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 tell us 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 measurement.
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 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 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 the following 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, in essence, the transformations were 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.
Lesson topic:
« Direct coordinates»
The purpose of the lesson:
Introduce students to the coordinate line and negative numbers.
Lesson objectives:
Educational: introduce students to the coordinate line and negative numbers.
Developmental: development of logical thinking, expansion of horizons.
Educational: development of cognitive interest, education of information culture.
Lesson plan:
Org moment. Checking students and their readiness for the lesson.
Updating basic knowledge. Oral survey of students on the topic covered.
Explanation of new material.
4. Reinforcing the material learned.
5. Summarizing. A summary of what was learned in the lesson. Questions from students.
6. Conclusions. Summarizing the main points of the lesson. Knowledge assessment. Making marks.
7. Homework. Independent work of students with the studied material.
Equipment: chalk, board, slides.
Detailed outline plan
Stage name and contents
Activity
Activity
students
Stage I
Org moment. Greetings.
Filling out the log.
greets the class, the class leader gives a list of those absent.
say hello to
teacher
Stage II
Updating basic knowledge.
The ancient Greek scientist Pythagoras said: “Numbers rule the world.” You and I live in this world of numbers, and during our school years we learn to work with different numbers.
1 What numbers do we already know for today’s lesson?
2 What problems do these numbers help us solve?
Today we move on to the study of the second chapter of our textbook “Rational Numbers”, where we will expand our knowledge about numbers, and after studying the entire chapter “Rational Numbers” we will learn to perform all the actions you know with them and start with the topic of the coordinate line.
1.natural, ordinary fractions, decimals
2.addition, subtraction, multiplication, division, finding fractions from a number and a number from its fraction, solve various equations and problems
Stage III
Explanation of new material.
Let's take straight line AB and split it with point O into two additional rays - OA and OB. Let us select a unit segment on a straight line and take point O as the origin and direction.
Definitions:
A straight line with a reference point, a unit segment and a direction chosen on it is called a coordinate line.
The number showing the position of a point on a line is called the coordinate of this point.
How to construct a coordinate line?
make a direct
set a unit segment
indicate direction
The coordinate line can be depicted in different ways: horizontally, vertically and at any other angle to the horizon, and has a beginning, but no end.
Exercise 1. Which of the following lines are not coordinate lines? (slide)
Let's draw a coordinate line, mark the origin, a unit segment and plot points 1,2,3,4 and so on to the left and right.
Let's look at the resulting coordinate line. Why is such a straight line inconvenient?
The direction to the right from the origin is called positive, and the direction on the straight line is indicated by an arrow. Numbers located to the right of point O are called positive. Negative numbers are placed to the left of point O, and the direction to the left of point O is called negative (the negative direction is not indicated). If the coordinate line is located vertically, then the numbers above the origin are positive, and the numbers below the origin are negative. Negative numbers are written with a “-” sign. They read: “Minus one”, “Minus two”, “Minus three”, etc. The number 0 – the origin is neither a positive nor a negative number. It separates positive from negative numbers.
Solving equations and the concept of “debt” in trade calculations led to the appearance of negative numbers.
Negative numbers appeared much later than natural numbers and ordinary fractions. The first information about negative numbers was found by Chinese mathematicians in the 2nd century. BC e. Positive numbers were then interpreted as property, and negative numbers as debt, shortage. In Europe, recognition came a thousand years later, and even then, for a long time, negative numbers were called “false,” “imaginary,” or “absurd.” In the 17th century, negative numbers received a visual geometric representation on the number axis
You can also give examples of a coordinate line: a thermometer, a comparison of mountain peaks and depressions (sea level is taken as zero), a distance on a map, an elevator shaft, houses, cranes.
Think Do you know any other examples of a coordinate line?
Tasks.
Task2. Name the coordinates of the points.
Task 3. Plot points on a coordinate line
Task4 . Draw a horizontal line and mark point O on it. Mark points A, B, C, K on this line if you know that:
A is 9 cells to the right of O;
B is to the left of O by 6.5 cells;
C is 3½ squares to the right of O;
K is 3 squares to the left of O .
Recorded in supporting notes.
They listen and complement.
They complete the task in their notebook and then explain their answers out loud.
Draw and mark the origin of a unit segment
Such a straight line is inconvenient because the same number corresponds to two points on the straight line.
History BC and our era.
Stage IV
Consolidation of the studied material.
1.What is a coordinate line?
2.How to construct a coordinate line?
1. A straight line with a reference point, a unit segment and a direction selected on it is called a coordinate line
2) make a direct
mark the beginning of the countdown on it
set a unit segment
indicate direction
Stage V
Summarizing
What new did we learn today?
