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.
• Arabs and Indians argued for the right to be called the progenitors of zero for a long time.
• 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, not even a minimal one.
• It literally means nothing, the absence of things to count.

In the primitive system 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 it comes to 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 calculations will find practical applications, 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:

Division by zero in mathematics, a division at which the divisor is zero. Such a division can be formally written as ⁄ 0, where is the dividend.

In ordinary arithmetic (with real numbers), this expression does not make sense, because:

• at ≠ 0, there is no number that, when multiplied by 0, gives, therefore, no number can be taken as a quotient ⁄ 0;
• at = 0, division by zero is also undefined, since any number, when multiplied by 0, gives 0 and can be taken as a quotient 0 ⁄ 0.

Historically, one of the first references to the mathematical impossibility of assigning the value ⁄ 0 is in George Berkeley's criticism of infinitesimal calculus.

Logic errors

Since when multiplying any number by zero, we always get zero as a result, when dividing both parts of the expression × 0 = × 0, which is true regardless of the value of and, by 0, we get the expression = , which is incorrect in the case of arbitrarily given variables. Since zero can be given implicitly, but in the form of a rather complex mathematical expression, for example, in the form of the difference between two values ​​reduced to each other by algebraic transformations, such a division can be a rather unobvious mistake. The imperceptible introduction of such a division into the proof process in order to show the identity of obviously different quantities, thereby proving any absurd statement, is one of the varieties of mathematical sophism.

In computer science

In programming, depending on the programming language, data type, and value of the dividend, an attempt to divide by zero can lead to different consequences. The consequences of division by zero in integer and real arithmetic are fundamentally different:

• Attempt integer division by zero is always a critical error that makes it impossible to continue executing the program. It leads either to throwing an exception (which the program can handle itself, thereby avoiding an emergency stop), or to immediately stop the program with a fatal error message and, possibly, the contents of the call stack. In some programming languages, such as Go, an integer division by a zero constant is considered a syntax error and will cause the program to compile abort.
• AT real arithmetic consequences can be different in different languages:

• throwing an exception or stopping the program, as with integer division;
• obtaining a special non-numeric value as a result of the operation. In this case, the calculations are not interrupted, and their result can subsequently be interpreted by the program itself or by the user as a meaningful value or as evidence of incorrect calculations. The principle is widely used, according to which, when dividing the form ⁄ 0, where ≠ 0 is a floating point number, the result is equal to positive or negative (depending on the sign of the dividend) infinity - or, and when = 0, the result is a special value NaN (abbreviated from English not a number - “not a number”). This approach is adopted in the IEEE 754 standard, which is supported by many modern programming languages.

Random division by zero in a computer program can sometimes cause costly or dangerous failures in the equipment controlled by the program. For example, on September 21, 1997, a division by zero in the computerized control system of the USS Yorktown (CG-48) US Navy cruiser shut down all electronic equipment in the system, causing the ship's power plant to stop working.

see also

Notes

Function = 1 ⁄ . When tends to zero from the right, tends to infinity; when tends to zero from the left, tends to minus infinity

If you divide any number by zero on a conventional calculator, then it will give you the letter E or the word Error, that is, “error”.

The computer calculator in a similar case writes (in Windows XP): "Division by zero is prohibited."

Everything is consistent with the rule known from school that you cannot divide by zero.

Let's see why.

Division is the mathematical operation that is the inverse of multiplication. Division is defined through multiplication.

Divide a number a(dividend, for example 8) by a number b(divisor, for example, the number 2) - means to find such a number x(quotient), when multiplied by a divisor b it turns out divisible a(4 2 = 8), i.e. a divide by b means to solve the equation x · b = a.

The equation a: b = x is equivalent to the equation x · b = a.

We replace division with multiplication: instead of 8: 2 = x we ​​write x 2 = 8.

8: 2 = 4 is equivalent to 4 2 = 8

18: 3 = 6 is equivalent to 6 3 = 18

20: 2 = 10 is equivalent to 10 2 = 20

The result of division can always be checked by multiplication. The result of multiplying a divisor by a quotient must be the dividend.

Similarly, let's try to divide by zero.

For example, 6: 0 = ... We need to find a number that, when multiplied by 0, will give 6. But we know that when multiplied by zero, zero is always obtained. There is no number that, when multiplied by zero, would give something other than zero.

When they say that it is impossible or forbidden to divide by zero, it means that there is no number corresponding to the result of such a division (it is possible to divide by zero, but not to divide :)).

