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1. Introduce special cases of multiplication with 0 and 1.
2. To consolidate the meaning of multiplication and the transposable property of multiplication, to practice computational skills.
3. Develop attention, memory, thought operations, speech, Creative skills, interest in mathematics.

Equipment: Slide presentation: Appendix 1.

1. Organizational moment.

Today is an unusual day for us. Guests are present at the lesson. Please me, friends, guests with your successes. Open your notebooks, write down the number, Classwork... In the margin, mark your mood at the beginning of the lesson. Slide 2.

The whole class orally repeats the multiplication table on flashcards with speaking out loud (children mark wrong answers with claps).

Physical education (“Brain gymnastics”, “Hat for thinking”, breathing).

2. Statement of the educational problem.

2.1. Tasks for the development of attention.

On the chalkboard and on the table the children have a two-color picture with numbers:

- What is interesting about the recorded numbers? (Written in different colors; all the "red" numbers are even, and the "blue" ones are odd.)
- What number is superfluous? (10 - round, and the rest are not; 10 - two-digit, and the rest are single-valued; 5 - repeated two times, and the rest - one at a time.)
- I will close the number 10. Is there any superfluous among the rest of the numbers? (3 - he doesn't have a pair until 10, while the others do.)
- Find the sum of all the “red” numbers and write it down in the red square. (30.)
- Find the sum of all the “blue” numbers and write it down in the blue square. (23.)
- How much is 30 more than 23? (At 7.)
- How much is 23 less than 30? (Also at 7.)
- What action were you looking for? (Subtraction.) Slide 3.

2.2. Tasks for the development of memory and speech. Knowledge update.

a) - Repeat in order the words that I will name: term, term, sum, decreased, subtracted, difference. (Children try to reproduce the word order.)
- What actions components have you named? (Addition and subtraction.)
- What action are you still familiar with? (Multiplication, division.)
- Name the components of the multiplication. (Multiplier, multiplier, product.)
- What does the first factor mean? (Equal terms in the sum.)
- What does the second factor mean? (The number of such terms.)

Write down the definition of multiplication.

b) - Consider the records. What task will you perform?

12 + 12 + 12 + 12 + 12
33 + 33 + 33 + 33
a + a + a

(Replace the amount with the product.)

What happens? (In the first expression there are 5 terms, each of which is equal to 12, so it is equal to 12 5. Similarly - 33 4, and 3)

c) - Name the reverse operation. (Replace the product with the sum.)

- Replace the product with the sum in expressions: 99 2. 8 4. B 3. (99 + 99, 8 + 8 + 8 + 8, b + b + b). Slide 4.

d) The equalities are written on the board:

81 + 81 = 81 – 2
21 3 = 21 + 22 + 23
44 + 44 + 44 + 44 = 44 + 4
17 + 17 – 17 + 17 – 17 = 17 5

Pictures are placed next to each equality.

- The animals of the forest school performed the task. Did they do it correctly?

Children find out that the elephant, tiger, hare and squirrel made a mistake, explain what their mistakes are. Slide 5.

e) Compare expressions:

8 5. 5 8
5 6. 3 6
34 9… 31 2
a 3.a 2 + a

(8 5 = 5 8, since the sum does not change from the permutation of the terms;
5 6> 3 6, since there are 6 terms on the left and on the right, but there are more terms on the left;
34 9> 31 2. since there are more terms on the left and the terms themselves are larger;
a 3 = a 2 + a, since on the left and on the right there are 3 terms equal to a.)

- What property of multiplication was used in the first example? (Traveling.) Slide 6.

2.3. Formulation of the problem. Goal setting.

Are the equalities true? Why? (True, since the sum of 5 + 5 + 5 = 15. then the sum becomes one more term 5, and the sum increases by 5.)

5 3 = 15
5 4 = 20
5 5 = 25
5 6 = 30

- Continue this pattern to the right. (5 7 = 35; 5 8 = 40.)
- Continue it now to the left. (5 2 = 10; 5 1=5; 5 0 = 0.)
- And what does the expression 5 1 mean? 50? (? Problem!)

However, the expressions 5 1 and 5 0 are meaningless. We can agree to consider these equalities to be true. But for this it is necessary to check whether we will violate the transposition property of multiplication.

So, the purpose of our tutorial is - establish whether we can count the equalities 5 1 = 5 and 5 0 = 0 correct?

- Lesson problem! Slide 7.

3. “Discovery” of new knowledge by children.

a) - Follow the steps: 1 7, 1 4, 1 5.

Children solve examples with comments in the notebook and on the blackboard:

1 7 = 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7
1 4 = 1 + 1 + 1 + 1 = 4
1 5 = 1 + 1 + 1 + 1 +1 = 5

- Make a conclusion: 1 a -? (1 a = a.) The card is exposed: 1 a = a

b) - Do the expressions 7 1, 4 1, 5 1 make sense? Why? (No, since the sum cannot contain one term.)

