Is it possible to divide by zero? The mathematician answers. Division by zero. Fascinating mathematics Any number multiplied by 0 is how much

If we can rely on other laws of arithmetic, then this single fact can be proven.

Suppose that 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 number x for which x * 0 would not be equal to zero.

the assumption cannot be true because it is just an assumption! nobody in simple language cannot explain or finds it difficult! if 0 * x= 0 then 0 *x=(0+0)*x=0*x + 0*x and as a result they reduced right to left 0=0*x this is like a mathematical proof! but this kind of nonsense with this zero is terribly contradictory and in my opinion 0 should not be a number, but only an abstract concept! So that the fact that the physical presence of objects, when miraculously multiplied by nothing, does not give birth to nothing, does not cause a burning sensation in the brain!

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

I'm crazy about reasoning like there is some kind of X and let it be any number

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

therefore X is numeric 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 to increase his capital, but the day turned out to be rainy, the trade did not work out and the cripple returned home with nothing. In mathematical language, the story about Kolya and apples looks like this

5 apples * 0 sales = received 0 profit 5*0=0

Before going to the market, Kolya went and picked 5 apples from the tree, and tomorrow he went to pick them but didn’t get there for some reason of his own...

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 labor on the second day)

The scourge of mathematics is the word “Suppose”

Answer

But to put it another way, 5 apples to 0 apples = how many apples, according to mathematics it should be zero, so here it is

In fact, any numbers make sense only when they are associated with material objects, such as 1 cow, 2 cows, or whatever, and a count appeared in order to count objects and not just like that, and there is a paradox if I don’t 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 was generally invented to facilitate the addition of large quantities of identical objects, when they are difficult to count using the addition method, for example, money was folded into columns of 10 coins, and then the number of columns was multiplied by the number of coins in the column, much easier than adding. but if the number of columns is multiplied by zero coins, then naturally the result will be zero, but if there are columns and coins, then no matter how you multiply them by zero, the coins will not go anywhere because there are them, and even if it is one coin, then the column is consisting of one coin, so there’s no getting around it, so when multiplied by zero, zero is obtained only under certain conditions, that is, in the absence of a material component, and if I have 2 socks, no matter how you multiply them by zero, they won’t go anywhere .

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

Since our elementary school years, we have all clearly learned the rule “you cannot divide by zero.” But if in childhood you take a lot of things 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, 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 remaining operations are considered to be derivatives. Let's look at a simple example.

Tell me, how much do you get if you subtract 18 from 20? Naturally, the answer immediately arises in our head: it will be 2. How did we come to this result? This question will seem strange to some - after all, everything is clear that the result will be 2, someone will explain that he took 18 from 20 kopecks and got two kopecks. Logically, all these answers are not in doubt, but from a mathematical point of view, this problem should be solved differently. Let us recall once again that the main operations in mathematics are multiplication and addition, and therefore in our case the answer lies in solving 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 so simple. However, without this it is difficult to explain why you cannot divide by zero.

Now let's see what happens if we want to divide 18 by zero. Let's create 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 place of X when multiplied by zero will give 0 and we will not be able to get 18. Now it becomes extremely clear why you cannot divide by zero. Zero itself can be divided by any number, but vice versa - alas, it’s impossible.

What happens if you divide zero by itself? This can be written as follows: 0: 0 = x, or x * 0 = 0. This equation has an infinite number of solutions. Therefore, the end result is infinity. Therefore, the operation in this case also does not make sense.

Division by 0 is at the root of many imaginary 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 4 out of brackets on the left side and 7 on the right. We get: 4*(x - 5) = 7*(x - 5). Now let's multiply the left and right sides of the equation by the fraction 1 / (x - 5). The equation will take the following form: 4*(x - 5)/(x - 5) = 7*(x - 5)/ (x - 5). Let's reduce the fractions by (x - 5) and it turns out 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 that a zero does not accidentally end up in the denominator, otherwise the result will be completely unpredictable.

MKOU Sarybalyk Secondary School

Teacher primary classes: Makoveeva Marina Valentinovna

Math lesson in 4th grade. (textbook for special (correctional) educational institutionsVIIIspecies, author M. N. Perova)

Topic: “Multiplying the number zero and by zero. Divide zero."

