b + a | a × b = b × a |
| existence of a neutral element: | a + 0 = a | a× 1 = a |
| existence of the opposite element: | a + (−a) = 0 | a≠ ±1 ⇒ 1/ a is not integer |
| distributivity of multiplication relative to addition: | a × ( b + c) = (a × b) + (a × c) | |
number systems |heading4= Hierarchy of numbers |list4=
|
|||||||||||||
| Complex numbers | |||||||||||||
number systems
|list5=Cardinal numbers – You definitely need to move it to the bed, it won’t be possible here...
The patient was so surrounded by doctors, princesses and servants that Pierre no longer saw that red-yellow head with a gray mane, which, despite the fact that he saw other faces, did not leave his sight for a moment during the entire service. Pierre guessed from the careful movement of the people surrounding the chair that the dying man was being lifted and carried.
“Hold on to my hand, you’ll drop me like this,” he heard the frightened whisper of one of the servants, “from below... there’s another one,” said the voices, and the heavy breathing and stepping of the people’s feet became more hasty, as if the weight they were carrying was beyond their strength .
The carriers, among whom was Anna Mikhailovna, drew level with the young man, and for a moment, from behind the backs and backs of the people’s heads, he saw a high, fat, open chest, the fat shoulders of the patient, raised upward by the people holding him under the arms, and a gray-haired, curly, lion's head. This head, with an unusually wide forehead and cheekbones, a beautiful sensual mouth and a majestic cold gaze, was not disfigured by the proximity of death. She was the same as Pierre knew her three months ago, when the count let him go to Petersburg. But this head swayed helplessly from the uneven steps of the carriers, and the cold, indifferent gaze did not know where to stop.
Several minutes of fussing around the high bed passed; the people carrying the sick man dispersed. Anna Mikhailovna touched Pierre's hand and told him: “Venez.” [Go.] Pierre walked with her to the bed on which the sick man was laid in a festive pose, apparently related to the sacrament that had just been performed. He lay with his head high on the pillows. His hands were laid out symmetrically on the green silk blanket, palms down. When Pierre approached, the count looked straight at him, but he looked with a look whose meaning and meaning cannot be understood by a person. Either this look said absolutely nothing except that as long as you have eyes, you must look somewhere, or it said too much. Pierre stopped, not knowing what to do, and looked questioningly at his leader Anna Mikhailovna. Anna Mikhailovna made a hasty gesture to him with her eyes, pointing to the patient’s hand and blowing her a kiss with her lips. Pierre, diligently craning his neck so as not to get caught in the blanket, followed her advice and kissed the big-boned and fleshy hand. Not a hand, not a single muscle of the count’s face trembled. Pierre again looked questioningly at Anna Mikhailovna, now asking what he should do. Anna Mikhailovna pointed him with her eyes to the chair that stood next to the bed. Pierre obediently began to sit down on the chair, his eyes continuing to ask whether he had done what was necessary. Anna Mikhailovna nodded her head approvingly. Pierre again assumed the symmetrically naive position of an Egyptian statue, apparently regretting that his clumsy and fat body occupied such a large space, and using all his mental strength to appear as small as possible. He looked at the count. The Count looked at the place where Pierre's face was while he stood. Anna Mikhailovna in her position showed an awareness of the touching importance of this last minute of the meeting between father and son. This lasted two minutes, which seemed like an hour to Pierre. Suddenly a tremor appeared in the large muscles and wrinkles of the count’s face. The shuddering intensified, the beautiful mouth became contorted (only then Pierre realized how close his father was to death), and an indistinct hoarse sound was heard from the contorted mouth. Anna Mikhailovna carefully looked into the patient’s eyes and, trying to guess what he needed, pointed first to Pierre, then to the drink, then in a questioning whisper called Prince Vasily, then pointed to the blanket. The patient's eyes and face showed impatience. He made an effort to look at the servant, who stood relentlessly at the head of the bed.