- What should they be equal to in order not to violate the displacement property of multiplication? (7 1 should also equal 7, so 7 1 = 7.)

4 1 = 4 are considered similarly; 5 1 = 5.

- Make a conclusion: a 1 =? (a 1 = a.)

A card is displayed: a 1 = a. The first card is superimposed on the second: a 1 = 1 a = a.

- Does our conclusion coincide with what we got on the number ray? (Yes.)
- Translate this equality into Russian. (When you multiply a number by 1 or 1 by a number, you get the same number.)
- Well done! So, we will consider: a 1 = 1 a = a. Slide 8.

2) The case of multiplication with 0 is investigated in a similar way. Conclusion:

- when a number is multiplied by 0 or 0 by a number, zero is obtained: a 0 = 0 a = 0. Slide 9.
- Compare both equalities: what remind you of 0 and 1?

Children express their versions. You can draw their attention to the images:

1 - "mirror", 0 - "terrible beast" or "invisible hat".

Well done! So, when multiplied by 1, you get the same number (1 - "mirror"), and when multiplied by 0, we get 0 ( 0 - "invisible hat").

4. Physical education (for eyes - "circle", "up - down", for hands - "lock", "cams").

5. Primary anchoring.

Examples are written on the board:

Children solve them in a notebook and on a blackboard, pronouncing the received rules in loud speech, for example:

3 1 = 3, since multiplying a number by 1 gives the same number (1 is a “mirror”), and so on.

a) 145 x = 145; b) x 437 = 437.

- When multiplying 145 by an unknown number, it turned out to be 145. So, multiplied by 1 x = 1. Etc.

- When multiplying 8 by an unknown number, it turned out to be 0. So, we multiplied by 0 x = 0. And so on.

6. Independent work with a test in the classroom. Slide 10.

Children solve the recorded examples on their own. Then on finished

to the sample they check their answers with pronunciation in loud speech, mark correctly solved examples with a plus, correct mistakes. Those who made mistakes receive a similar task on the card and modify it individually while the class solves the revision problems.

7. Repetition tasks. (Work in pairs). Slide 11.

a) - Do you want to know what awaits you in the future? You will find out by decrypting the entry:

xn - i1abbnckbmcl9fb.xn - p1ai

Multiplication by 1 and 0 rule

According to the generally accepted definition, zero Is the number that separates positive from negative numbers on the number line. Zero- this is the most problematic place in mathematics, which does not obey logic, and all mathematical actions with zero are not based on logic, but on generally accepted definitions.

First example of problematic scratch Are natural numbers. In Russian schools zero is not natural number, in other schools, zero is a natural number. Since the concept of "natural numbers" is an artificial separation of some numbers from all other numbers according to certain criteria, there can be no mathematical proof of the naturalness or non-naturalness of zero. Zero is considered to be neutral with respect to addition and subtraction operations.

Zero counts as an integer, unsigned number. Also zero counts an even number, since dividing zero by 2 gives an integer zero.

Zero is the first digit in all standard number systems. In positional number systems, to which the familiar to us belongs decimal system reckoning, digit zero denote the absence of the value of this digit when recording a number. The Maya Indians used zero in their 12-digit number system a thousand years before Indian mathematicians. Each month began with day zero in the Mayan calendar. I wonder what the same sign zero Mayan mathematicians also designated infinity - the second problem of modern mathematics.

Word " zero"In Arabic sounds like" syfr ". From arabic word zero(syfr) the word "digit" occurred.

How to spell it correctly - zero or zero? The words zero and zero have the same meaning, but differ in use. Usually, zero used in everyday speech and in a number of stable combinations, zero- in terminology, in scientific speech. Both spellings of this word will be correct. For example: Division by zero. Zero integers. Zero attention. Zero without a stick. Absolute zero. Zero point five.

In grammar, words are derived from words zero and zero are written like this: zero or zero, zero or zero, zero or zero, zero or less commonly encountered zero, zero-zero. For example: Below zero. Equal to zero. Reduce to zero. Zero meredian. Zero mileage. At twelve zero-zero.

