Negative numbers. Coordinate line (number line), coordinate ray

Topic: “Coordinates on a straight line.”

• Give a comprehensive understanding of new numbers.
• Learn to read and write positive and negative numbers, and represent them as points on a line.
• Determine the coordinates of points, find the coordinate of a point, mark a point on a coordinate line by its coordinate.
• To develop thinking skills, attentiveness, a culture of reading, a culture of mathematical speech, and to develop student activity.

Equipment: demonstration coordinate line, demonstration thermometer, tables, tools (ruler with divisions), cards.

2. Oral counting.“Soft landing method.”

Did Dunno solve the examples correctly?

Which ray is called the coordinate ray?

Does the coordinate ray have an end? Start?

What numbers correspond to points A, E, C, D on the coordinate ray?

What points on the coordinate ray correspond to the numbers 2, 4, 5, 8?

2. Preparation for studying new material.

Problem 1. The squirrel has crawled out of the hollow and is running up and down the tree trunk.

What do you need to know to determine the position of a squirrel on a tree? Is it enough to know only the distance of the squirrel from the hollow?

Problem 2. “Meteor” left the village of Gornopravdinsk and is moving at a speed of 40 km/h.

Where will Meteor be in 2 hours?

Is it enough to know just the distance? ( Answer: no, you also need to know the direction).

3. Presentation of new material.

Practical work with the class. (Student work at the blackboard and class work in a notebook).

Draw a horizontal line.

Mark point O on it (origin).

Select a single segment and move it to the right and left from the origin once, twice, three, etc. once.

Under each dot, label the corresponding number.

Why is this scale inconvenient? (The same number appears under two different dots).

How to get out of this difficulty?

In mathematics, it is customary to write numbers that go to the left of the origin with a minus sign “-”.

Introduction of the concept of positive and negative numbers.

The direction to the right from the origin is called positive, and the direction on the straight line is indicated by an arrow. The 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).

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.

Coordinate line.

Definition: a straight line with a reference point, a unit segment and a direction selected on it is called coordinate line.

Task: name a line among these lines that is a coordinate line.

Point coordinate.

Definition: a number indicating the position of a point on a line is called the coordinate of this point.

Work according to the textbook. Repeat the definition of the coordinate line; point coordinates.

Introduce the concept of a vertical coordinate line.

Work according to the table.

They say: “Point A has coordinate 2”; “Point C has coordinate – 4.”

They write: A (2); V (3.5); C (- 4); D(-2).

They read: “Point A with coordinate 2”; “Point C with coordinate – 4”, etc.

Psychological relief:(The soundtrack “sound of the sea” sounds).

Against the background of the “noise of the waves,” a fragment from M. Gorky’s work “Song of the Falcon” sounds:

“... The huge sea, lazily sighing near the shore, fell asleep and motionless in the distance, bathed in the blue radiance of the moon. Soft and silvery, it merges there with the blue, tender sky and sleeps soundly, reflecting the transparent fabric of cirrus clouds, motionless and not hiding the golden patterns of stars. It seems that the sky is leaning lower and lower over the sea, wanting to understand what the restless waves are whispering about, sleepily gliding to the shore...”

4. Consolidation of new material.

Game moment.(Demonstration board with a coordinate line).

The teacher reinforces the point. Students name its coordinate.

The teacher calls the number. Students strengthen a point with a given coordinate.

Practical work:(On the tables there are cards with a coordinate line on which points are marked).

Write the coordinates of points A, B, C, D, E, K, O, M.

Game moment:“Find the mistake.”

Points A, B, C, D are marked on the coordinate line.

Dunno wrote down the coordinates of the points like this: A (2), B (- 3), C (- 2), D (- 4). Did he write it down correctly?

5. Lesson summary.

Students answer the teacher's questions:

Which line is called the coordinate line?

What numbers are the coordinates of points on the coordinate line to the right of the origin? To the left of the origin?

What is the coordinate of the origin?

6. Grading.

7. Homework: clause 26, No. 902 - orally, No. 903, No. 904.

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, to reach a common opinion about the essence of paradoxes scientific community so far it has not been possible... 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? Stay in constant units measurements of time and do not go to reciprocal quantities. 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 it is not complete solution Problems. 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 draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.

Wednesday, July 4, 2018

The differences between set and multiset are described very well on Wikipedia. Let's see.

As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Such absurd logic sentient beings never understand. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.

Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.

No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.

We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.

First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: on different coins there is different quantities dirt, crystal structure and atomic arrangement of each coin is unique...

And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.

Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.

To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."

Sunday, March 18, 2018

The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.

Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.

Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.

1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.

2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.

