What happens if you divide by. Division by zero. Fascinating mathematics. Non-standard methods of forbidden division

Evgeniy SHIRYAEV, teacher and head of the Mathematics Laboratory of the Polytechnic Museum, told AiF about division by zero:

1. Jurisdiction of the issue

Agree, what makes the rule especially provocative is the ban. How can this not be done? Who banned? What about our civil rights?

Neither the Constitution, nor the Criminal Code, nor even the charter of your school objects to the intellectual action that interests us. This means that the ban has no legal force, and nothing prevents you from trying to divide something by zero right here, on the pages of AiF. For example, a thousand.

2. Let's divide as taught

Remember, when you first learned how to divide, the first examples were solved with a multiplication check: the result multiplied by the divisor had to coincide with the dividend. It didn’t match - they didn’t decide.

Example 1. 1000: 0 =...

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

Incorrect ones will be cut off by the check. Try the following 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

By multiplying zero, everything turns into itself and never into a thousand. The conclusion is easy to formulate: no number will pass the test. That is, no number can be the result of dividing a non-zero 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 non-zero number cannot be divided by 0. But maybe 0 itself can?

Example 2. 0: 0 = ...

What are your suggestions for private? 100? Please: the quotient of 100 multiplied by the divisor 0 is equal to the dividend 0.

More options! 1? Fits too. And −23, and 17, and that’s it. 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 doesn’t take long to agree that Alice is not Alice, but Mary Ann, and both of them are a rabbit’s dream.

4. What about higher mathematics?

The problem has been resolved, the nuances have been taken into account, the dots have been placed, everything has become clear - the answer to the example with division by zero cannot be a single number. Solving such problems is hopeless and impossible. Which means... interesting! Take two.

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

But no way. But 1000 can be easily divided by other numbers. Well, let's at least do what works, even if we change the task. And then, you see, we get carried away, and the answer will appear by itself. Let’s 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.

The dynamics are obvious: the closer the divisor is to zero, the larger the quotient. The trend can be observed further by moving to fractions and continuing to reduce the numerator:

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

In this process there is no zero and there is no last quotient. We indicated the movement towards them by replacing the number with a sequence converging to the number we are interested in:

This implies a similar replacement for the dividend:

1000 ↔ { 1000, 1000, 1000,... }

It’s not for nothing that the arrows are double-sided: some sequences can converge to numbers. Then we can associate the sequence with its numerical limit.

Let's look at the sequence of quotients:

It grows unlimitedly, not striving for any number and surpassing any. Mathematicians add symbols to numbers ∞ to be able to put a double-sided arrow next to such a sequence:

Comparison with the numbers of sequences that have a limit allows us to propose a solution to the third example:

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

5. And here is the nuance with two zeros

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

Uncertain situation. And that’s what it’s called: uncertainty of type 0/0 . When mathematicians see sequences that fit 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, voltage and resistance in a circuit. It is often written in this form:

Let's allow ourselves to ignore the neat physical understanding and formally look at the right-hand side as the quotient of two numbers. Let's imagine that we are solving a school problem on electricity. The condition gives the voltage in volts and resistance in ohms. The question is obvious, the solution is in one action.

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 a superconducting circuit? Just set it up R= 0 If it doesn’t work out, physics throws up an interesting problem, behind which, obviously, there is a scientific discovery. And the people who managed to divide by zero in this situation received the Nobel Prize. It’s useful to be able to bypass any prohibitions!

If you break generally accepted rules in the world of science, you can get the most unexpected results.

Ever since school, teachers told us that in mathematics there is one rule that cannot be broken. It sounds like this: "You can't divide by zero!"

Why does such a familiar number 0, which we so often encounter in everyday life, cause so many difficulties when carrying out a simple arithmetic operation such as division?

Let's look into this issue.

If we divide one number by ever smaller numbers, the result will be increasingly larger values. For example

Thus, it turns out that if we divide by a number tending to zero, we will get the greatest result tending to infinity.

Does this mean that if we divide our number by zero, we will get infinity?

This sounds logical, but all we know is that if we divide by a number close in value to zero, then the result will only tend to infinity and this does not mean that when divided by zero we will end up with infinity . Why is this so?

First, we need to understand what the arithmetic operation of division is. So, if we divide 20 by 10, this will mean how many times we will need to add the number 10 to get 20 as a result, or what number we need to take twice to get 20.