The coordinate line and negative numbers.
Stage VI
Knowledge assessment. Making marks.
Homework.
Make up questions on the topic covered (know the answers to them)
So a unit segment and its tenth, hundredth, and so on parts allow us to get to the points of the coordinate line, which will correspond to the final decimal fractions (as in the previous example). However, there are points on the coordinate line that we cannot get to, but to which we can get as close as we like, using smaller and smaller ones down to an infinitesimal fraction of a unit segment. These points correspond to infinite periodic and non-periodic decimal fractions. Let's give a few examples. One of these points on the coordinate line corresponds to the number 3.711711711...=3,(711) . To approach this point, you need to set aside 3 unit segments, 7 tenths, 1 hundredth, 1 thousandth, 7 ten-thousandths, 1 hundred thousandth, 1 millionth of a unit segment, and so on. And another point on the coordinate line corresponds to pi (π=3.141592...).
Since the elements of the set of real numbers are all numbers that can be written in the form of finite and infinite decimal fractions, then all the information presented above in this paragraph allows us to state that we have assigned a specific real number to each point of the coordinate line, and it is clear that different the points correspond to different real numbers.
It is also quite obvious that this correspondence is one-to-one. That is, we can assign a real number to a specified point on a coordinate line, but we can also, using a given real number, indicate a specific point on a coordinate line to which a given real number corresponds. To do this, we will have to set aside a certain number of unit segments, as well as tenths, hundredths, and so on, of fractions of a unit segment from the beginning of the countdown in the desired direction. For example, the number 703.405 corresponds to a point on the coordinate line, which can be reached from the origin by plotting in the positive direction 703 unit segments, 4 segments constituting a tenth of a unit, and 5 segments constituting a thousandth of a unit.
So, to each point on the coordinate line there is a real number, and each real number has its place in the form of a point on the coordinate line. This is why the coordinate line is often called number line.
Coordinates of points on a coordinate line
The number corresponding to a point on a coordinate line is called coordinate of this point.
In the previous paragraph, we said that each real number corresponds to a single point on the coordinate line, therefore, the coordinate of a point uniquely determines the position of this point on the coordinate line. In other words, the coordinate of a point uniquely defines this point on the coordinate line. On the other hand, each point on the coordinate line corresponds to a single real number - the coordinate of this point.
All that remains to be said is about the accepted notation. The coordinate of the point is written in parentheses to the right of the letter that represents the point. For example, if point M has coordinate -6, then you can write M(-6), and notation of the form means that point M on the coordinate line has coordinate.
Bibliography.
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics: textbook for 5th grade. educational institutions.
- Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8th grade. educational institutions.
This article is devoted to the analysis of such concepts as a coordinate ray and a coordinate line. We will dwell on each concept and look at examples in detail. Thanks to this article, you can refresh your knowledge or become familiar with the topic without the help of a teacher.
In order to define the concept of a coordinate ray, you should have an idea of what a ray is.
Definition 1
Ray- this is a geometric figure that has an origin of the coordinate ray and a direction of movement. The straight line is usually depicted horizontally, indicating the direction to the right.
In the example we see that O is the beginning of the ray.
Example 1
The coordinate ray is depicted according to the same scheme, but is significantly different. We set a starting point and measure a single segment.
Example 2
Definition 2
Unit segment is the distance from 0 to the point chosen for measurement.
Example 3
From the end of a single segment you need to put a few strokes and make markings.
Thanks to the manipulations that we did with the beam, it became coordinate. Label the strokes with natural numbers in sequence from 1 - for example, 2, 3, 4, 5...
Example 4
Definition 3
is a scale that can last indefinitely.
It is often depicted as a ray starting at point O, and a single unit segment is plotted. An example is shown in the figure.
Example 5
In any case, we will be able to continue the scale to the number we need. You can write numbers as convenient as possible - under the beam or above it.
Example 6
Both uppercase and lowercase letters can be used to display ray coordinates.
The principle of depicting a coordinate line is practically no different from depicting a ray. It's simple - draw a ray and add it to a straight line, giving it a positive direction, which is indicated by an arrow.
Example 7
Draw the beam in the opposite direction, extending it to a straight line
Example 8
Set aside single segments according to the example above
On the left side write down the natural numbers 1, 2, 3, 4, 5... with the opposite sign. Pay attention to the example.
Example 9
You can only mark the origin and single segments. See the example of how it will look.
Example 10
Definition 4
- this is a straight line, which is depicted with a certain reference point, which is taken as 0, a unit segment and a given direction of movement.