In mathematical operations with zero, the following results have been determined to date:

addition- if you add to any number zero, the number will remain unchanged; if to zero add any number the result of addition will be the same any number:

subtraction- if you subtract from any number zero, the number will remain unchanged; if from scratch subtract any number and the result will be the same any number with the opposite sign:

multiplication- if any number is multiplied by zero, the result is zero; if zero is multiplied by any number, the result is zero:

division- division by zero forbidden because the result does not exist; the generally accepted view of the problem of division by zero is presented in the work of Alexander Sergeev “ Why can't you divide by zero?"; for the curious, another article was written, which discusses the possibility of division by zero:

a: 0 = no division by zero, wherein a not equal to zero

zero divided by zero- the expression has no meaning, since it cannot be defined:

0: 0 = expression is meaningless

zero divided by number- if zero divided by a number, the result will always be zero, no matter what number is in the denominator (the exception to this rule is the number zero, see above):

0: a = 0, wherein a not equal to zero

zero to the power — zero is equal to any degree zero:

0 a = 0, wherein a not equal to zero

exponentiation- any number to a degree zero equals one (a number to the power of 0):

a 0 = 1, wherein a not equal to zero

zero to the power of zero- the expression does not make sense, since it cannot be determined (zero to the zero degree, 0 to the degree 0):

0 0 = expression is meaningless

root extraction- root of any degree from scratch is equal to zero:

0 1 / a = 0, wherein a not equal to zero

factorial- factorial of zero, or zero factorial, equals one:

distribution of numbers- when calculating the distribution of digits zero is considered insignificant. Changing the approach in the rules for calculating the distribution of digits when zero is considered a SIGNIFICANT figure will allow you to get more accurate results distribution of digits in all standard number systems, including the binary number system.

Who is interested in the question of occurrence scratch, I suggest you read the article "The History of Zero" by J. J. O'Connor and EF Robertson translated by I. Yu. Osmolovsky.

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How to multiply by 0.1

Let's analyze the rule and look at examples of how to multiply any number by 0.1.

Therefore, multiplying a number by 0.1 can be replaced by dividing it by 10. In general, it can be written as follows:

Hence the rule follows.

The rule for multiplying by 0.1

To multiply a number by 0.1, you need to move the comma in the entry of this number one digit to the left.

In recording a natural number, the comma at the end is not written:

Multiplying a natural number by 0.1 means, move this comma one sign to the left:

If the last digit in the record of a natural number is zero, as a result of multiplying this number by 0.1, we get a natural number (since a zero after the decimal point at the end of the number is not written):

To multiply by 0.1 common fraction, it is necessary to bring both fractions to the same form - either convert an ordinary fraction into decimal, or decimal - into an ordinary one.

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The rule of multiplying any number by zero

Back in school, teachers tried to hammer the simplest rule into our heads: "Any number multiplied by zero equals zero!"- but all the same, a lot of controversy constantly arises around him. Someone just memorized the rule and does not bother with the question “why?”. "You can't and that's it, because they said so at school, a rule is a rule!" Someone can write half a notebook with formulas, proving this rule or, conversely, its illogicality.

Who is right in the end

During these disputes, both people who have opposite points of view look at each other like a ram, and prove with all their might their innocence. Although, if you look at them from the side, you can see not one, but two rams resting their horns against each other. The only difference between them is that one is slightly less educated than the other.

This is interesting: bit terms - what are they?

More often than not, those who believe this rule to be incorrect try to invoke logic in this way:

I have two apples on my table, if I put zero apples to them, that is, I don’t put a single one, then my two apples will not disappear from this! The rule is illogical!

Indeed, apples will not disappear anywhere, but not because the rule is illogical, but because a slightly different equation is used here: 2 + 0 = 2. So we discard such a conclusion right away - it is illogical, although it has the opposite purpose - to call to logic.

This is interesting: How to find the difference of numbers in mathematics?

What is multiplication

The original rule of multiplication was defined only for natural numbers: multiplication is a number added to itself a certain number of times, which implies the naturalness of the number. Thus, any number with multiplication can be reduced to this equation:

1. 25 × 3 = 75
2. 25 + 25 + 25 = 75
3. 25 × 3 = 25 + 25 + 25

The conclusion follows from this equation, that multiplication is simplified addition.

This is interesting: what is a chord of a circle in geometry, definition and properties.

What is zero

Any person from childhood knows: zero is emptiness, Despite the fact that this emptiness has a designation, it does not carry anything at all. Ancient oriental scientists thought differently - they approached the issue philosophically and drew some parallels between emptiness and infinity and saw deep meaning in this number. After all, zero, which has the meaning of emptiness, standing next to any natural number, multiplies it ten times. Hence all the controversy over multiplication - this number carries so much inconsistency that it becomes difficult not to get confused. In addition, zero is constantly used to define empty digits in decimal fractions, this is done both before and after the comma.

Is it possible to multiply by emptiness

You can multiply by zero, but it's useless, because, whatever one may say, but even when multiplying negative numbers, you will still get zero. It is enough just to remember this simple rule and never ask this question again. In fact, everything is simpler than it seems at first glance. There are no hidden meanings and secrets, as the ancient scientists believed. The most logical explanation will be given below that this multiplication is useless, because when a number is multiplied by it, the same thing will still be obtained - zero.

This is interesting: what is the module of a number?