“They want to turn over on the other side,” the servant whispered and stood up to turn the count’s heavy body over to face the wall.
Pierre stood up to help the servant.
While the count was being turned over, one of his arms fell helplessly back, and he made a vain effort to drag it. Did the count notice the look of horror with which Pierre looked at this lifeless hand, or what other thought flashed through his dying head at that moment, but he looked at the disobedient hand, at the expression of horror in Pierre’s face, again at the hand, and on the face a weak, suffering smile that did not suit his features appeared, expressing a kind of mockery of his own powerlessness. Suddenly, at the sight of this smile, Pierre felt a shudder in his chest, a pinch in his nose, and tears blurred his vision. The patient was turned on his side against the wall. He sighed.
“Il est assoupi, [He dozed off," said Anna Mikhailovna, noticing the princess coming to replace her. – Аllons. [Let's go to.]
Pierre left.
There are many types of numbers, one of them is integers. Integers appeared in order to facilitate counting not only in the positive direction, but also in the negative direction.
Let's look at an example:
During the day the temperature outside was 3 degrees. By evening the temperature dropped by 3 degrees.
3-3=0
It became 0 degrees outside. And at night the temperature dropped by 4 degrees and the thermometer began to show -4 degrees.
0-4=-4
A series of integers.
We cannot describe such a problem using natural numbers; we will consider this problem on a coordinate line.
We got a series of numbers:
…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …
This series of numbers is called series of integers.
Positive integers. Negative integers.
The series of integers consists of positive and negative numbers. To the right of zero are the natural numbers, or they are also called positive integers. And to the left of zero they go negative integers.
Zero is neither a positive nor a negative number. It is the boundary between positive and negative numbers.
is a set of numbers consisting of natural numbers, negative integers and zero.
A series of integers in a positive and negative direction is an infinite number.
If we take any two integers, then the numbers between these integers will be called finite set.
For example:
Let's take integers from -2 to 4. All numbers between these numbers are included in the finite set. Our final set of numbers looks like this:
-2, -1, 0, 1, 2, 3, 4.
Natural numbers are denoted by the Latin letter N.
Integers are denoted by the Latin letter Z. The entire set of natural numbers and integers can be depicted in a picture. 
Non-positive integers in other words, they are negative integers.
Non-negative integers are positive integers.
Simply put, these are vegetables cooked in water according to a special recipe. I will consider two initial components (vegetable salad and water) and the finished result - borscht. Geometrically, it can be thought of as a rectangle, with one side representing lettuce and the other side representing water. The sum of these two sides will indicate borscht. The diagonal and area of such a “borscht” rectangle are purely mathematical concepts and are never used in borscht recipes.
How do lettuce and water turn into borscht from a mathematical point of view? How can the sum of two line segments become trigonometry? To understand this, we need linear angular functions.
You won't find anything about linear angular functions in math textbooks. But without them there can be no mathematics. The laws of mathematics, like the laws of nature, work regardless of whether we know about their existence or not.
Linear angular functions are addition laws. See how algebra turns into geometry and geometry turns into trigonometry.
Is it possible to do without linear angular functions? It’s possible, because mathematicians still manage without them. The trick of mathematicians is that they always tell us only about those problems that they themselves know how to solve, and never talk about those problems that they cannot solve. Look. If we know the result of addition and one term, we use subtraction to find the other term. All. We don’t know other problems and we don’t know how to solve them. What should we do if we only know the result of the addition and do not know both terms? In this case, the result of the addition must be decomposed into two terms using linear angular functions. Next, we ourselves choose what one term can be, and linear angular functions show what the second term should be so that the result of the addition is exactly what we need. There can be an infinite number of such pairs of terms. In everyday life, we get along just fine without decomposing the sum; subtraction is enough for us. But in scientific research into the laws of nature, decomposing a sum into its components can be very useful.
Another law of addition that mathematicians don't like to talk about (another of their tricks) requires that the terms have the same units of measurement. For salad, water, and borscht, these could be units of weight, volume, value, or unit of measure.