Target: introduce the rule of multiplying the number 0 and by 0, dividing 0; consolidate knowledge of the multiplication table, the ability to solve problems of the types studied; learn to reason and draw conclusions.

Planned results: Students will learn to multiply 0 by a number, a number by 0, and divide 0; use multiplication and division tables; solve problems of the studied types; evaluate the correctness of actions.

Equipment: cards for the game “Postman”; table with geometric shapes, handouts,personal computer, media projector, textbook “Mathematics” by M. N. Perov(4th grade).

Lesson type: new topic.

Lesson type: lesson-game.

Lesson progress

I . Org. moment:

Checking homework.

II . Oral counting.

Teacher: remember table multiplication and division. Now we will play the game “Postmen”. Sveta, you will be a postman. There are houses with numbers on the board. Your task is to take an example letter, solve it correctly and determine which house we need to take the letter to.

3x4 2x2 9x2 3x1 3x8 25:5

6x2 16:4 3x6 9:3 6x4 5:1

4:1 3:1

Teacher: Insert the missing action sign.

4…0=4 1…3=4 5…1=6

4…4=0 1…3=3 5…1=5

3…3=0 1…0=1 9…0=0

III . Getting to know new material

ABOUT ZERO

In vain they think that it is zero

Plays a small role

Many people once thought

That zero means nothing

And, oddly enough, they thought

That he is not a number at all.

But about its special properties

We will now tell the story

When you add zero to a number

Or you take it away from him

In response you immediately receive

Same number again

Finding itself as a multiplier among the numbers

He brings everything to naught in an instant

And therefore in the work

One for all bears the answer

And regarding division

We need to firmly remember that

What a long time ago in the scientific world

Dividing by zero is prohibited

Indeed: which of the famous

We take the number as the quotient

When with a zero in a product

All numbers can only give zero

Teacher: Let's check if everything in the poem is correct:

7+0=7 7-0=7 7 0=0 7:0

Teacher: we apply the commutative property of multiplication and replace multiplication with addition: 7·0=0·7=0+0+0+0+0+0+0=0

What happened?

Teacher: we know that division is checked by multiplication: then we multiply the quotient by 0 - we should get 7, but this is not possible! Whatever number we multiply by 0, there will always be 0 in the product.

IV . Fizminutka

V . Reinforcing the material learned

1. Solving the problem (p. 143 No. 7)

Teacher: What does the problem say?

Student: about repairs, foundations, bricks.

Teacher: what do you need to know?

Student: How many bricks are left to lay?

Teacher: Can we answer this question right away?

Student: no.

Teacher: Why?

Student: Because we don't know how many bricks the worker used.

Teacher: will we be able to find out?

Student: yes.

Teacher: what action?

Student: division.

Teacher: Can we now answer the question of the problem?

Student: yes.

Teacher: what action?

Student: by subtraction.

Teacher: How many bricks does the worker have left to lay?

Student: (40:5=8, 40-8=32) 32 bricks.

2.Independent work(p. 144 no. 18)

7*0 7:1 3*0 8:1

7*1 0*7 0*3 0:8

1*6 0*1 3*1 0*8

0*6 0:1 1*3 0*1

3. Work at the board (p. 144 No. 11)

7*0 0*8 0:5 1*3 5+0

7+1 0:8 6*0 1+3 5*0

7-1 8+0 8-0 4-1 5-1

VI. Repetition

1.Circular examples

Teacher: We will be foresters. We need to determine the height of some trees; for this we need to solve circular examples.

2. Arithmetic dictation

Teacher: And now we will be stenographers. I dictate, and you write down - you take shorthand with the help of cards.

Sum of numbers 45 and 18 (45+18=63)

Product of numbers 8 and 3 (8*3=24)

Difference of numbers 35 and 7 (35-7=22)

The quotient of 20 and 4 (20:4=5)

3.Geometric material.

Teacher: last task. Which geometric shapes you see?

Count and say how many times each figure appears.

(Circle - 12, square - 6, triangle - 6, rectangle - 5.)