3. Convert individual graphic symbols into numbers. This is not a mathematical operation.

4. Add the resulting numbers. Now this is mathematics.

The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.

From a mathematical point of view, it does not matter in which number system we write a number. So, in different systems In calculus, 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 mathematical operation does not depend on the size of the number, the unit of measurement used and who performs the action.

Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?

Female... The halo on top and the arrow down are male.

If such a work of design art flashes before your eyes several times a day,

Then it’s not surprising that you suddenly find a strange icon in your car:

Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.

1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.

Negative numbers are numbers with a minus sign (−), for example −1, −2, −3. Reads like: minus one, minus two, minus three.

Application example negative numbers is a thermometer that shows the temperature of the body, air, soil or water. IN winter time, when it is very cold outside, the temperature can be negative (or, as people say, “minus”).

For example, −10 degrees cold:

The ordinary numbers that we looked at earlier, such as 1, 2, 3, are called positive. Positive numbers are numbers with a plus sign (+).

When writing positive numbers, the + sign is not written down, which is why we see the numbers 1, 2, 3 that are familiar to us. But we should keep in mind that these positive numbers look like this: +1, +2, +3.

This is a straight line on which all numbers are located: both negative and positive. As follows:

The numbers shown here are from −5 to 5. In fact, the coordinate line is infinite. The figure shows only a small fragment of it.

Numbers on the coordinate line are marked as dots. In the figure, the thick black dot is the origin. The countdown starts from zero. Negative numbers are marked to the left of the origin, and positive numbers to the right.

The coordinate line continues indefinitely on both sides. Infinity in mathematics is symbolized by the symbol ∞. The negative direction will be indicated by the symbol −∞, and the positive direction by the symbol +∞. Then we can say that all numbers from minus infinity to plus infinity are located on the coordinate line:

Each point on the coordinate line has its own name and coordinate. Name is any Latin letter. Coordinate is a number that shows the position of a point on this line. Simply put, a coordinate is the very number that we want to mark on the coordinate line.

For example, point A(2) reads as "point A with coordinate 2" and will be denoted on the coordinate line as follows:

Here A is the name of the point, 2 is the coordinate of the point A.

Example 2. Point B(4) reads as "point B with coordinate 4"

Here B is the name of the point, 4 is the coordinate of the point B.

Example 3. Point M(−3) reads as "point M with coordinate minus three" and will be denoted on the coordinate line as follows:

Here M is the name of the point, −3 is the coordinate of point M .

Points can be designated by any letters. But it is generally accepted to denote them in capital Latin letters. Moreover, the beginning of the report, which is otherwise called origin usually denoted by the capital Latin letter O

It is easy to notice that negative numbers lie to the left relative to the origin, and positive numbers lie to the right.

There are phrases such as “the further to the left, the less” And "the further to the right, the more". You probably already guessed what we are talking about. With each step to the left, the number will decrease downward. And with each step to the right the number will increase. An arrow pointing to the right indicates a positive reference direction.

Comparing negative and positive numbers

Rule 1. Any negative number is less than any positive number.

For example, let's compare two numbers: −5 and 3. Minus five less than three, despite the fact that five strikes the eye first of all as a number greater than three.

This is due to the fact that −5 is a negative number, and 3 is positive. On the coordinate line you can see where the numbers −5 and 3 are located

It can be seen that −5 lies to the left, and 3 to the right. And we said that “the further to the left, the less” . And the rule says that any negative number is less than any positive number. It follows that

−5 < 3

"Minus five is less than three"

Rule 2. Of two negative numbers, the one that is located to the left on the coordinate line is smaller.

For example, let's compare the numbers −4 and −1. Minus four less, than minus one.

This is again due to the fact that on the coordinate line −4 is located to the left than −1

It can be seen that −4 lies to the left, and −1 to the right. And we said that “the further to the left, the less” . And the rule says that of two negative numbers, the one that is located to the left on the coordinate line is smaller. It follows that

Minus four is less than minus one

Rule 3. Zero is greater than any negative number.

For example, let's compare 0 and −3. Zero more than minus three. This is due to the fact that on the coordinate line 0 is located more to the right than −3

It can be seen that 0 lies to the right, and −3 to the left. And we said that "the further to the right, the more" . And the rule says that zero is greater than any negative number. It follows that

Zero is greater than minus three

Rule 4. Zero is less than any positive number.

For example, let's compare 0 and 4. Zero less, than 4. This is in principle clear and true. But we will try to see this with our own eyes, again on the coordinate line:

It can be seen that on the coordinate line 0 is located to the left, and 4 to the right. And we said that “the further to the left, the less” . And the rule says that zero is less than any positive number. It follows that

Zero is less than four

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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 we're talking about about 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 are all the numbers that more number 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.