Going back to the very beginning, to the argument about two apples, 2 times 0 looks like this:

• If you eat two apples five times, then you eat 2 × 5 = 2 + 2 + 2 + 2 + 2 = 10 apples
• If you eat them two three times, then 2 × 3 = 2 + 2 + 2 = 6 apples are eaten
• If you eat two apples zero times, then nothing will be eaten - 2 × 0 = 0 × 2 = 0 + 0 = 0

After all, to eat an apple 0 times means not to eat a single one. Even the smallest child will understand this. Whatever one may say - 0 will come out, a two or three can be replaced with absolutely any number and absolutely the same will come out. To put it simply, then zero is nothing and when you have there is nothing, then no matter how much you multiply, it doesn't matter will be zero... There is no magic, and nothing will come out of an apple, even if you multiply 0 by a million. This is the simplest, most understandable and logical explanation of the rule of multiplication by zero. For a person far from all formulas and mathematics, such an explanation will be enough for the dissonance in the head to dissipate, and everything falls into place.

Another important rule follows from all of the above:

You cannot divide by zero!

This rule has also been stubbornly hammered into our heads since childhood. We just know that it's impossible and that's all, without stuffing our heads with unnecessary information. If you are unexpectedly asked the question why it is forbidden to divide by zero, then the majority will be confused and will not be able to clearly answer the simplest question from school curriculum because there are not so many controversies and contradictions around this rule.

Everyone just memorized the rule and did not divide by zero, not suspecting that the answer lies on the surface. Addition, multiplication, division and subtraction are unequal, only multiplication and addition are complete from the above, and all other manipulations with numbers are built from them. That is, writing 10: 2 is an abbreviation of the equation 2 * x = 10. So, writing 10: 0 is the same abbreviation from 0 * x = 10. It turns out that dividing by zero is a task to find a number, multiplying it by 0, you get 10 And we have already figured out that such a number does not exist, which means that this equation has no solution, and it will be incorrect a priori.

Let me tell you

To not divide by 0!

Cut 1 as you want, lengthwise,

Just don't divide by 0!

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Multiplication with 0 and 1.2 grade

Lesson presentation

Attention! Slide previews are for informational purposes only and may not represent all the presentation options. If you are interested in this work, please download the full version.

Lesson objectives:

• Educational:
  • to form the ability to perform multiplication with zero and one;
  • form the ability to correctly read mathematical expressions, name the components of multiplication;
  • to consolidate the ability to replace the product of numbers with a sum and orally calculate their value; shape initial skills work with the test.
• Developing:
  • promote the development of mathematical speech, working memory, voluntary attention, visual-active thinking.
• Educational:
  • foster a culture of behavior when frontal work, individual work; interest in the subject.

Lesson type- a lesson in the discovery of new knowledge.

The formation of new skills is possible only in activity, therefore, in the development of the lesson, the technology of the activity method was used. The use of this technology is a significant factor in increasing the efficiency of students' mastering of subject knowledge, the formation of educational universal action: regulatory, communicative, cognitive.

The developed lesson has the following structure:

1. Acquisition of primary experience of performing an action and motivation.
2. Formation of a new method (algorithm) of action, the establishment of primary connections with the available methods.
3. Training, clarification of connections, self-control and correction.
4. Control.

Equipment for the lesson:

• Standard: a textbook, a table for filling out test answers, stars made of colored paper, memos for students.
• Innovative: multimedia projector, interactive whiteboard, multimedia presentation "Journey to the Planet of Multiplication"

The use of multimedia components in the lesson introduces an element of novelty, makes the process of work visual, helps the teacher to focus on the main points. The work on each stage of the lesson is built as a kind of dialogue between the teacher and the students, in which the interactive board serves as a demonstrator for solving questions. Its use in educational process allows you to achieve high degree effectiveness.

Class: 3

Lesson presentation

Back forward

Attention! Slide previews are for informational purposes only and may not represent all the presentation options. If you are interested in this work, please download the full version.

Target:

1. Introduce special cases of multiplication with 0 and 1.
2. To consolidate the meaning of multiplication and the transposable property of multiplication, to practice computational skills.
3. Develop attention, memory, mental operations, speech, creativity, interest in mathematics.

Equipment: Slide presentation: Appendix 1.

During the classes

1. Organizational moment.

Today is an unusual day for us. Guests are present at the lesson. Please me, friends, guests with your successes. Open notebooks, write down the number, great work. In the margin, mark your mood at the beginning of the lesson. Slide 2.

The whole class orally repeats the multiplication table on flashcards with speaking out loud (children mark wrong answers with claps).

Physical education (“Brain gymnastics”, “Hat for thinking”, breathing).