VII . Reflection

Independent execution p. 144 No. 17 (1.2 art.). The answers are written on the board: 0,0,0;5,5,5.

Appreciate your work in class with a smiley face.

VIII. Homework

P. 144 No. 12.

Which of these sums do you think can be replaced by a product?

Let's think like this. In the first sum, the terms are the same, the number five is repeated four times. This means we can replace addition with multiplication. The first factor shows which term is repeated, the second factor shows how many times this term is repeated. We replace the sum with the product.

Let's write down the solution.

In the second sum, the terms are different, so it cannot be replaced by a product. Add the terms and get the answer 17.

Let's write down the solution.

Can a product be replaced by a sum of identical terms?

Let's look at the works.

Let's carry out the actions and draw a conclusion.

1*2=1+1=2

1*4=1+1+1+1=4

1*5=1+1+1+1+1=5

We can conclude: The number of unit terms is always equal to the number by which the unit is multiplied.

Means, When you multiply the number one by any number, you get the same number.

1 * a = a

Let's look at the works.

These products cannot be replaced by a sum, since a sum cannot have one term.

The products in the second column differ from the products in the first column only in the order of the factors.

This means that in order not to violate the commutative property of multiplication, their values ​​must also be equal to the first factor, respectively.

Let's conclude: When you multiply any number by the number one, you get the number that was multiplied.

Let's write this conclusion as an equality.

a * 1= a

Solve examples.

Hint: Don't forget the conclusions we made in the lesson.

Test yourself.

Now let's observe products where one of the factors is zero.

Let's consider products where the first factor is zero.

Let us replace the products with the sum of identical terms. Let's carry out the actions and draw a conclusion.

0*3=0+0+0=0

0*6=0+0+0+0+0+0=0

0*4=0+0+0+0=0

The number of zero terms is always equal to the number by which zero is multiplied.

Means, When you multiply zero by a number, you get zero.

Let's write this conclusion as an equality.

0 * a = 0

Let's consider products where the second factor is zero.

These products cannot be replaced by a sum, since a sum cannot have zero terms.

Let's compare the works and their meanings.

0*4=0

The products of the second column differ from the products of the first column only in the order of the factors.

This means that in order not to violate the commutative property of multiplication, their values ​​must also be equal to zero.

Let's conclude: When any number is multiplied by zero, the result is zero.

Let's write this conclusion as an equality.

a * 0 = 0

But you can't divide by zero.

Solve examples.

Hint: Don't forget the conclusions you made in the lesson. When calculating the values ​​of the second column, be careful when determining the order of actions.

Test yourself.

Today in class we met special cases multiplying by 0 and 1, practiced multiplying by 0 and 1.

References

1. M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 1. - M.: “Enlightenment”, 2012.
2. M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. 3rd grade: in 2 parts, part 2. - M.: “Enlightenment”, 2012.
3. M.I. Moro. Math lessons: Methodical recommendations for the teacher. 3rd grade. - M.: Education, 2012.
4. Regulatory document. Monitoring and evaluation of learning outcomes. - M.: “Enlightenment”, 2011.
5. "School of Russia": Programs for primary school. - M.: “Enlightenment”, 2011.
6. S.I. Volkova. Mathematics: Test work. 3rd grade. - M.: Education, 2012.
7. V.N. Rudnitskaya. Tests. - M.: “Exam”, 2012.

1. Nsportal.ru ().
2. Prosv.ru ().
3. Do.gendocs.ru ().

Homework

1. Find the meanings of the expressions.

2. Find the meanings of the expressions.

3. Compare the meanings of the expressions.

(56-54)*1 … (78-70)*1

4. Create an assignment on the topic of the lesson for your friends.

Class: 3

Presentation for the lesson

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

Target:

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

Equipment: Slide presentation: Appendix 1.

Lesson progress

1. Organizational moment.

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

The whole class orally repeats the multiplication table on cards, saying it out loud. (children mark incorrect answers with clapping).

