Why can't you divide by zero? An illustrative example. Why can't you divide by zero?

Mathematicians have a specific sense of humor and some issues related to calculations have not been taken seriously for a long time. It is not always clear whether they are trying to explain to you in all seriousness why it is impossible to divide by zero, or is this another joke. But the question itself is not so obvious, if in elementary mathematics it is possible to reach its solution purely logically, then in higher mathematics there may well be other initial conditions.

When did zero appear?

The number zero is fraught with many mysteries:

• IN Ancient Rome this number was not known, the reference system began with I.
• For the right to be called the progenitors of zero for a long time Arabs and Indians argued.
• Studies of the Maya culture have shown that this ancient civilization could well be the first in terms of the use of zero.
• Zero has no numerical value, even minimal.
• It literally means nothing, the absence of things to count.

IN primitive order there was no special need for such a figure, the absence of something could be explained with the help of words. But with the rise of civilizations, human needs have also increased, in terms of architecture and engineering.

To carry out more complex calculations and derive new functions, it took a number that would indicate the complete absence of something.

Is it possible to divide by zero?

On this account, there are two diametrically opposed opinions:

At school, even in elementary grades, they teach that division by zero is impossible in any case. This is explained very simply:

1. Imagine that you have 20 tangerine slices.
2. By dividing them by 5, you will distribute 4 slices to five friends.
3. Dividing by zero will not work, because the process of division between someone will not.

Of course, this is a figurative explanation, largely simplified and not entirely consistent with reality. But it explains in the most accessible way the meaninglessness of dividing something by zero.

After all, in fact, in this way it is possible to denote the fact of the absence of division. And why complicate mathematical calculations and write down also the absence of division?

Can zero be divided by a number?

From the point of view of applied mathematics, any division in which zero takes part does not make much sense. But school textbooks are unequivocal in their opinion:

• Zero can be divided.
• Any number should be used for division.
• You can't divide zero by zero.

The third point may cause slight bewilderment, because just a few paragraphs above it was indicated that such a division is quite possible. In fact, it all depends on the discipline in which you conduct calculations.

In this case, it is really better for schoolchildren to write that expression cannot be determined and, therefore, it does not make sense. But in some branches of algebraic science it is allowed to write such an expression, with the division of zero by zero. Especially when we are talking about computers and programming languages.

The need to divide zero by a number may arise during the solution of any equalities and the search for initial values. But in that case, the answer will always be zero. Here, as with multiplication, no matter what number you divide zero by, you will not end up with more than zero. Therefore, if this cherished number is noticed in a huge formula, try to quickly “estimate” whether all the calculations will be reduced to a very simple solution.

If infinity is divided by zero

It was necessary to mention infinitely large and infinitely small values ​​a little earlier, because this also opens up some loopholes for division, including using zero. That's true, and there is a small snag, because infinitesimal value and the complete absence of value are different concepts.

But this small difference in our conditions can be neglected, in the end, the calculations are carried out using abstract quantities:

• The numerator must have an infinity sign.
• The denominators are a symbolic image of a value tending to zero.
• The answer will be infinity, representing an infinitely large function.

It should be noted that we are still talking about the symbolic display of an infinitesimal function, and not about using zero. Nothing has changed with this sign, it still cannot be divided into it, only as very, very rare exceptions.

For the most part, zero is used to solve problems that are in purely theoretical plane. Perhaps, after decades or even centuries, all modern computing will have practical use, and they will provide some kind of grandiose breakthrough in science.

In the meantime, most mathematical geniuses only dream of world recognition. An exception to these rules is our compatriot, Perelman. But he is known thanks to the solution of a truly epoch-making problem with the proof of the Poinquere conjecture and extravagant behavior.

Paradoxes and the meaninglessness of division by zero

Division by zero, for the most part, makes no sense:

• division is represented as function inverse to multiplication.
• We can multiply any number by zero and get zero in the answer.
• By the same logic, one could divide any number by zero.
• Under such conditions, it would not be difficult to conclude that any number multiplied or divided by zero is equal to any other number on which this operation was carried out.
• We discard the mathematical action and get an interesting conclusion - any number is equal to any number.