2. Statement of the educational problem.

2.1. Tasks for the development of attention.

On the chalkboard and on the table the children have a two-color picture with numbers:

- What is interesting about the recorded numbers? (Written in different colors; all the "red" numbers are even, and the "blue" ones are odd.)
- What number is superfluous? (10 - round, and the rest are not; 10 - two-digit, and the rest are single-valued; 5 - repeated two times, and the rest - one at a time.)
- I will close the number 10. Is there any superfluous among the rest of the numbers? (3 - he doesn't have a pair until 10, while the others do.)
- Find the sum of all the “red” numbers and write it down in the red square. (30.)
- Find the sum of all the “blue” numbers and write it down in the blue square. (23.)
- How much is 30 more than 23? (At 7.)
- How much is 23 less than 30? (Also at 7.)
- What action were you looking for? (Subtraction.) Slide 3.

2.2. Tasks for the development of memory and speech. Knowledge update.

a) - Repeat in order the words that I will name: term, term, sum, decreased, subtracted, difference. (Children try to reproduce the word order.)
- What actions components have you named? (Addition and subtraction.)
- What action are you still familiar with? (Multiplication, division.)
- Name the components of the multiplication. (Multiplier, multiplier, product.)
- What does the first factor mean? (Equal terms in the sum.)
- What does the second factor mean? (The number of such terms.)

Write down the definition of multiplication.

a + a+… + a= аn

b) - Consider the records. What task will you perform?

12 + 12 + 12 + 12 + 12
33 + 33 + 33 + 33
a + a + a

(Replace the amount with the product.)

What happens? (In the first expression there are 5 terms, each of which is equal to 12, so it is equal to 12 5. Similarly - 33 4, and 3)

c) - Name the reverse operation. (Replace the product with the sum.)

- Replace the product with the sum in expressions: 99 2. 8 4. B 3.(99 + 99, 8 + 8 + 8 + 8, b + b + b). Slide 4.

d) The equalities are written on the board:

81 + 81 = 81 – 2
21 3 = 21 + 22 + 23
44 + 44 + 44 + 44 = 44 + 4
17 + 17 – 17 + 17 – 17 = 17 5

Pictures are placed next to each equality.

- The animals of the forest school performed the task. Did they do it correctly?

Children find out that the elephant, tiger, hare and squirrel made a mistake, explain what their mistakes are. Slide 5.

e) Compare expressions:

8 5... 5 8
5 6... 3 6
34 9… 31 2
a 3 ... a 2 + a

(8 5 = 5 8, since the sum does not change from the permutation of the terms;
5 6> 3 6, since there are 6 terms on the left and on the right, but there are more terms on the left;
34 9> 31 2. since there are more terms on the left and the terms themselves are larger;
a 3 = a 2 + a, since on the left and on the right there are 3 terms equal to a.)

- What property of multiplication was used in the first example? (Traveling.) Slide 6.

2.3. Formulation of the problem. Goal setting.

Are the equalities true? Why? (They are true, since the sum of 5 + 5 + 5 = 15. then the sum becomes one more term 5, and the sum increases by 5.)

5 3 = 15
5 4 = 20
5 5 = 25
5 6 = 30

- Continue this pattern to the right. (5 7 = 35; 5 8 = 40...)
- Continue it now to the left. (5 2 = 10; 5 1=5; 5 0 = 0.)
- And what does the expression 5 1 mean? 50? (? Problem!)

Outcome of discussion:

However, the expressions 5 1 and 5 0 are meaningless. We can agree to consider these equalities to be true. But for this it is necessary to check whether we will violate the transposition property of multiplication.

So, the purpose of our tutorial is - establish whether we can count the equalities 5 1 = 5 and 5 0 = 0 correct?

- Lesson problem! Slide 7.

3. “Discovery” of new knowledge by children.

a) - Follow the steps: 1 7, 1 4, 1 5.

Children solve examples with comments in the notebook and on the blackboard:

1 7 = 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7
1 4 = 1 + 1 + 1 + 1 = 4
1 5 = 1 + 1 + 1 + 1 +1 = 5

- Make a conclusion: 1 a -? (1 a = a.) The card is exposed: 1 a = a

b) - Do the expressions 7 1, 4 1, 5 1 make sense? Why? (No, since the sum cannot contain one term.)

- What should they be equal to in order not to violate the displacement property of multiplication? (7 1 should also equal 7, so 7 1 = 7.)

4 1 = 4 are considered similarly; 5 1 = 5.

- Make a conclusion: a 1 =? (a 1 = a.)

A card is displayed: a 1 = a. The first card is superimposed on the second: a 1 = 1 a = a.

- Does our conclusion coincide with what we got on the number ray? (Yes.)
- Translate this equality into Russian. (When you multiply a number by 1 or 1 by a number, you get the same number.)
- Well done! So, we will consider: a 1 = 1 a = a. Slide 8.

2) The case of multiplication with 0 is investigated in a similar way. Conclusion:

- when a number is multiplied by 0 or 0 by a number, zero is obtained: a 0 = 0 a = 0. Slide 9.
- Compare both equalities: what remind you of 0 and 1?