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

2. Statement of the educational task.

2.1. Tasks for the development of attention.

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

– What is interesting about the written numbers? (Write in different colors; all “red” numbers are even, and “blue” numbers are odd.)
– Which number is the odd one out? (10 is round, and the rest are not; 10 is two-digit, and the rest are single-digit; 5 is repeated twice, and the rest - one at a time.)
– I’ll close the number 10. Is there an extra one among the other numbers? (3 – he doesn’t have a pair until 10, but the rest do.)
– Find the sum of all the “red” numbers and write it in the red square. (30.)
– Find the sum of all the “blue” numbers and write it in the blue square. (23.)
– How much more is 30 than 23? (On 7.)
– How much is 23 less than 30? (Also at 7.)
– What action did you use to search for? (Subtraction.) Slide 3.

2.2. Tasks for the development of memory and speech. Updating knowledge.

a) – Repeat in order the words that I will name: addend, addend, sum, minuend, subtrahend, difference. (Children try to reproduce the order of words.)
– What components of actions were named? (Addition and subtraction.)
– What action are you still familiar with? (Multiplication, division.)
– Name the components of 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= an

b) – Look at the notes. What task will you be doing?

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

(Replace the sum with the product.)

What will happen? (The first expression has 5 terms, each of which is equal to 12, so it is equal to 12 5. Similarly - 33 4, and 3)

c) – Name the inverse operation. (Replace the product with the sum.)

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

d) 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 were completing a task. Did they do it correctly?

Children establish that the elephant, tiger, hare and squirrel were mistaken, and explain what their mistakes were. Slide 5.

e) Compare the 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 rearranging the terms;
5 6 > 3 6, since there are 6 terms on the left and 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 right there are 3 terms equal to a.)

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

2.3. Statement of the problem. Goal setting.

Are the equalities true? Why? (Correct, since the sum is 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.)
– What does the expression 5 1 mean? 5 0? (? Problem!)

Summary of the discussion:

However, the expressions 5 1 and 5 0 do not make sense. We can agree to consider these equalities true. But to do this, we need to check whether we will violate the commutative property of multiplication.

So, the goal of our lesson is determine whether we can count equalities 5 1 = 5 and 5 0 = 0 true?

- Lesson problem! Slide 7.

3. “Discovery” of new knowledge by children.

a) – Follow steps: 1 7, 1 4, 1 5.

Children solve examples with comments in their notebooks and on the board:

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

– Draw a conclusion: 1 a – ? (1 a = a.) The card is displayed: 1 a = a

b) – Do the expressions 7 1, 4 1, 5 1 make sense? Why? (No, because the sum cannot have one term.)

– What should they be equal to so that the commutative property of multiplication is not violated? (7 1 must also equal 7, so 7 1 = 7.)

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

– Conclude: a 1 = ? (a 1 = a.)

The 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 line? (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 assume: a 1 = 1 a = a. Slide 8.

2) The case of multiplication with 0 is studied similarly. Conclusion:

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

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

1 – “mirror”, 0 – “terrible beast” or “invisible hat”.

Well done! So, multiplying by 1 gives the same number (1 – “mirror”), and when multiplied by 0 it turns out 0 ( 0 – “invisibility cap”).

4. Physical education (for the eyes – “circle”, “up and down”, for the hands – “lock”, “fists”).

5. Primary consolidation.

Examples written on the board:

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

Children solve them in a notebook and on the board, pronouncing the resulting rules out loud, for example:

3 1 = 3, since when a number is multiplied by 1, the same number is obtained (1 is a “mirror”), etc.

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

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

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

– When multiplying 8 by an unknown number, the result was 0. So, multiplied by 0 x = 0. Etc.

6. Independent work with testing in class. Slide 10.

Children independently solve written examples. Then according to the finished

Following the example, they check their answers by pronouncing them out loud, mark correctly solved examples with a plus, and correct any mistakes made. Those who made mistakes receive a similar task on a card and work on it individually while the class solves repetition 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 deciphering the recording:

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

-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 did I think of?”

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

8. Lesson summary.Slide 12.

What new rules have you met?
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.
Fill out the self-assessment table:

I want to know more
Okay, but I can do better
I'm still experiencing difficulties

Thanks for your work, you did a great job!

9. Homework

pp. 72–73 Rule, No. 6.