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. 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 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. 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 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 the help of 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, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
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.
Goals and objectives of the lesson:
- General lesson on mathematics in 6th grade "Addition and Subtraction positive and negative numbers»
- Summarize and systematize students’ knowledge on this topic.
- Develop subject and general academic skills and abilities, the ability to use acquired knowledge to achieve a goal; establish patterns of diversity of connections to achieve a level of systematic knowledge.
- Developing self-control and mutual control skills; develop desires and needs to generalize the facts received; develop independence and interest in the subject.
During the classes
Guys, we are traveling through the country of “Rational Numbers”, where positive, negative numbers and zero live. While traveling, we learn a lot of interesting things about them, get acquainted with the rules and laws by which they live. This means that we must follow these rules and obey their laws.
What rules and laws have we become familiar with? (addition and subtraction rules rational numbers, laws of addition)
And so the topic of our lesson is “Adding and subtracting positive and negative numbers.”(Students write down the date and topic of the lesson in their notebooks)
II. Examination homework
III. Updating knowledge.
Let's start the lesson with oral work. There is a series of numbers in front of you.
8,6; 21,8; -0,5; 6,6; 4,7; 7; -18; 0.
Answer the questions:
Which number in the series is the largest?
What number has the greatest modulus?
Which number is the smallest in the series?
What number has the smallest modulus?
How to compare two positive numbers?
How to compare two negative numbers?
How to compare numbers with different signs?
Which numbers in the series are opposites?
List the numbers in ascending order.
IV. Find the mistake
a) -47 + 25+ (-18)= 30
c) - 7.2+(- 3.5) + 10.6= - 0.1
d) - 7.2+ (- 2.9) + 7.2= 2.4
V.Task “Guess the word”
In each group, I distributed tasks in which words were encrypted.
After completing all the tasks, you will guess the key words(flowers, gift, girls)
| 1 row | Answer | Letter |
|
| Answer | Letter |
||
| 54-(-74) | |||
| 2,5-3,6 | |||
| 23,7+23,7 | |||
| 11,2+10,3 | |||
| 3rd row | Answer | Letter |
|
| 2,03-7,99 | |||
| 67,34-45,08 | |||
| 10,02 | 112,42 | 50,94 | 50,4 |
VI. Fizminutka
Well done, you have worked hard, I think it’s time to relax, concentrate, relieve fatigue, return peace of mind simple exercises will help
PHYSICAL MINUTE (If the statement is correct, clap your hands; if not, shake your head from side to side):
When adding two negative numbers, the modules of the terms must be subtracted -
The sums of two negative numbers are always negative +
When adding two opposite numbers always turns out 0 +
When adding numbers with different signs, you need to add their modules -
The sum of two negative numbers is always less than each of the terms +
When adding numbers with different signs, you need to subtract the smaller module from the larger module +
VII.Solving tasks according to the textbook.
No. 1096(a,d,i)
VIII. Homework
Level 1 “3”-No. 1132
Level 2 - “4” - No. 1139, 1146
IX. Independent work according to options.
Level 1, "3"
| 1 option | Option 2 |
Level 2, “4”
| 1 option | Option 2 |
| 1 - (- 3 )+(- 2 ) |
Level 3, "5"
| 1 option | 2nd option |
| 4,2-3,25-(-0,6) | 2,4-1,75-(-2,6) |
Mutual check on the board, change desk neighbors
X. Summing up the lesson. Reflection
Let's remember the beginning of our lesson, guys.
What lesson goals did we set for ourselves?
Do you think we managed to achieve our goals?
Guys, now evaluate your work in class. In front of you is a card with a picture of a mountain. If you think you did a good job in class, you'll be fine.Clearly, then draw yourself on the top of the mountain. If anything is unclear, draw yourself below, and decide for yourself on the left or right.
Give me your drawings along with your score card, you will learn the final grade for your work in the next lesson.
Now we'll figure it out positive and negative numbers. First, we will give definitions, introduce notation, and then give examples of positive and negative numbers. We will also focus on semantic load, which is carried by positive and negative numbers.
Page navigation.
Positive and Negative Numbers - Definitions and Examples
Give identifying positive and negative numbers will help us. For convenience, we will assume that it is located horizontally and directed from left to right.
Definition.
Numbers that correspond to points of the coordinate line lying to the right of the origin are called positive.
Definition.
The numbers that correspond to the points of the coordinate line lying to the left of the origin are called negative.
The number zero, which corresponds to the origin, is neither a positive nor a negative number.
From the definition of negative and positive numbers it follows that the set of all negative numbers is the set of numbers opposite all positive numbers (if necessary, see the article opposite numbers). Therefore, negative numbers are always written with a minus sign.
Now, knowing the definitions of positive and negative numbers, we can easily give examples of positive and negative numbers. Examples of positive numbers are the natural numbers 5, 792 and 101,330, and indeed any natural number is positive. Examples of positive rational numbers are the numbers , 4.67 and 0,(12)=0.121212... , and negative ones are the numbers , −11 , −51.51 and −3,(3) . Examples of positive irrational numbers include pi, e, and the infinite non-periodic decimal 809.030030003…, and examples of negative irrational numbers are the numbers minus pi, minus e and the number equal to . It should be noted that in last example It is by no means obvious that the value of the expression is a negative number. To find out for sure, you need to get the value of this expression in the form of a decimal fraction, and we will tell you how to do this in the article comparison of real numbers.
Sometimes positive numbers are preceded by a plus sign, just as negative numbers are preceded by a minus sign. In these cases, you should know that +5=5, 
and so on. That is, +5 and 5, etc. - this is the same number, but designated differently. Moreover, you can come across definitions of positive and negative numbers based on the plus or minus sign.