Children express their versions. You can draw their attention to the images:

1 - "mirror", 0 - "terrible beast" or "invisible hat".

Well done! So, when multiplied by 1, you get the same number (1 - "mirror"), and when multiplied by 0, we get 0 ( 0 - "invisible hat").

4. Physical education (for eyes - "circle", "up - down", for hands - "lock", "cams").

5. Primary anchoring.

Examples are written on the board:

23 1 =
1 89 =
0 925 =
364 1 =
156 0 =
0 1 =

Children solve them in a notebook and on a blackboard, pronouncing the received rules in loud speech, for example:

3 1 = 3, since multiplying a number by 1 gives the same number (1 is a “mirror”), and so on.

a) 145 x = 145; b) x 437 = 437.

- When multiplying 145 by an unknown number, it turned out to be 145. So, multiplied by 1 x = 1. Etc.

a) 8 x = 0; b) x 1 = 0.

- When multiplying 8 by an unknown number, it turned out to be 0. So, we multiplied by 0 x = 0. And so on.

6. Independent work with a test in the classroom. Slide 10.

Children solve the recorded examples on their own. Then on finished

to the sample, they check their answers with pronunciation in loud speech, mark correctly solved examples with a plus, correct mistakes. Those who made mistakes receive a similar task on the card and modify it individually while the class solves the revision problems.

7. Repetition tasks. (Work in pairs). Slide 11.

a) - Do you want to know what awaits you in the future? You will find out by decrypting the entry:

G – 49:7 O – 9 8 n – 9 9 v – 45:5 th – 6 6 d – 7 8 NS – 24:3

81	72	5	8	36	7	72	56

-So what awaits us? (New Year.)

b) - “I thought of a number, subtracted 7 from it, added 15, then added 4 and got 45. What number am I thinking?"

Reverse operations must be done in reverse order: 45 – 4 – 15 + 7 = 31.

8. Lesson summary.Slide 12.

What new rules did you meet?
What did you like? What was difficult?
Can this knowledge be applied in life?
In the margins, you can express your mood at the end of the lesson.
Complete the self-assessment table:

I want to know more
Ok, but I can do better
While I am experiencing difficulties

Thanks for your work, you did a good job!

9. Homework

Pp. 72–73 Rule, No. 6.

Very often, many people ask the question why division by zero cannot be used? In this article, we will go into great detail about where this rule came from, as well as what actions can be performed with zero.

In contact with

Zero can be called one of the most interesting numbers. This figure has no meaning, it means emptiness in the literal sense of the word. However, if you put zero next to any digit, then the value of this digit will increase several times.

The number is very mysterious in itself. Used it still ancient people Mayan. In the Maya, zero meant "beginning", and the countdown of calendar days also began from zero.

Highly interesting fact is that the sign of zero and the sign of uncertainty were similar. By this, the Maya wanted to show that zero is the same identical sign as uncertainty. In Europe, the designation of zero appeared relatively recently.

Also, many people know the ban associated with zero. Anyone will say that you cannot divide by zero... This is what the teachers say at school, and the children usually take their word for it. Usually, children are either simply not interested in knowing this, or they know what will happen if, after hearing an important prohibition, they immediately ask “Why can't you divide by zero?”. But when you get older, then interest wakes up, and I want to know more about the reasons for such a ban. However, there is reasonable evidence.

Zero actions

First, you need to determine what actions can be performed with zero. Exists several types of actions:

• Addition;
• Multiplication;
• Subtraction;
• Division (zero by number);
• Exponentiation.

Important! If you add zero to any number during addition, then this number will remain the same and will not change its numerical value. The same will happen if zero is subtracted from any number.

With multiplication and division, things are a little different. If multiply any number by zero, then the product will also become zero.

Let's consider an example:

Let's write this as addition:

There are five zeros added in total, so it turns out that

Let's try to multiply one by zero... The result will also be zero.

Zero can also be divided by any other number not equal to it. In this case, it will turn out, the value of which will also be zero. The same rule applies to negative numbers. If zero is divided by a negative number, you get zero.

You can also build any number to zero... In this case, the result will be 1. It is important to remember that the expression "zero to zero" is absolutely meaningless. If you try to raise zero to any power, you get zero. Example:

Using the rule of multiplication, we get 0.

So is it possible to divide by zero

So, here we come to the main question. Can you divide by zero generally? And why is it not possible to divide the number by zero, given that all other actions with zero quite exist and are applied? To answer this question, one must turn to higher mathematics.

Let's start with the definition of the concept, what is zero? School teachers claim that zero is nothing. Emptiness. That is, when you say that you have 0 pens, it means that you have no pens at all.