The figure shows two levels of difference for mathematical . The first level is the differences in the field of numbers, which are indicated a, b, c. This is what mathematicians do. The second level is the differences in the field of units of measurement, which are shown in square brackets and indicated by the letter U. This is what physicists do. We can understand the third level - differences in the area of the objects being described. Different objects can have the same number of identical units of measurement. How important this is, we can see in the example of borscht trigonometry. If we add subscripts to the same unit designation for different objects, we can say exactly what mathematical quantity describes a particular object and how it changes over time or due to our actions. Letter W I will designate water with a letter S I'll designate the salad with a letter B- borsch. This is what linear angular functions for borscht will look like.
If we take some part of the water and some part of the salad, together they will turn into one portion of borscht. Here I suggest you take a little break from borscht and remember your distant childhood. Remember how we were taught to put bunnies and ducks together? It was necessary to find how many animals there would be. What were we taught to do then? We were taught to separate units of measurement from numbers and add numbers. Yes, any one number can be added to any other number. This is a direct path to the autism of modern mathematics - we do it incomprehensibly what, incomprehensibly why, and very poorly understand how this relates to reality, because of the three levels of difference, mathematicians operate with only one. It would be more correct to learn how to move from one unit of measurement to another.
Bunnies, ducks, and little animals can be counted in pieces. One common unit of measurement for different objects allows us to add them together. This is a children's version of the problem. Let's look at a similar task for adults. What do you get when you add bunnies and money? There are two possible solutions here.
First option. We determine the market value of the bunnies and add it to the available amount of money. We got the total value of our wealth in monetary terms.
Second option. You can add the number of bunnies to the number of banknotes we have. We will receive the amount of movable property in pieces.
As you can see, the same addition law allows you to get different results. It all depends on what exactly we want to know.
But let's get back to our borscht. Now we can see what will happen for different angle values of linear angular functions.
The angle is zero. We have salad, but no water. We can't cook borscht. The amount of borscht is also zero. This does not mean at all that zero borscht is equal to zero water. There can be zero borscht with zero salad (right angle).
For me personally, this is the main mathematical proof of the fact that . Zero does not change the number when added. This happens because addition itself is impossible if there is only one term and the second term is missing. You can feel about this as you like, but remember - all mathematical operations with zero were invented by mathematicians themselves, so throw away your logic and stupidly cram the definitions invented by mathematicians: “division by zero is impossible”, “any number multiplied by zero equals zero” , “beyond the puncture point zero” and other nonsense. It is enough to remember once that zero is not a number, and you will never again have a question whether zero is a natural number or not, because such a question loses all meaning: how can something that is not a number be considered a number? It's like asking what color an invisible color should be classified as. Adding a zero to a number is the same as painting with paint that is not there. We waved a dry brush and told everyone that “we painted.” But I digress a little.
The angle is greater than zero but less than forty-five degrees. We have a lot of lettuce, but not enough water. As a result, we will get thick borscht.
The angle is forty-five degrees. We have equal quantities of water and salad. This is the perfect borscht (forgive me, chefs, it's just math).
The angle is greater than forty-five degrees, but less than ninety degrees. We have a lot of water and little salad. You will get liquid borscht.
Right angle. We have water. All that remains of the salad are memories, as we continue to measure the angle from the line that once marked the salad. We can't cook borscht. The amount of borscht is zero. In this case, hold on and drink water while you have it)))
Here. Something like this. I can tell other stories here that would be more than appropriate here.
Two friends had their shares in a common business. After killing one of them, everything went to the other.
The emergence of mathematics on our planet.
All these stories are told in the language of mathematics using linear angular functions. Some other time I will show you the real place of these functions in the structure of mathematics. In the meantime, let's return to borscht trigonometry and consider projections.
Saturday, October 26, 2019
I watched an interesting video about Grundy series One minus one plus one minus one - Numberphile. Mathematicians lie. They did not perform an equality check during their reasoning.