Definition.
Numbers with a plus sign are called positive, and with a minus sign – negative.
There is another definition of positive and negative numbers based on comparison of numbers. To give this definition, it is enough just to remember that the point on the coordinate line corresponding to the larger number lies to the right of the point corresponding to the smaller number.
Definition.
Positive numbers are numbers that Above zero, A negative numbers are numbers less than zero.
Thus, zero seems to separate positive numbers from negative.
Of course, we should also dwell on the rules for reading positive and negative numbers. If a number is written with a + or − sign, then pronounce the name of the sign, after which the number is pronounced. For example, +8 is read as plus eight, and - as minus one point two fifths. The names of the signs + and − are not declined by case. Example correct pronunciation is the phrase “a equals minus three” (not minus three).
Interpretation of positive and negative numbers
We have been describing positive and negative numbers for quite some time. However, it would be nice to know what meaning they carry? Let's look at this issue.
Positive numbers can be interpreted as an arrival, as an increase, as an increase in some value, and the like. Negative numbers, in turn, mean exactly the opposite - expense, deficiency, debt, reduction of some value, etc. Let's understand this with examples.
We can say that we have 3 items. Here the positive number 3 indicates the number of items we have. How can you interpret the negative number −3? For example, the number −3 could mean that we have to give someone 3 items that we don't even have in stock. Similarly, we can say that at the cash register we were given 3.45 thousand rubles. That is, the number 3.45 is associated with our arrival. In turn, a negative number -3.45 will indicate a decrease in money in the cash register that issued this money to us. That is, −3.45 is the expense. Another example: a temperature increase of 17.3 degrees can be described with a positive number of +17.3, and a temperature decrease of 2.4 can be described with a negative number, as a temperature change of -2.4 degrees.
Positive and negative numbers are often used to describe the values of certain quantities in different measuring instruments. The most accessible example is a device for measuring temperatures - a thermometer - with a scale on which both positive and negative numbers are written. Often negative numbers are depicted in blue (it symbolizes snow, ice, and at temperatures below zero degrees Celsius, water begins to freeze), and positive numbers are written in red (the color of fire, the sun, at temperatures above zero degrees Celsius, ice begins to melt). Writing positive and negative numbers in red and blue is also used in other cases when you need to highlight the sign of the numbers.
Bibliography.
- Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.
In this article we will talk about adding negative numbers. First we give the rule for adding negative numbers and prove it. After this, we will look at typical examples of adding negative numbers.
Page navigation.
Rule for adding negative numbers
Before formulating the rule for adding negative numbers, let us turn to the material in the article: positive and negative numbers. There we mentioned that negative numbers can be perceived as debt, and in this case determines the amount of this debt. Therefore, the addition of two negative numbers is the addition of two debts.
This conclusion allows us to understand rule for adding negative numbers. To add two negative numbers, you need:
- fold their modules;
- put a minus sign in front of the received amount.
Let's write down the rule for adding negative numbers −a and −b in letter form: (−a)+(−b)=−(a+b).
It is clear that the stated rule reduces the addition of negative numbers to the addition of positive numbers (the modulus of a negative number is a positive number). It is also clear that the result of adding two negative numbers is a negative number, as evidenced by the minus sign that is placed in front of the sum of the modules.
The rule for adding negative numbers can be proven based on properties of operations with real numbers(or the same properties of operations with rational or integer numbers). To do this, it is enough to show that the difference between the left and right parts equality (−a)+(−b)=−(a+b) is equal to zero.
Since subtracting a number is the same as adding the opposite number (see the rule for subtracting integers), then (−a)+(−b)−(−(a+b))=(−a)+(−b)+(a+b). Due to the commutative and associative properties of addition, we have (−a)+(−b)+(a+b)=(−a+a)+(−b+b). Since the sum of opposite numbers is equal to zero, then (−a+a)+(−b+b)=0+0, and 0+0=0 due to the property of adding a number with zero. This proves the equality (−a)+(−b)=−(a+b) , and hence the rule for adding negative numbers.
All that remains is to learn how to apply the rule of adding negative numbers in practice, which we will do in the next paragraph.
Examples of adding negative numbers
Let's sort it out examples of adding negative numbers. Let's start with the simplest case - the addition of negative integers; we will carry out the addition according to the rule discussed in the previous paragraph.
Example.
Add the negative numbers −304 and −18,007.
Solution.
Let's follow all the steps of the rule for adding negative numbers.
First we find the modules of the numbers being added: and 
. Now you need to add the resulting numbers; here it is convenient to perform column addition: 
Now we put a minus sign in front of the resulting number, as a result we have −18,311.
Let's write the whole solution in short form: (−304)+(−18 007)= −(304+18 007)=−18 311 .
Answer:
−18 311 .
The addition of negative rational numbers, depending on the numbers themselves, can be reduced either to the addition of natural numbers, or to the addition of ordinary fractions, or to the addition of decimal fractions.
Example.
Add a negative number and a negative number −4,(12) .
Solution.
According to the rule for adding negative numbers, you first need to calculate the sum of the modules. The modules of the negative numbers being added are equal to 2/5 and 4, (12) respectively. The addition of the resulting numbers can be reduced to the addition of ordinary fractions. To do this, we convert the periodic decimal fraction into an ordinary fraction: . Thus, 2/5+4,(12)=2/5+136/33. Now let's do it
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.
Lesson contentThis 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. Bold in the picture black dot is the starting point. 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 downwards. 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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