1 6 on the coordinate line. Coordinate line (number line), coordinate ray

On this lesson We will get acquainted with the concept of a coordinate line and derive its main characteristics and properties. Let's formulate and learn to solve the main problems. Let's solve several examples of combining these problems.

From the geometry course we know what a straight line is, but what needs to be done with an ordinary straight line for it to become a coordinate line?

1) Select the starting point;

2) Choose a direction;

3) Select scale;

Figure 1 shows a regular line, and Figure 2 shows a coordinate line.

A coordinate line is a line l on which the starting point O is chosen - the origin of reference, the scale is a unit segment, that is, a segment whose length is considered equal to one, and a positive direction.

The coordinate line is also called the coordinate axis or X-axis.

Let's find out why the coordinate line is needed; to do this, we'll define its main property. The coordinate line establishes a one-to-one correspondence between the set of all numbers and the set of all points on this line. Here are some examples:

Two numbers are given: (sign “+”, modulus is equal to three) and (sign “-”, modulus is equal to three). Let’s depict these numbers on the coordinate line:

Here the number is called coordinate A, the number is called coordinate B.

They also say that the image of a number is point C with coordinate , and the image of a number is point D with coordinate:

So, since the main property of the coordinate line is the establishment of a one-to-one correspondence between points and numbers, two main tasks arise: to indicate a point by a given number, we have already done this above, and to indicate a number by given point. Let's look at an example of the second task:

Let point M be given:

To determine a number from a given point, you must first determine the distance from the origin to the point. In this case, the distance is two. Now you need to determine the sign of the number, that is, in which ray of the straight line the point M lies. In this case, the point lies to the right of the origin, in the positive ray, which means the number will have a “+” sign.

Let's take another point and use it to determine the number:

The distance from the origin to the point is similar to the previous example, equal to two, but in this case the point lies to the left of the origin, on the negative ray, which means point N characterizes the number

All typical problems associated with the coordinate line are in one way or another connected with its main property and the two main problems that we formulated and solved.

TO typical tasks relate:

-be able to place points and their coordinates;

-understand comparison of numbers:

the expression means that point C with coordinate 4 lies to the right of the point M with coordinate 2:

And vice versa, if we are given the location of points on a coordinate line, we must understand that their coordinates are related by a certain relationship:

Let the points M(x M) and N(x N) be given:

We see that point M lies to the right of point n, which means their coordinates are related as

-Determining the distance between points.

We know that the distance between points X and A is equal to the modulus of the number. let two points be given:

Then the distance between them will be equal to:

Another very important task is geometric description of number sets.

Consider a ray that lies on the coordinate axis, does not include its origin, but includes all other points:

So, we are given a set of points located on the coordinate axis. Let us describe the set of numbers that is characterized by this set of points. There are countless such numbers and points, so this entry looks like this:

Let us make an explanation: in the second recording option, if you put a parenthesis “(”, then the extreme number - in this case, the number 3, is not included in the set, but if you put a square bracket “[”, then the extreme number is included in the set.

So, we wrote analytically number set, which characterizes a given set of points. analytical notation, as we said, is performed either in the form of an inequality or in the form of an interval.

A set of points is given:

In this case, the point a=3 is included in the set. Let us describe analytically the set of numbers:

Please note that a parenthesis is always placed after or before the infinity sign, since we will never reach infinity, and there can be either a parenthesis or a square bracket next to the number, depending on the conditions of the task.

Let's consider an example of an inverse problem.

A coordinate line is given. Draw on it a set of points corresponding to the numerical set and:

The coordinate line establishes a one-to-one correspondence between any point and a number, and therefore between numerical sets and sets of points. We looked at rays directed in both positive and negative directions, including their vertex and not including it. Now let's look at the segments.

Example 10:

A set of numbers is given. Draw the corresponding set of points

Example 11:

A set of numbers is given. Draw a set of points:

Sometimes, to show that the ends of a segment are not included in the set, arrows are drawn:

Example 12:

A number set is given. Construct its geometric model:

Find smallest number from between:

Find greatest number from the interval if it exists:

We can subtract an arbitrarily small number from eight and say that the result will be the greatest a large number, but we will immediately find an even smaller number, and the result of the subtraction will increase, so that it is impossible to find the largest number in this interval.

Let us pay attention to the fact that it is impossible to select the closest number to any number on the coordinate line, because there is always a number even closer.

How many natural numbers contained within a given interval?

From the interval we select the following natural numbers: 4, 5, 6, 7 - four natural numbers.