This echoes my thoughts about .
Let's take a closer look at the signs that mathematicians are deceiving us. At the very beginning of the argument, mathematicians say that the sum of a sequence DEPENDS on whether it has an even number of elements or not. This is an OBJECTIVELY ESTABLISHED FACT. What happens next?
Next, mathematicians subtract the sequence from unity. What does this lead to? This leads to a change in the number of elements of the sequence - an even number changes to an odd number, an odd number changes to an even number. After all, we added one element equal to one to the sequence. Despite all the external similarity, the sequence before the transformation is not equal to the sequence after the transformation. Even if we are talking about an infinite sequence, we must remember that an infinite sequence with an odd number of elements is not equal to an infinite sequence with an even number of elements.
By putting an equal sign between two sequences with different numbers of elements, mathematicians claim that the sum of the sequence DOES NOT DEPEND on the number of elements in the sequence, which contradicts an OBJECTIVELY ESTABLISHED FACT. Further reasoning about the sum of an infinite sequence is false, since it is based on a false equality.
If you see that mathematicians, in the course of proofs, place brackets, rearrange elements of a mathematical expression, add or remove something, be very careful, most likely they are trying to deceive you. Like card magicians, mathematicians use various manipulations of expression to distract your attention in order to ultimately give you a false result. If you cannot repeat a card trick without knowing the secret of deception, then in mathematics everything is much simpler: you don’t even suspect anything about deception, but repeating all the manipulations with a mathematical expression allows you to convince others of the correctness of the result obtained, just like when -they convinced you.
Question from the audience: Is infinity (as the number of elements in the sequence S) even or odd? How can you change the parity of something that has no parity?
Infinity is for mathematicians, like the Kingdom of Heaven is for priests - no one has ever been there, but everyone knows exactly how everything works there))) I agree, after death you will be absolutely indifferent whether you lived an even or odd number of days, but... Adding just one day into the beginning of your life, we will get a completely different person: his last name, first name and patronymic are exactly the same, only the date of birth is completely different - he was born one day before you.
Now let’s get to the point))) Let’s say that a finite sequence that has parity loses this parity when going to infinity. Then any finite segment of an infinite sequence must lose parity. We don't see this. The fact that we cannot say for sure whether an infinite sequence has an even or odd number of elements does not mean that parity has disappeared. Parity, if it exists, cannot disappear without a trace into infinity, like in a sharpie’s sleeve. There is a very good analogy for this case.
Have you ever asked the cuckoo sitting in the clock in which direction the clock hand rotates? For her, the arrow rotates in the opposite direction to what we call “clockwise”. As paradoxical as it may sound, the direction of rotation depends solely on which side we observe the rotation from. And so, we have one wheel that rotates. We cannot say in which direction the rotation occurs, since we can observe it both from one side of the plane of rotation and from the other. We can only testify to the fact that there is rotation. Complete analogy with the parity of an infinite sequence S.
Now let's add a second rotating wheel, the plane of rotation of which is parallel to the plane of rotation of the first rotating wheel. We still can't say for sure in which direction these wheels rotate, but we can absolutely tell whether both wheels rotate in the same direction or in the opposite direction. Comparing two infinite sequences S And 1-S, I showed with the help of mathematics that these sequences have different parities and putting an equal sign between them is a mistake. Personally, I trust mathematics, I don’t trust mathematicians))) By the way, to fully understand the geometry of transformations of infinite sequences, it is necessary to introduce the concept "simultaneity". This will need to be drawn.
