Zero is a very interesting figure in itself. By itself, it means emptiness, lack of meaning, and next to another number increases its significance 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.

The number 0 can be thought of as a kind of border that separates the world of real numbers from imaginary or negative ones. Due to the ambiguous position, many operations with this numerical value do not obey mathematical logic. The impossibility of dividing by zero is a prime example of this. And permitted arithmetic operations with zero can be performed using generally accepted definitions.

The story of zero

Zero is the point of reference in all standard numeric systems. Europeans began to use this number relatively recently, but the sages of ancient India used zero a thousand years before the empty number was regularly used by European mathematicians. Even before the Indians, zero was a mandatory value in the Mayan number system. This American people used the duodecimal system of number, and they began with a zero on the first day of each month. Interestingly, the Maya sign for "zero" exactly coincided with the sign for "infinity." Thus, the ancient Maya concluded that these values ​​were identical and unknowable.

Math operations with zero

Standard math operations with zero can be boiled down to a few rules.

Addition: if you add zero to an arbitrary number, then it will not change its value (0 + x = x).

Subtraction: When subtracting zero from any number, the value of the subtracted remains unchanged (x-0 = x).

Multiplication: Any number multiplied by 0 gives 0 in the product (a * 0 = 0).

Division: zero can be divided by any number other than zero. In this case, the value of such a fraction will be 0. And division by zero is prohibited.

Exponentiation. This action can be performed with any number. An arbitrary number raised to the zero power will give 1 (x 0 = 1).

Zero to any power is 0 (0 a = 0).

In this case, a contradiction immediately arises: the expression 0 0 has no meaning.

Paradoxes of mathematics

Many people know that division by zero is impossible from school. But for some reason it is not possible to explain the reason for such a ban. Indeed, why does the formula for division by zero not exist, but other actions with this number are quite reasonable and possible? The answer to this question is given by mathematicians.

The thing is that the usual arithmetic operations that schoolchildren learn in primary grades, in fact, are not nearly as equal as we think. All simple operations with numbers can be reduced to two: addition and multiplication. These actions are the essence of the very concept of number, and the rest of the operations are based on the use of these two.

Addition and multiplication

Let's take a standard example of subtraction: 10-2 = 8. At school, it is considered simply: if two are taken away from ten subjects, eight will remain. But mathematicians look at this operation in a completely different way. After all, such an operation as subtraction does not exist for them. This example can be written in another way: x + 2 = 10. For mathematicians, the unknown difference is simply a number that needs to be added to two to make eight. And no subtraction is required here, you just need to find a suitable numeric value.

Multiplication and division are treated the same way. In example 12: 4 = 3, you can understand that we are talking about dividing eight objects into two equal piles. But in reality this is just an inverted formula for writing 3x4 = 12. There are endless examples of division.

Division by 0 examples

This is where it becomes a little clear why it is impossible to divide by zero. Multiplication and division by zero obeys their own rules. All examples of the division of this quantity can be formulated as 6: 0 = x. But this is the inverted notation of the expression 6 * x = 0. But, as you know, any number multiplied by 0 gives in the product only 0. This property is inherent in the very concept of a zero value.

It turns out that such a number that, when multiplied by 0, gives some tangible value, does not exist, that is, this problem has no solution. You should not be afraid of such an answer, it is a natural answer for problems of this type. It's just that the 6-0 record doesn't make any sense, and it can't explain anything. In short, this expression can be explained by the immortal "division by zero is impossible."

Is there a 0: 0 operation? Indeed, if the operation of multiplying by 0 is legal, can zero be divided by zero? After all, the equation of the form 0x 5 = 0 is completely legal. Instead of the number 5, you can put 0, the product will not change from this.

Indeed, 0x0 = 0. But you still can't divide by 0. As said, division is simply the inverse of multiplication. Thus, if in the example 0x5 = 0, you need to determine the second factor, we get 0x0 = 5. Or 10. Or infinity. Dividing infinity by zero - how do you like it?

But if any number fits into the expression, then it does not make sense, we cannot choose one from the infinite set of numbers. And if so, it means the expression 0: 0 does not make sense. It turns out that even zero itself cannot be divided by zero.