In higher mathematics, the concept of "zero" is broader. It does not mean emptiness at all. Here zero is called uncertainty, because if you do a little research, it turns out that when zero is divided by zero, we can end up with any other number, which may not necessarily be zero.

Did you know that those simple arithmetic operations that you learned in school are not so equal with each other? The most basic actions are addition and multiplication.

For mathematicians, there is no such thing as "" and "subtraction". Let's say: if you subtract three from five, then there will be two. This is what subtraction looks like. However, mathematicians will write it this way:

Thus, it turns out that the unknown difference is a certain number that needs to be added to 3 to get 5. That is, you do not need to subtract anything, you just need to find a suitable number. This rule applies to addition.

Things are a little different with rules of multiplication and division. It is known that multiplication by zero produces a zero result. For example, if 3: 0 = x, then if you flip the record, you get 3 * x = 0. And the number that is multiplied by 0 will give zero in the product. It turns out that the number that would give in the product with zero any value other than zero does not exist. This means that division by zero is meaningless, that is, it fits our rule.

But what happens if you try to divide zero by itself? Let us take as x some indefinite number. It turns out the equation 0 * x = 0. It can be solved.

If we try to take zero instead of x, then we get 0: 0 = 0. It would seem logical? But if we try to take any other number instead of x, for example, 1, then we end up with 0: 0 = 1. The same situation will be if you take any other number and substitute it in the equation.

In this case, it turns out that we can take any other number as a factor. The result will be an infinite variety of different numbers. Sometimes, nevertheless, division by 0 in higher mathematics makes sense, but then usually a certain condition appears, thanks to which we can still choose one suitable number. This action is called "disclosing uncertainty." In ordinary arithmetic, division by zero again loses its meaning, since we cannot choose any one number from the set.

Important! Zero cannot be divided by zero.

Zero and infinity

Infinity is very common in higher mathematics. Since it is simply not important for schoolchildren to know that there are still mathematical operations with infinity, teachers cannot explain to children why it is impossible to divide by zero.

Students begin to learn basic mathematical secrets only in the first year of the institute. Higher mathematics provides a large set of tasks that have no solution. The most famous problems are those with infinity. They can be solved with mathematical analysis.

You can also apply to infinity elementary mathematical operations: addition, multiplication by a number. Usually, subtraction and division are still used, but in the end they still boil down to two simple operations.

But what will happen if you try:

• Infinity times zero. In theory, if we try to multiply any number by zero, we will get zero. But infinity is an indefinite set of numbers. Since we cannot choose one number from this set, the expression ∞ * 0 has no solution and is absolutely meaningless.
• Zero divided by infinity. The same story happens here as above. We can't choose one number, which means we don't know what to divide by. The expression doesn't make sense.

Important! Infinity is a little different from uncertainty! Infinity is a type of uncertainty.

Now let's try to divide infinity by zero. It would seem that there should be uncertainty. But if we try to replace division with multiplication, we get a very definite answer.

For example: ∞ / 0 = ∞ * 1/0 = ∞ * ∞ = ∞.

It turns out like this mathematical paradox.

The answer why you can't divide by zero

Thought experiment, trying to divide by zero

Output

So, now we know that zero obeys almost all operations that are performed with, except for one single one. You cannot divide by zero just because the result is uncertainty. We also learned how to perform actions with zero and infinity. Uncertainty will be the result of such actions.

Zero is a very interesting figure in itself. By itself, it means emptiness, lack of meaning, and next to another number increases its significance by 10 times. Any numbers in the zero degree always give 1. This sign was used in the Mayan civilization, and it also denoted the concept of "beginning, cause". Even the calendar started from day zero. And this figure is also associated with a strict ban.

Ever since the elementary school years, we all have clearly learned the rule “you cannot divide by zero”. But if in childhood you take a lot on faith and the words of an adult rarely raise doubts, then over time sometimes you still want to understand the reasons, to understand why certain rules were established.

Why can't you divide by zero? I would like to get a clear logical explanation for this question. In the first grade, the teachers could not do this, because in mathematics the rules are explained using equations, and at that age we had no idea what it was. And now it's time to figure it out and get a clear logical explanation of why you can't divide by zero.

The fact is that in mathematics, only two of the four basic operations (+, -, x, /) with numbers are recognized as independent: multiplication and addition. The rest of the operations are considered to be derivatives. Let's look at a simple example.

Tell me, how much will it turn out if you subtract 18 from 20? Naturally, the answer immediately arises in our head: it will be 2. And how did we come to such a result? To some, this question will seem strange - after all, everything is clear that it will turn out 2, someone will explain that he took 18 from 20 kopecks and he got two kopecks. Logically, all these answers are not in doubt, but from the point of view of mathematics, this problem should be solved in a different way. Let us remind once again that the main operations in mathematics are multiplication and addition, and therefore in our case the answer lies in the solution of the following equation: x + 18 = 20. From which it follows that x = 20 - 18, x = 2. It would seem, why describe everything in such detail? After all, everything is elementary simple. However, without this, it is difficult to explain why one cannot divide by zero.