Wednesday, August 7, 2019
Concluding the conversation about, we need to consider an infinite set. The point is that the concept of “infinity” affects mathematicians like a boa constrictor affects a rabbit. The trembling horror of infinity deprives mathematicians of common sense. Here's an example:
The original source is located. Alpha stands for real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take the infinite set of natural numbers as an example, then the considered examples can be represented in this form:
To clearly prove that they were right, mathematicians came up with many different methods. Personally, I look at all these methods as shamans dancing with tambourines. Essentially, they all boil down to the fact that either some of the rooms are unoccupied and new guests are moving in, or that some of the visitors are thrown out into the corridor to make room for guests (very humanly). I presented my view on such decisions in the form of a fantasy story about the Blonde. What is my reasoning based on? Relocating an infinite number of visitors takes an infinite amount of time. After we have vacated the first room for a guest, one of the visitors will always walk along the corridor from his room to the next one until the end of time. Of course, the time factor can be stupidly ignored, but this will be in the category of “no law is written for fools.” It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.
What is an “endless hotel”? An infinite hotel is a hotel that always has any number of empty beds, regardless of how many rooms are occupied. If all the rooms in the endless "visitor" corridor are occupied, there is another endless corridor with "guest" rooms. There will be an infinite number of such corridors. Moreover, the “infinite hotel” has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians are not able to distance themselves from banal everyday problems: there is always only one God-Allah-Buddha, there is only one hotel, there is only one corridor. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to “shove in the impossible.”
I will demonstrate the logic of my reasoning to you using the example of an infinite set of natural numbers. First you need to answer a very simple question: how many sets of natural numbers are there - one or many? There is no correct answer to this question, since we invented numbers ourselves; numbers do not exist in Nature. Yes, Nature is great at counting, but for this she uses other mathematical tools that are not familiar to us. I’ll tell you what Nature thinks another time. Since we invented numbers, we ourselves will decide how many sets of natural numbers there are. Let's consider both options, as befits real scientists.
Option one. “Let us be given” one single set of natural numbers, which lies serenely on the shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and nowhere to take them. We cannot add one to this set, since we already have it. What if you really want to? No problem. We can take one from the set we have already taken and return it to the shelf. After that, we can take one from the shelf and add it to what we have left. As a result, we will again get an infinite set of natural numbers. You can write down all our manipulations like this:
I wrote down the actions in algebraic notation and in set theory notation, with a detailed listing of the elements of the set. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one is subtracted from it and the same unit is added.
Option two. We have many different infinite sets of natural numbers on our shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. Let's take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. This is what we get:
The subscripts "one" and "two" indicate that these elements belonged to different sets. Yes, if you add one to an infinite set, the result will also be an infinite set, but it will not be the same as the original set. If you add another infinite set to one infinite set, the result is a new infinite set consisting of the elements of the first two sets.
The set of natural numbers is used for counting in the same way as a ruler is for measuring. Now imagine that you added one centimeter to the ruler. This will be a different line, not equal to the original one.
You can accept or not accept my reasoning - it is your own business. But if you ever encounter mathematical problems, think about whether you are following the path of false reasoning trodden by generations of mathematicians. After all, studying mathematics, first of all, forms a stable stereotype of thinking in us, and only then adds to our mental abilities (or, conversely, deprives us of free-thinking).
pozg.ru
Sunday, August 4, 2019
I was finishing a postscript to an article about and saw this wonderful text on Wikipedia:
We read: "... the rich theoretical basis of the mathematics of Babylon did not have a holistic character and was reduced to a set of disparate techniques, devoid of a common system and evidence base."
Wow! How smart we are and how well we can see the shortcomings of others. Is it difficult for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, I personally got the following:
The rich theoretical basis of modern mathematics is not holistic in nature and is reduced to a set of disparate sections, devoid of a common system and evidence base.
I won’t go far to confirm my words - it has a language and conventions that are different from the language and conventions of many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole series of publications to the most obvious mistakes of modern mathematics. See you soon.
Saturday, August 3, 2019
How to divide a set into subsets? To do this, you need to enter a new unit of measurement that is present in some of the elements of the selected set. Let's look at an example.
May we have plenty A consisting of four people. This set is formed on the basis of “people.” Let us denote the elements of this set by the letter A, the subscript with a number will indicate the serial number of each person in this set. Let's introduce a new unit of measurement "gender" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set A based on gender b. Notice that our set of “people” has now become a set of “people with gender characteristics.” After this we can divide the sexual characteristics into male bm and women's bw sexual characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, no matter which one - male or female. If a person has it, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we use regular school mathematics. Look what happened.