Higher mathematics

Division by zero is headache for school mathematics. Studied in technical universities mathematical analysis slightly expands the concept of problems that have no solution. For example, to the already known expression 0: 0, new ones are added that have no solution in school courses mathematics:

• infinity divided by infinity: ∞: ∞;
• infinity minus infinity: ∞ − ∞;
• one raised to an infinite power: 1 ∞;
• infinity times 0: ∞ * 0;
• some others.

It is impossible to solve such expressions by elementary methods. But higher mathematics, thanks to the additional possibilities for a number of similar examples, gives final solutions. This is especially evident in the consideration of problems from the theory of limits.

Disclosure of uncertainty

In the theory of limits, the value 0 is replaced by the conditional infinitesimal variable... And expressions in which, when the desired value is substituted, division by zero is obtained, are converted. Below is a standard example of limit expansion using ordinary algebraic transformations:

As you can see in the example, a simple reduction of the fraction leads its value to a completely rational answer.

When considering the limits trigonometric functions their expressions tend to be reduced to the first remarkable limit. When considering the limits in which the denominator goes to 0 when the limit is substituted, a second remarkable limit is used.

Lopital's method

In some cases, the limits of expressions can be replaced by the limit of their derivatives. Guillaume L'Hôpital - French mathematician, founder of the French school mathematical analysis... He proved that the limits of expressions are equal to the limits of the derivatives of these expressions. In mathematical notation, his rule is as follows.

Evgeny Shiryaev, Lecturer and Head of the Mathematics Laboratory of the Polytechnic Museum, told AiF.ru about division by zero:

1. Jurisdiction of the issue

Agree, the ban gives a special provocation to the rule. How is it impossible? Who banned it? What about our civil rights?

Neither the Constitution of the Russian Federation, nor the Criminal Code, nor even the charter of your school object to the intellectual action of interest to us. This means that the ban does not have legal force, and nothing prevents right here, on the pages of AiF.ru, to try to divide something by zero. For example, a thousand.

2. Divide as taught

Remember, when you just learned how to divide, the first examples were solved by checking by multiplication: the result multiplied by the divisor should have been the same doable. Didn't match - didn't decide.

Example 1. 1000: 0 =...

Let's forget about the forbidden rule for a minute and make a few attempts to guess the answer.

The check will cut off the wrong ones. Go through the options: 100, 1, −23, 17, 0, 10,000. For each of them, the check will give the same result:

100 0 = 1 0 = - 23 0 = 17 0 = 0 0 = 10 000 0 = 0

Zero by multiplication turns everything into itself and never into a thousand. The conclusion is not difficult to formulate: no number will pass the test. That is, no number can be the result of dividing a nonzero number by zero. Such division is not prohibited, but simply has no result.

3. Nuance

We almost missed one opportunity to refute the ban. Yes, we admit that a nonzero number cannot be divisible by 0. But maybe 0 itself can?

Example 2. 0: 0 = ...

Your suggestions for a private? 100? Please: quotient 100 multiplied by a divisor of 0 is equal to a dividend of 0.

More options! 1? Also fits. And −23, and 17, and all-all-all. In this example, the test will be positive for any number. And to be honest, the solution in this example should be called not a number, but a set of numbers. Everyone. And it won't take long to come to an agreement to the point that Alice is not Alice, but Mary Ann, and both of them are a rabbit's dream.

4. What about higher mathematics?

The problem was resolved, the nuances were taken into account, the dots were placed, everything became clear - the answer for the example with division by zero can not be a single number. To solve such problems is a hopeless and impossible task. Which means ... interesting! Take two.

Example 3. Figure out how to divide 1000 by 0.

But in no way. But 1000 can be easily divided by other numbers. Well, let's at least do what we get, even if we change the task. And there, you see, we will get carried away, and the answer will appear by itself. Forget about zero for a minute and divide by one hundred:

A hundred is far from zero. Let's take a step towards it by decreasing the divisor:

1000: 25 = 40,
1000: 20 = 50,
1000: 10 = 100,
1000: 8 = 125,
1000: 5 = 200,
1000: 4 = 250,
1000: 2 = 500,
1000: 1 = 1000.

Obvious dynamics: the closer the divisor to zero, the larger the quotient. The trend can be observed further, moving on to fractions and continuing to decrease the numerator:

It remains to note that we can approach zero as close as we like, making the quotient arbitrarily large.