In addition to creating such incidents, division by zero has no practical value, from the word in general. Even if you can perform this action, you will not get any new information.

From the point of view of elementary mathematics, during division by zero, the whole object is divided zero times, that is, not even once. Simply put - no division process, therefore, the result of this event cannot be.

Being in the same society with a mathematician, you can always ask a couple of banal questions, for example, why you can’t divide by zero and get an interesting and understandable answer. Or irritability, because this is probably not the first time a person has been asked this. And not even ten. So take care of your mathematician friends, do not make them repeat one explanation hundreds of times.

Video: divide by zero

In this video, mathematician Anna Lomakova will tell you what happens if you divide a number by zero and why this cannot be done, from the point of view of mathematics:

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

1. Jurisdiction of the issue

Agree, the ban gives a special provocativeness to the rule. How is it impossible? Who banned? But 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 that interests us. This means that the ban has no legal force, and nothing prevents right here, on the pages of AiF.ru, from trying to divide something by zero. For example, a thousand.

2. Divide as taught

Remember, when you first learned how to divide, the first examples were solved by checking by multiplication: the result multiplied by the divisor had to match the divisible. Did not match - did not decide.

Example 1 1000: 0 =...

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

Incorrect will cut off the check. Iterate over the options: 100, 1, −23, 17, 0, 10,000. For each of them, the test 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 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 a division is not forbidden, but simply has no result.

3. Nuance

Almost missed one opportunity to refute the ban. Yes, we recognize that a non-zero number will not be divisible by 0. But maybe 0 itself can?

Example 2 0: 0 = ...

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

More options! 1? Also suitable. And -23, and 17, and all-all-all. In this example, the result check will be positive for any number. And to be honest, the solution in this example should not be called a number, but a set of numbers. Everyone. And it won’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 is solved, the nuances are taken into account, the dots are placed, everything is clear - no number can be the answer for the example with division by zero. Solving such problems is hopeless and impossible. So... interesting! Double 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 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 is to zero, the greater the quotient. The trend can be observed further, moving to fractions and continuing to reduce the numerator:

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

There is no zero in this process and no last quotient. We indicated the movement towards them by 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 associate a sequence with its numerical limit.

Let's look at the sequence of quotients:

It grows indefinitely, striving for no number and surpassing any. Mathematicians add symbols to numbers ∞ to be able to put a double-sided arrow next to such a sequence:

Comparing the numbers of sequences with a limit allows us to propose a solution to the third example:

Dividing a sequence converging to 1000 element-wise by a sequence of positive numbers converging to 0, we get a sequence converging to ∞.

5. And here is the 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 a sequence-dividend converges to zero faster, then in a particular sequence with a zero limit. And when the elements of the divisor decrease much faster than the dividend, the quotient sequence will grow strongly:

Uncertain situation. And so it is called: the uncertainty of the form 0/0 . When mathematicians see sequences that fall under such uncertainty, they do not rush to divide two identical numbers by each other, but figure out which of the sequences runs to zero faster and how. 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 us neglect accurate physical understanding and formally look at the right side as a quotient of two numbers. Imagine that we are solving a school problem on electricity. The condition is given voltage in volts and resistance in ohms. The question is obvious, the decision in one action.

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

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

So, the children were puzzled, I had to delve into tyrnets, find a bunch of obviously delusional explanations and blind my own, also apparently imperfect, successfully tested on the younger ten-year-old. Might be useful to someone:
"Everyone knows from school that you cannot divide by zero. And why? The teacher won't let you?

Maybe you need to act on a joke:

Why are you drinking cognac? The doctor told you not to.

I gave him money and he let me.

It is surprising why the school does not immediately explain that division by zero is a mathematical operation from the field of higher mathematics, but impossible in elementary mathematics because of the uncertainty that arises in this case. By the way, atmultiplication by zero is also from higher mathematics, that is, again from the series "children, this cannot be understood, you just need to remember it."