After multiplication, reduction and rearrangement, we ended up with two subsets: the subset of men Bm and a subset of women Bw. Mathematicians reason in approximately the same way when they apply set theory in practice. But they don’t tell us the details, but give us the finished result - “a lot of people consist of a subset of men and a subset of women.” Naturally, you may have a question: how correctly has the mathematics been applied in the transformations outlined above? I dare to assure you that essentially everything was done correctly; it is enough to know the mathematical basis of arithmetic, Boolean algebra and other branches of mathematics. What it is? Some other time I will tell you about this.
As for supersets, you can combine two sets into one superset by selecting the unit of measurement present in the elements of these two sets.
As you can see, units of measurement and ordinary mathematics make set theory a relic of the past. A sign that all is not well with set theory is that mathematicians have come up with their own language and notation for set theory. Mathematicians acted as shamans once did. Only shamans know how to “correctly” apply their “knowledge.” They teach us this “knowledge”.
In conclusion, I want to show you how mathematicians manipulate
Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells about a flying arrow:
A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
I'll show you the process with an example. We select the “red solid in a pimple” - this is our “whole”. At the same time, we see that these things are with a bow, and there are without a bow. After that, we select part of the “whole” and form a set “with a bow”. This is how shamans get their food by tying their set theory to reality.
Now let's do a little trick. Let’s take “solid with a pimple with a bow” and combine these “wholes” according to color, selecting the red elements. We got a lot of "red". Now the final question: are the resulting sets “with a bow” and “red” the same set or two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so it will be.
This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We formed a set of "red solid with a pimple and a bow." The formation took place in four different units of measurement: color (red), strength (solid), roughness (pimply), decoration (with a bow). Only a set of units of measurement allows us to adequately describe real objects in the language of mathematics. This is what it looks like.
The letter "a" with different indices denotes different units of measurement. The units of measurement by which the “whole” is distinguished at the preliminary stage are highlighted in brackets. The unit of measurement by which the set is formed is taken out of brackets. The last line shows the final result - an element of the set. As you can see, if we use units of measurement to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not the dancing of shamans with tambourines. Shamans can “intuitively” come to the same result, arguing that it is “obvious,” because units of measurement are not part of their “scientific” arsenal.
Using units of measurement, it is very easy to split one set or combine several sets into one superset. Let's take a closer look at the algebra of this process.
What does a whole number mean?
So, let's look at what numbers are called integers.
Thus, the following numbers will be denoted by integers: $0$, $±1$, $±2$, $±3$, $±4$, etc.
The set of natural numbers is a subset of the set of integers, i.e. Any natural number will be an integer, but not every integer is a natural number.
Positive integers and negative integers
Definition 2
plus.
The numbers $3, 78, 569, 10450$ are positive integers.
Definition 3
are signed integers minus.
The numbers $−3, −78, −569, -10450$ are negative integers.
Note 1
The number zero is neither a positive nor a negative integer.
Positive integers are integers greater than zero.
Negative integers are integers less than zero.
The set of natural integers is the set of all positive integers, and the set of all opposite natural numbers is the set of all negative integers.
Non-positive and non-negative integers
All positive integers and zero are called non-negative integers.
Non-positive integers are all negative integers and the number $0$.
Note 2
Thus, non-negative integer are integers greater than zero or equal to zero, and non-positive integer– integers less than zero or equal to zero.
For example, non-positive integers: $−32, −123, 0, −5$, and non-negative integers: $54, 123, 0, 856,342.$
Describing changes in quantities using integers
Integers are used to describe changes in the number of objects.
Let's look at examples.