Recall that natural numbers are numbers used for counting.

Let's take another set.

Example 13:

Set of numbers given

Construct its geometric model:

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 familiarize yourself with a topic without the help of a teacher.

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In order to define the concept of a coordinate ray, you should have an idea of what a ray is.

Definition 1

Ray- This geometric figure, which has the origin of the coordinate ray and the 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 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

– this 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...s 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 below 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 decimals. In order to look for such a point, you will have to set aside a unit segment, a tenth, a hundredth, a thousandth, ten-thousandth 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 us 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 the opposite direction. 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.

In this lesson we will get acquainted with the concept of a coordinate line, we will derive its main characteristics and properties. Let's formulate and learn to solve the main problems. Let's solve several examples of combining these problems.

From the geometry course we know what a straight line is, but what needs to be done with an ordinary straight line for it to become a coordinate line?

1) Select the starting point;

2) Choose a direction;

3) Select scale;

Figure 1 shows a regular line, and Figure 2 shows a coordinate line.

The coordinate line is also called the coordinate axis or X-axis.

Two numbers are given: (sign “+”, modulus is equal to three) and (sign “-”, modulus is equal to three). Let’s depict these numbers on the coordinate line:

Here the number is called coordinate A, the number is called coordinate B.

They also say that the image of a number is point C with coordinate , and the image of a number is point D with coordinate:

So, since the main property of the coordinate line is the establishment of a one-to-one correspondence between points and numbers, two main tasks arise: to indicate a point by a given number, we have already done this above, and to indicate a number by a given point. Let's look at an example of the second task:

Let point M be given:

Let's take another point and use it to determine the number:

All typical problems associated with the coordinate line are in one way or another connected with its main property and the two main problems that we formulated and solved.

Typical tasks include:

-be able to place points and their coordinates;

-understand comparison of numbers:

the expression means that point C with coordinate 4 lies to the right of point M with coordinate 2:

And vice versa, if we are given the location of points on a coordinate line, we must understand that their coordinates are related by a certain relationship:

Let the points M(x M) and N(x N) be given:

We see that point M lies to the right of point n, which means their coordinates are related as

-Determining the distance between points.

We know that the distance between points X and A is equal to the modulus of the number. let two points be given:

Then the distance between them will be equal to:

Another very important task is geometric description of number sets.

Consider a ray that lies on the coordinate axis, does not include its origin, but includes all other points:

So, we have written analytically a numerical set that characterizes a given set of points. analytical notation, as we said, is performed either in the form of an inequality or in the form of an interval.

A set of points is given:

In this case, the point a=3 is included in the set. Let us describe analytically the set of numbers:

Let's consider an example of an inverse problem.

A coordinate line is given. Draw on it a set of points corresponding to the numerical set and:

Example 10:

A set of numbers is given. Draw the corresponding set of points

Example 11:

A set of numbers is given. Draw a set of points:

Sometimes, to show that the ends of a segment are not included in the set, arrows are drawn:

Example 12:

A number set is given. Construct its geometric model:

Find the smallest number from the interval:

Find the largest number in the interval if it exists:

We can subtract an arbitrarily small number from eight and say that the result will be the largest number, but we will immediately find an even smaller number, and the result of the subtraction will increase, so that it is impossible to find the largest number in this interval.

Let us pay attention to the fact that it is impossible to select the closest number to any number on the coordinate line, because there is always a number even closer.

How many natural numbers are there in a given interval?

From the interval we select the following natural numbers: 4, 5, 6, 7 - four natural numbers.

Recall that natural numbers are numbers used for counting.

Let's take another set.

Example 13:

Set of numbers given

Construct its geometric model:

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...were involved in the study of the issue mathematical analysis, set theory, new physical and philosophical approaches; 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. WITH physical point From a perspective, it 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 jump to reciprocals. 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 logical paradox it can be 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 point out Special attention, 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. Applicable mathematical theory sets to the 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: on different coins there is different quantities dirt, crystal structure and atomic arrangement of each coin is unique...

And now I have the most interest Ask: 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, it means it 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: minus sign, number four, 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.

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.

Educational: development logical thinking, broadening your horizons.

Educational: development cognitive interest, education of information culture.

Lesson plan:

Org moment. Checking students and their readiness for the lesson.

Update background knowledge. Oral survey 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 students with the material studied.

Equipment: chalk, board, slides.

Detailed outline plan

Stage name and contents

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 in school years learning 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 studying 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. To the left of point O is located negative numbers, 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 then they were interpreted as property, and negative ones - 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.