In fact, all this is not so difficult to understand. In elementary mathematics, quite definite results are obtained, for example, 2x3=6, and if we divide the result by one of the factors, then we will clearly get the second factor: 6:3=2 or 6:2=3.

But actions with zero are not so unambiguous. Any number "y" is multiplied by zero: Yх0=0. Now we divide the result by one of the factors 0:Y=0 or 0:0=Y getting any number, that is, an indefinite result.

Why is this happening? You can get closer to understanding this without even getting into the jungle of higher mathematics with set theories, operations with infinity, complex numbers, and so on.

Surprisingly, just like with the wrong "multiplication table", for some reason the elementary things are not explained in school: there are quantitative (cardinal) and ordinal (ordinal) numbers. For example, concepts" 10 apartments" - quantitative and "apartment number 10" - ordinal, quite obviouslydiffer sharply. Quantitative "10 apartments" can be divided, added and perform other actions according to the rules of elementary mathematics, which will give a completely definite quantitative result.

But the ordinal number 10 (apartment No. 10) with the same actions will not give any quantitative result, the apartment will still be one, only another. Math Actions with ordinal numbers are needed, for example, when you need to immediately calculate on which floor the apartment you need is located and not ride the elevator "at random". We look at the last apartment number in the previous entrance, subtract from the apartment number we need and divide the result by the number of apartments on the floor. Profit!

Figuratively speaking, if you do not understand the difference between quantitative and ordinal numbers, then when adding 10 apartments and apartment number 10, you can get 20 apartments and apartment number 20.

So zero is very specialordinal (ordinal) number, which by definition cannot be quantitative.Zero is the main reference point, a border that has no size.Moreover, it is a point, not a segment.

The geometric representation of any natural and imaginary (negative) numbers is segments, that is, parts of a straight line, limited by points that do not have a size. If they, like segments, can be divided into arbitrarily small segments, then it is no longer possible to divide a point in elementary mathematics by its definition as having no size.

Hence, by the way, the nuances over time. Shouldto distinguish between the designation of the moment, the point on the time scale and the time interval - the segment on this scale between zero and the designated point of the moment of time. For example, when we talk about age, we mean at the same timehow many years he lived, and which year goes on, in what year of life. But to ask current time necessary " what time is it now " (ordinal), and not "how long" (quantitative), because "how long" refers to the duration of some processes - cooking, movement, etc.

"You can't divide by zero!" - most students memorize this rule by heart, without asking questions. All children know what “No” is and what will happen if you ask in response to it: “why? But it’s actually very interesting and important to know why it’s impossible.

The thing is that the four operations of arithmetic - addition, subtraction, multiplication and division - are actually unequal. Mathematicians recognize only two of them as full-fledged - addition and multiplication. These operations and their properties are included in the very definition of the concept of number. All other actions are built in one way or another from these two.

We will consider, for example, subtraction. What does 5 - 3 mean? The student will answer this simply: you need to take five items, take away (remove) three of them and see how many remain. But mathematicians look at this problem in a completely different way. There is no subtraction, only addition. Therefore, writing 5 - 3 means a number that, when added to the number 3, will give the number 5. That is, 5 - 3 is just an abbreviated notation of the equation: x 3 \u003d 5. There is no subtraction in this equation. There is only a task - to find a suitable number.

The same is true with multiplication and division. Record 8: 4 can be understood as the result of the division of eight objects into four equal piles. But in reality, this is just a shortened form of the equation 4 * x = 8.

This is where it becomes clear why it is impossible (or rather impossible) to divide by zero. Record 5: 0 is short for 0 * x = 5. That is, this task is to find a number that, when multiplied by 0, will give 5. but we know that when multiplied by 0, it always turns out to be 0. this is an inherent property of zero, strictly speaking , part of its definition.