In this process, there is no zero and no last quotient. We designated the movement towards them, replacing the number with a sequence converging to the number of interest to us:

This implies a similar replacement for the dividend:

1000 ↔ { 1000, 1000, 1000,... }

The arrows are double-sided for a reason: some sequences can converge to numbers. Then we can assign the sequence to its numerical limit.

Let's look at the sequence of quotients:

It grows indefinitely, not striving for any number and surpassing any. Mathematicians add the symbol to numbers ∞ to be able to put a double-headed arrow next to such a sequence:

Comparison of the numbers of sequences with a limit allows us to offer a solution to the third example:

When a sequence converging to 1000 is divided elementwise by a sequence of positive numbers converging to 0, we obtain a sequence converging to ∞.

5. And here is a nuance with two zeros

What will be the result of dividing two sequences of positive numbers that converge to zero? If they are the same, then the identical unit. If the dividend sequence converges to zero faster, then in the particular the sequence has a zero limit. And when the elements of the divisor decrease much faster than that of the dividend, the sequence of quotients will grow strongly:

Uncertain situation. And so it is called: the uncertainty of the species 0/0 ... When mathematicians see sequences suitable for such uncertainty, they do not rush to divide two identical numbers by each other, but figure out which of the sequences runs faster to zero and how exactly. And each example will have its own specific answer!

6. In life

Ohm's Law relates current strength, voltage and resistance in a circuit. It is often written in this form:

Let us neglect the accurate physical understanding and formally look at the right side as a quotient of two numbers. Imagine solving a school electricity problem. The condition gives voltage in volts and resistance in ohms. The question is obvious, the solution is in one step.

Now let's look at the definition of superconductivity: this is the property of some metals to have zero electrical resistance.

Well, let's solve the problem for the superconducting circuit? Just substitute R = 0 will not work, physics throws up an interesting problem, behind which, obviously, there is a scientific discovery. And people who managed to divide by zero in this situation received Nobel Prize... It is useful to be able to bypass any prohibitions!

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

Let's reason like this. In the first sum, the terms are the same, the number five is repeated four times. So, you can replace addition by 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 you cannot replace it with the product. Add the terms and get the answer 17.

Let's write down the solution.

Can the product be replaced by the sum of the same terms?

Consider the works.

We will perform 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: always the number of units-terms is 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

Consider the works.

These products cannot be replaced by a sum, since the sum cannot contain 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 travel property of multiplication, their values ​​must also be equal, respectively, to the first factor.

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

Let's write this conclusion in the form of equality.

a * 1 = a

Solve examples.

Hint: do not forget the conclusions we made in the lesson.

Check yourself.

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

Consider products where the first factor is zero.

We replace the products with the sum of the same terms. We will perform 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

Always the number of zeros-terms is equal to the number by which zero is multiplied.

Means, multiplying zero by a number results in zero.

Let's write this conclusion in the form of equality.

0 * a = 0

Consider products where the second factor is zero.

These products cannot be replaced by a sum, since the 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 factors.

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

Let's conclude: multiplying any number by zero results in zero.

Let's write this conclusion in the form of equality.

a * 0 = 0

But you cannot divide by zero.

Solve examples.

Hint: Don't forget the lessons learned from the lesson. When calculating the values ​​of the second column, be careful when determining the order of actions.

Check yourself.

Today in the lesson we met with special cases multiplication by 0 and 1, practiced multiplying by 0 and 1.

Bibliography

1. M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Education", 2012.
2. M.I. Moreau, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Education", 2012.
3. M.I. Moreau. Mathematics lessons: Guidelines for the teacher. Grade 3. - M .: Education, 2012.
4. Normative legal document. Monitoring and evaluation of learning outcomes. - M .: "Education", 2011.
5. "School of Russia": Programs for primary school... - M .: "Education", 2011.
6. S.I. Volkova. Maths: Verification work... Grade 3. - 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 values ​​of the expressions.

2. Find the values ​​of the expressions.

3. Compare the values ​​of the expressions.

(56-54)*1 … (78-70)*1

4. Make an assignment on the topic of the lesson for your peers.

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 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, 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 ...