In general, division is the inverse arithmetic operation of multiplication. For example, when multiplying any number by X, we can ask the question: “Is there a number that we need to multiply by the result to find out the original value of X?” And if there is such a number, then it will be the inverse value for X. For example, if we multiply 2 by 5, we get 10. If after this we multiply 10 by one fifth, we again get 2:

Thus, 1/5 is the reciprocal of 5, the reciprocal of 10 is 1/10.

As you have already noticed, when multiplying a number by its reciprocal, the answer will always be one. And if you want to divide a number by zero, you will need to find its inverse number, which should be equal to one divided by zero.

This will mean that when multiplied by zero the result must be one, and since it is known that if you multiply any number by 0 you get 0, then this is impossible and zero has no reciprocal number.

Is it possible to come up with something to get around this contradiction?

Previously, mathematicians had already found ways to bypass mathematical rules, because in the past, according to mathematical rules, it was impossible to obtain the value of the square root of a negative number, then it was proposed to denote such square roots by imaginary numbers. As a result, a new branch of mathematics about complex numbers appeared.

So why don't we also try to introduce a new rule, according to which one divided by zero would be denoted by an infinity sign and see what happens?

Let us assume that we know nothing about infinity. In this case, if we start from the reciprocal number zero, then multiplying zero by infinity, we should get one. And if we add to this one more value of zero divided by infinity, the result should be the number two:

In accordance with the distributive law of mathematics, the left side of the equation can be represented as:

and since 0+0=0, then our equation will take the form 0*∞=2, due to the fact that we have already defined 0*∞=1, it turns out that 1=2.

This sounds ridiculous. However, this answer also cannot be considered completely incorrect, since such calculations simply do not work for ordinary numbers. For example, in the Riemann sphere, division by zero is used, but in a completely different way, and this is a completely different story...

In short, dividing by zero in the usual way does not end well, but nevertheless this should not become an obstacle for us to experiment in the field of mathematics, in case we manage to open up new areas for research.

Everyone remembers from school that you cannot divide by zero. Primary schoolchildren are never explained why this should not be done. They simply offer to take this as a given, along with other prohibitions like “you can’t put your fingers in sockets” or “you shouldn’t ask stupid questions to adults.” AiF.ru decided to find out whether the school teachers were right.

Algebraic explanation of the impossibility of division by zero

From an algebraic point of view, you can't divide by zero because it doesn't make any sense. Let's take two arbitrary numbers, a and b, and multiply them by zero. a × 0 is equal to zero and b × 0 is equal to zero. It turns out that a × 0 and b × 0 are equal, because the product in both cases is equal to zero. Thus, we can create the equation: 0 × a = 0 × b. Now let's assume that we can divide by zero: we divide both sides of the equation by it and get that a = b. It turns out that if we allow the operation of division by zero, then all the numbers coincide. But 5 is not equal to 6, and 10 is not equal to ½. Uncertainty arises, which teachers prefer not to tell inquisitive junior high school students.

Explanation of the impossibility of dividing by zero from the point of view of mathematical analysis

In high school they study the theory of limits, which also talks about the impossibility of dividing by zero. This number is interpreted there as an “undefined infinitesimal quantity.” So if we consider the equation 0 × X = 0 within the framework of this theory, we will find that X cannot be found because to do this we would have to divide zero by zero. And this also does not make any sense, since both the dividend and the divisor in this case are indefinite quantities, therefore, it is impossible to draw a conclusion about their equality or inequality.

When can you divide by zero?

Unlike schoolchildren, students of technical universities can divide by zero. An operation that is impossible in algebra can be performed in other areas of mathematical knowledge. New additional conditions of the problem appear in them that allow this action. Dividing by zero will be possible for those who listen to a course of lectures on non-standard analysis, study the Dirac delta function and become familiar with the extended complex plane.

A strict ban on dividing by zero is imposed even in the lower grades of school. Children usually don’t think about its reasons, but in fact, knowing why something is prohibited is both interesting and useful.

Arithmetic operations

The arithmetic operations that are studied at school are not equivalent from the point of view of mathematicians. They recognize only two of these operations as valid - addition and multiplication. They are included in the very concept of number, and all other actions with numbers are in one way or another built on these two. That is, not only division by zero is impossible, but division in general is impossible.

Subtraction and division

What is missing from the rest of the actions? Again, we know from school that, for example, subtracting four from seven means taking seven sweets, eating four of them and counting the ones that remain. But mathematicians, when they eat sweets and in general, perceive them completely differently. For them, there is only addition, that is, the notation 7 - 4 means a number that, when added to the number 4, will be equal to 7. That is, for mathematicians, 7 - 4 is a short notation of the equation: x + 4 = 7. This is not a subtraction, but a problem - find the number that needs to be put in place of x.