Example 1
Let a store sell a certain number of product names. When the store receives $520$ of items, the number of items in the store will increase, and the number $520$ shows a change in the number in a positive direction. When the store sells $50$ of product items, the number of product items in the store will decrease, and the number $50$ will express a change in the number in the negative direction. If the store neither delivers nor sells goods, then the number of goods will remain unchanged (i.e., we can talk about a zero change in the number).
In the above example, the change in the number of goods is described using the integers $520$, $−50$ and $0$, respectively. A positive value of the integer $520$ indicates a change in the number in a positive direction. A negative value of the integer $−50$ indicates a change in the number in a negative direction. The integer $0$ indicates that the number is immutable.
Integers are convenient to use because... there is no need for an explicit indication of an increase or decrease in the number - the sign of the integer indicates the direction of the change, and the value indicates the quantitative change.
Using integers you can express not only a change in quantity, but also a change in any quantity.
Let's consider an example of a change in the cost of a product.
Example 2
An increase in value, for example, by $20$ rubles is expressed using a positive integer $20$. A decrease in price, for example, by $5$ rubles is described using a negative integer $−5$. If there is no change in value, then such change is determined using the integer $0$.
Let us separately consider the meaning of negative integers as the amount of debt.
Example 3
For example, a person has $5,000$ rubles. Then, using the positive integer $5,000$, you can show the number of rubles he has. A person must pay rent in the amount of $7,000$ rubles, but he does not have that kind of money, in which case such a situation is described by a negative integer $−7,000$. In this case, the person has $−7,000$ rubles, where “–” indicates debt, and the number $7,000$ indicates the amount of debt.
If we add the number 0 to the left of a series of natural numbers, we get series of positive integers:
0, 1, 2, 3, 4, 5, 6, 7, ...
Negative integers
Let's look at a small example. The picture on the left shows a thermometer that shows a temperature of 7 °C. If the temperature drops by 4 °C, the thermometer will show 3 °C of heat. A decrease in temperature corresponds to the action of subtraction:
Note: all degrees are written with the letter C (Celsius), the degree sign is separated from the number by a space. For example, 7 °C.
If the temperature drops by 7 °C, the thermometer will show 0 °C. A decrease in temperature corresponds to the action of subtraction:
If the temperature drops by 8 °C, the thermometer will show -1 °C (1 °C below zero). But the result of subtracting 7 - 8 cannot be written using natural numbers and zero.
Let's illustrate subtraction using a series of positive integers:
1) From the number 7, count 4 numbers to the left and get 3:
2) From the number 7, count 7 numbers to the left and get 0:
It is impossible to count 8 numbers from the number 7 to the left in a series of positive integers. To make actions 7 - 8 feasible, we expand the range of positive integers. To do this, to the left of zero, we write (from right to left) in order all the natural numbers, adding to each of them the sign - , indicating that this number is to the left of zero.
The entries -1, -2, -3, ... read minus 1, minus 2, minus 3, etc.:
5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...
The resulting series of numbers is called series of integers. The dots to the left and right in this entry mean that the series can be continued indefinitely to the right and left.
To the right of the number 0 in this row are numbers called natural or positive integers(briefly - positive).
To the left of the number 0 in this row are numbers called integer negative(briefly - negative).
The number 0 is an integer, but is neither a positive nor a negative number. It separates positive and negative numbers.
Hence, the series of integers consists of negative integers, zero and positive integers.
Integer Comparison
Compare two integers- means to find out which one is greater, which one is smaller, or determine that the numbers are equal.
You can compare integers using a row of integers, since the numbers in it are arranged from smallest to largest if you move along the row from left to right. Therefore, in a series of integers, you can replace commas with a less than sign:
5 < -4 < -3 < -2 < -1 < 0 < 1 < 2 < 3 < 4 < 5 < ...
Hence, of two integers, the greater is the number that is to the right in the series, and the smaller is the one that is to the left, Means:
1) Any positive number is greater than zero and greater than any negative number:
1 > 0; 15 > -16
2) Any negative number less than zero:
7 < 0; -357 < 0
3) Of two negative numbers, the one that is to the right in the series of integers is greater.