Now let's see what happens if we wish to divide 18 by zero. Let's make the equation again: 18: 0 = x. Since the division operation is a derivative of the multiplication procedure, transforming our equation we get x * 0 = 18. This is where the dead end begins. Any number in the place of x when multiplied by zero will give 0 and we will not be able to get 18 in any way. Now it becomes very clear why one cannot divide by zero. Zero itself can be divided by any number, but on the contrary - alas, it cannot be.

What happens if zero is divided by itself? It can be written like this: 0: 0 = x, or x * 0 = 0. This equation has countless solutions. So the end result is infinity. Therefore, the operation does not make sense in this case either.

Division by 0 is at the root of many supposed mathematical jokes that can be used to puzzle any ignorant person if desired. For example, consider the equation: 4 * x - 20 = 7 * x - 35. Let's take out 4 in the left part, and in the right part 7. We get: 4 * (x - 5) = 7 * (x - 5). Now we multiply the left and right sides of the equation by the fraction 1 / (x - 5). The equation will take this form: 4 * (x - 5) / (x - 5) = 7 * (x - 5) / (x - 5). Reduce the fractions by (x - 5) and we get that 4 = 7. From this we can conclude that 2 * 2 = 7! Of course, the catch here is that it is equal to 5 and it was impossible to cancel fractions, since this led to division by zero. Therefore, when reducing fractions, you must always check so that zero does not accidentally end up in the denominator, otherwise the result will turn out to be completely unpredictable.

If we can rely on other laws of arithmetic, then this separate fact can be proved.

Suppose there is a number x for which x * 0 = x ", and x" is not zero (for simplicity, we will assume that x "> 0)

Then, on the one hand, x * 0 = x ", on the other hand, x * 0 = x * (1 - 1) = x - x

It turns out that x - x = x ", whence x = x + x", that is, x> x, which cannot be true.

This means that our assumption leads to a contradiction and there is no such number x for which x * 0 would not be equal to zero.

the assumption cannot be true because it is just an assumption! no one can explain in simple language or is at a loss! if 0 * x = 0 then 0 * x = (0 + 0) * x = 0 * x + 0 * x and as a result we have reduced the right to left 0 = 0 * x this is like a mathematical proof! but such nonsense with this zero is terribly contradictory and in my opinion 0 should not be a number, but only just an abstract concept! So that mere mortals do not burn in the brain by the fact that the physical presence of objects, when miraculously multiplied by nothing, gave rise to nothing!

P / s is not entirely clear to me not a mathematician, but a mere mortal where did you get units in the equation-reasoning (like 0 is the same as 1-1)

I bastard with reasoning, like there is some kind of X and let it be any number

is in the equation 0 and when multiplying by it we reset all numerical values

hence X is a numerical value, and 0 is the number of actions performed on the number X (and actions, in turn, are also displayed in numerical format)

EXAMPLE on apples)):

Kolya had 5 apples, he took these apples and went to the market in order to increase the capital, but the day was rainy, the cloudy trade did not work out and Kalek returned home with nothing. In mathematical terms, the story about Kolya and apples looks like this

5 apples * 0 sales = got 0 profit 5 * 0 = 0

Before going to the bazaar, Kolya went and plucked 5 apples from the tree, and tomorrow he went to pick but didn't get there for some reason ...

Apples 5, tree 1, 5 * 1 = 5 (Kolya collected 5 apples on the 1st day)

Apples 0, tree 1, 0 * 1 = 0 (actually the result of Kolya's work on the second day)

The scourge of mathematics is the word "Suppose"

To answer

And if in another way, 5 apples to 0 apples = how many apples, according to mathematics, there should be zero, and so

In fact, any numbers make sense only when they are associated with material objects, such as 1 cow, 2 cows, or whatever, and an account has appeared in order to count objects and not just like that, and there is a paradox if I do not have a cow , and the neighbor has a cow, and we multiply my absence by the neighbor's cow, then his cow should disappear, multiplication is generally invented to facilitate the addition of large quantities of identical objects when it is difficult to count them by the addition method, for example, money was added in columns of 10 coins, and then the number of columns was multiplied by the number of coins in a column, much easier than adding. but if the number of columns is multiplied by zero of coins, then naturally it will turn out to be zero, but if there are columns and coins, then how do not multiply them by zero, the coins will not go anywhere because they are there, and even if it is one coin, then the column is composed of one coin, so you can't go anywhere, so zero when multiplied by zero is obtained only under certain conditions, that is, in the absence of a material component, and if I have 2 socks, don't multiply them by zero, they won't go anywhere ...