There is simply no such number that, when multiplied by 0, will give something other than zero. That is, our problem has no solution. (Yes, it happens, not every problem has a solution.) So, writing 5: 0 does not correspond to any specific number, and it simply does not stand for anything, and therefore does not make sense. The meaninglessness of this entry is briefly expressed by saying that you cannot divide by zero.

The most attentive readers at this point will certainly ask: is it possible to divide zero by zero? Indeed, the equation 0 * x = 0 is successfully solved. For example, you can take x = 0, and then we get 0 * 0 = 0. So, 0: 0=0? But let's not rush. Let's try to take x = 1. we get 0 * 1 = 0. right? So 0: 0 = 1? But you can take any number this way and get 0: 0 = 5, 0: 0 = 317, etc.

But if any number is suitable, then we have no reason to opt for any one of them. That is, we cannot say to which number the entry 0: 0 corresponds. And if so, then we are forced to admit that this entry also does not make sense. It turns out that even zero cannot be divided by zero. (In mathematical analysis, there are cases when, due to additional conditions of the problem, one can give preference to one of the options solution of the equation 0 * x = 0; in such cases mathematicians speak of "Uncertainty Revealing", but in arithmetic such cases do not occur. Here is a feature of the division operation. To be more precise, the multiplication operation and the number associated with it have zero.

Well, the most meticulous, having read up to this point, may ask: why is it so that you cannot divide by zero, but you can subtract zero? In a sense, this is where real mathematics begins. It can be answered only by getting acquainted with the formal mathematical definitions of numerical sets and operations on them. It is not so difficult, but for some reason it is not studied at school. But at lectures on mathematics at the university, in the first place, they will teach you exactly this.

In the course of school arithmetic, all mathematical operations are carried out with real numbers. The set of these numbers (or a continuous ordered field) has a number of properties (axioms): commutativity and associativity of multiplication and addition, the existence of zero, one, opposite and inverse elements. Also the axioms of order and continuity applied to comparative analysis, allow you to define all the properties of real numbers.

Since division is the inverse of multiplication, two unsolvable problems inevitably arise when dividing real numbers by zero. First, checking the result of division by zero using multiplication does not have a numeric expression. Whatever number the quotient is, if it is multiplied by zero, the dividend cannot be obtained. Secondly, in the 0:0 example, absolutely any number can serve as the answer, which, when multiplied with a divisor, always turns to zero.

Division by zero in higher mathematics

The enumerated difficulties of dividing by zero led to the taboo on this operation, at least within the framework of the school course. However, in higher mathematics find ways to circumvent this ban.

For example, by constructing another algebraic structure, different from the familiar number line. An example of such a structure is a wheel. There are laws and regulations here. In particular, division is not tied to multiplication and is converted from a binary operation (with two arguments) to a unary operation (with one argument), denoted by the symbol /x.

The expansion of the field of real numbers occurs due to the introduction of hyperreal numbers, which covers infinitely large and infinitely small quantities. This approach allows us to consider the term "infinity" as a certain number. Moreover, this number, when the number line expands, loses its sign, turning into an idealized point connecting the two ends of this line. This approach can be compared with the date change line, when, when moving between two time zones UTC + 12 and UTC-12, you can find yourself in the next day or in the previous one. At the same time, it becomes true statement x/0=∞ for any x≠0.

To eliminate the uncertainty 0/0, the wheel is introduced new element⏊=0/0. At the same time, this algebraic structure has its own nuances: 0 x≠0; x-x≠0 in the general case. Also x·/x≠1, since division and multiplication are no longer considered inverse operations. But these features of the wheel are well explained using the identities of the distributive law, which operates in such an algebraic structure somewhat differently. More detailed explanations can be found in specialized literature.

Algebra, to which everyone is accustomed, is, in fact, a special case of a more complex systems, for example, the same wheel. As you can see, it is possible to divide by zero in higher mathematics. This requires going beyond the boundaries of the usual ideas about numbers, algebraic operations and the laws that they obey. Although this is a completely natural process that accompanies any search for new knowledge.