Correspondence between points on a coordinate line and real numbers
A coordinate line can contain many points. They are directly related to real numbers. This can be defined as a one-to-one correspondence.
Definition 5
Each point on the coordinate line corresponds to a single real number, and each real number corresponds to a single point on the coordinate line.
In order to better understand the rule, you should mark a point on the coordinate line and see what natural number corresponds to the mark. If this point coincides with the origin, it will be marked zero. If the point does not coincide with the starting point, we postpone the required number of unit segments until we reach the specified mark. The number written under it will correspond to this point. Using the example below, we will show you this rule clearly.
Example 11
If we cannot find a point by plotting unit segments, we should also mark points that make up one tenth, hundredth or thousandth of a unit segment. An example can be used to examine this rule in detail.
By setting aside several similar segments, we can obtain not only an integer, but also a fractional number - both positive and negative.
The marked segments will help us find the required point on the coordinate line. These can be either whole or fractional numbers. However, there are points on a straight line that are very difficult to find using single segments. These points correspond to decimal fractions. In order to look for such a point, you will have to set aside a unit segment, a tenth, a hundredth, a thousandth, ten-thousandths and other parts of it. One point on the coordinate line corresponds to the irrational number π (= 3, 141592...).
The set of real numbers includes all numbers that can be written as a fraction. This allows you to identify the rule.
Definition 6
Each point on the coordinate line corresponds to a specific real number. Different points define different real numbers.
This correspondence is unique - each point corresponds to a certain real number. But this also works in reverse. We can also specify a specific point on the coordinate line that will relate to a specific real number. If the number is not an integer, then we need to mark several unit segments, as well as tenths and hundredths in a given direction. For example, the number 400350 corresponds to a point on the coordinate line, which can be reached from the origin by plotting in the positive direction 400 unit segments, 3 segments constituting a tenth of a unit, and 5 segments constituting a thousandth.
Coordinate line.
Let's take an ordinary straight line. Let's call it straight line x (Fig. 1). Let us select a reference point O on this straight line, and also indicate with an arrow the positive direction of this straight line (Fig. 2). Thus, we will have positive numbers to the right of point O, and negative numbers to the left. Let's choose a scale, that is, the size of a straight line segment, equal to one. We did it coordinate line(Fig. 3). Each number corresponds to a specific single point on this line. Moreover, this number is called the coordinate of this point. That's why the line is called a coordinate line. And the reference point O is called the origin.
For example, in Fig. 4 point B is located at a distance of 2 to the right of the origin. Point D is located at a distance of 4 to the left of the origin. Accordingly, point B has coordinate 2, and point D has coordinate -4. Point O itself, being a reference point, has coordinate 0 (zero). This is usually written like this: O(0), B(2), D(-4). And in order not to constantly say “point D with coordinate such and such,” they say more simply: “point 0, point 2, point -4.” And in this case it is enough to designate the point itself by its coordinate (Fig. 5).
Knowing the coordinates of two points on a coordinate line, we can always calculate the distance between them. Let's say we have two points A and B with coordinates a and b, respectively. Then the distance between them will be |a - b|. Notation |a - b| reads as “a minus b modulo” or “modulus of the difference between the numbers a and b.”
What is a module?
Algebraically, the modulus of a number x is a non-negative number. Denoted by |x|. Moreover, if x > 0, then |x| = x. If x< 0, то |x| = -x. Если x = 0, то |x| = 0.
Geometrically, the modulus of a number x is the distance between a point and the origin. And if there are two points with coordinates x1 and x2, then |x1 - x2| is the distance between these points.
The module is also called absolute value.
What else can we say when it comes to the coordinate line? Of course, about numerical intervals.
Types of numerical intervals.
Let's say we have two numbers a and b. Moreover, b > a (b is greater than a). On a coordinate line, this means that point b is to the right of point a. Let us replace b in our inequality with the variable x. That is x > a. Then x is all the numbers that are greater than a. On the coordinate line, these are, respectively, all points to the right of point a. This part of the line is shaded (Fig. 6). Such a set of points is called open beam, and this numerical interval is denoted by (a; +∞), where the sign +∞ is read as “plus infinity”. Please note that point a itself is not included in this interval and is indicated by a light circle.
Let us also consider the case when x ≥ a. Then x is all numbers that are greater than or equal to a. On the coordinate line, these are all points to the right of a, as well as point a itself (in Fig. 7, point a is already indicated by a dark circle). Such a set of points is called closed beam(or simply a beam), and this numerical interval is designated .
The coordinate line is also called coordinate axis. Or just the x axis.