The same applies to division and multiplication. Dividing ten by two, a junior student puts ten candies into two identical piles. The mathematician sees the equation here too: 2 x = 10.

This explains why division by zero is prohibited: it is simply impossible. The entry 6: 0 should turn into the equation 0 · x = 6. That is, you need to find a number that can be multiplied by zero and get 6. But it is known that multiplication by zero always gives zero. This is the essential property of zero.

Thus, there is no number that, when multiplied by zero, would give some number other than zero. This means that this equation has no solution, there is no number that would correlate with the notation 6: 0, that is, it does not make sense. They talk about its meaninglessness when division by zero is prohibited.

Is zero divisible by zero?

Is it possible to divide zero by zero? The equation 0 · x = 0 does not cause any difficulties, and you can take this very zero for x and get 0 · 0 = 0. Then 0: 0 = 0? But, if, for example, we take x to be one, we also get 0 1 = 0. You can take x to be any number at all and divide by zero, and the result will remain the same: 0: 0 = 9, 0: 0 = 51, and so on Further.

Thus, absolutely any number can be inserted into this equation, and it is impossible to choose any specific one, it is impossible to determine which number is denoted by the notation 0: 0. That is, this notation also makes no sense, and division by zero is still impossible: it not even divisible by itself.

This is an important feature of the division operation, that is, multiplication and the associated number zero.

The question remains: is it possible to subtract it? One could say that real mathematics begins with this interesting question. To find the answer to it, you need to learn the formal mathematical definitions of number sets and become familiar with the operations on them. For example, there are not only simple ones, but also the division of which differs from the division of ordinary ones. This is not included in the school curriculum, but university lectures in mathematics begin with this.

Even at school, teachers tried to hammer into our heads the simplest rule: “Any number multiplied by zero equals zero!”, - but still a lot of controversy constantly arises around him. Some people just remember the rule and don’t bother themselves with the question “why?” “You can’t and that’s it, because they said so at school, the rule is the rule!” Someone can fill half a notebook with formulas, proving this rule or, conversely, its illogicality.

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Who's right in the end?

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

Most often, those who consider this rule to be incorrect try to appeal to logic in this way:

I have two apples on my table, if I put zero apples on them, that is, I don’t put a single one, then my two apples will not disappear! 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 let’s discard this conclusion right away - it is illogical, although it has the opposite goal - to call to logic.

What is multiplication

Originally the multiplication rule was defined only for natural numbers: multiplication is a number added to itself a certain number of times, which implies that the number is natural. 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

From this equation it follows that that multiplication is a simplified addition.

What is zero

Any person knows from childhood: zero is emptiness. Despite the fact that this emptiness has a designation, it does not carry anything at all. Ancient Eastern scientists thought differently - they approached the issue philosophically and drew some parallels between emptiness and infinity and saw a 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 about 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 decimal point.

Is it possible to multiply by emptiness?

You can multiply by zero, but it is useless, because, whatever one may say, even when multiplying negative numbers, you will still get zero. It’s 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 ancient scientists believed. Below we will give the most logical explanation that this multiplication is useless, because when you multiply a number by it, you will still get the same thing - zero.

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

After all, eating an apple 0 times means not eating a single one. This will be clear to even the smallest child. Whatever one may say, the result will be 0, two or three can be replaced with absolutely any number and the result will be absolutely the same. And to put it simply, then zero is nothing, and when do you have there is nothing, then no matter how much you multiply, it’s still the same will be zero. There is no such thing as magic, and nothing will make 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 who is far from all formulas and mathematics, such an explanation will be enough for the dissonance in the head to resolve and everything to fall into place.

Division

From all of the above, another important rule follows:

You can't divide by zero!

This rule has also been persistently drilled into our heads since childhood. We just know that it’s impossible to do everything without filling our heads with unnecessary information. If you are unexpectedly asked the question why it is forbidden to divide by zero, then most will be confused and will not be able to clearly answer the simplest question from the school curriculum, because there are not so many disputes and contradictions surrounding this rule.

Everyone simply memorized the rule and did not divide by zero, not suspecting that the answer was hidden on the surface. Addition, multiplication, division and subtraction are unequal; of the above, only multiplication and addition are valid, and all other manipulations with numbers are built from them. That is, the notation 10: 2 is an abbreviation of the equation 2 * x = 10. This means that the notation 10: 0 is the same abbreviation for 0 * x = 10. It turns out that division by zero is a task to find a number, multiplying 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 a priori incorrect.

Let me tell you,

So as not to divide by 0!

Cut 1 as you want, lengthwise,

Just don't divide by 0!