Why do they say in school that you can't divide by zero?

Therefore, in definition operations of dividing a by b, it is immediately emphasized that b ≠ 0.

If everything written above seemed too complicated for you, then it’s completely on your fingers: Dividing 8 by 2 means finding out how many twos you need to take to get 8 (answer: 4). Dividing 18 by 3 means to find out how many triples you need to take to get 18 (answer: 6).

Dividing 6 by zero means finding out how many zeros you need to take to get 6. No matter how many zeros you take, you still get zero, but you never get 6, i.e. division by zero is not defined.

An interesting result is obtained if you try to divide the number by zero on the android calculator. The screen will display ∞ (infinity) (or - ∞ if you divide by a negative number). This result is incorrect, since there is no number ∞. Apparently, programmers have confused completely different operations - dividing numbers and finding the limit of a numerical sequence n / x, where x → 0. When dividing zero by zero, NaN (Not a Number - Not a number) will be written.

"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 in fact, it is very interesting and important to know why it is 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.

Consider, for example, subtraction. What means 5 - 3 ? 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, the entry 5 - 3 means a number that, when added to a number 3 will give the number 5 . I.e 5 - 3 is just a shorthand for the equation: x + 3 = 5. There is no subtraction in this equation.

Division by zero

There is only a task - to find a suitable number.

The same is true with multiplication and division. Recording 8: 4 can be understood as the result of the division of eight objects into four equal piles. But it's really 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. Recording 5: 0 is an abbreviation 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 always turns out 0 . This is an inherent property of zero, strictly speaking, part of its definition.

A number that, when multiplied by 0 will give something other than null, just doesn't exist. That is, our problem has no solution. (Yes, it happens, not every problem has a solution.) 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, since the equation 0 x = 0 successfully resolved. For example, you can take x=0, and then we get 0 0 = 0. It turns out 0: 0=0 ? But let's not rush. Let's try to take x=1. Get 0 1 = 0. Correctly? Means, 0: 0 = 1 ? But you can take any number 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 tell which number corresponds to the entry 0: 0 . And if so, then we are forced to admit that this record 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 possible options for solving the equation 0 x = 0; in such cases, mathematicians speak of "disclosure of indeterminacy", but in arithmetic such cases do not occur.)

This is the feature of the division operation. More precisely, 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 in lectures on mathematics at the university, you will be taught this in the first place.

The division function is not defined for a range where the divisor is zero. You can divide, but the result is not defined

You can't delt by zero. Mathematics 2 classes of high school.

If my memory serves me right, then zero can be represented as an infinitesimal value, so there will be infinity. And the school "zero - nothing" is just a simplification, there are so many of them in school mathematics. But without them in any way, everything in due time.

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Division by zero

Private from division by zero there is no number other than zero.

The reasoning here is as follows: since in this case no number can satisfy the definition of a quotient.

Let's write, for example,

whatever number you take for testing (say, 2, 3, 7), it is not good because:

\[ 2 0 = 0 \]

\[ 3 0 = 0 \]

\[ 7 0 = 0 \]

What happens if you divide by 0?

etc., but you need to get in the product 2,3,7.

We can say that the problem of dividing by zero a number other than zero has no solution. However, a number other than zero can be divided by a number arbitrarily close to zero, and the closer the divisor is to zero, the larger the quotient will be. So if we divide 7 by

\[ \frac(1)(10), \frac(1)(100), \frac(1)(1000), \frac(1)(10000) \]

then we get private 70, 700, 7000, 70,000, etc., which increase indefinitely.

Therefore, it is often said that the quotient of dividing 7 by 0 is "infinitely large", or "equal to infinity", and they write

\[7:0 = \infin\]

The meaning of this expression is that if the divisor approaches zero, and the dividend remains equal to 7 (or approaches 7), then the quotient increases indefinitely.

What kind of questions our kids don’t ask!.. But the question “Why can’t you divide by zero?” do not ask. Why? Because even at school the teacher said that it was IMPOSSIBLE. You can't, so you can't! Much later, already at the institutes, we learned that it is still possible to divide, and the result will be - infinity. But, admit it, our mind accepted this fact as a kind of assumption, convention, because we remember from childhood - it’s impossible. And, actually, why all the same?

To begin with, let's figure out where infinity comes from, the concept of which in the first years of the university we treated with some degree of distrust. Everything is surprisingly simple: if any number is divided by smaller and smaller, then more and more value will be obtained. The smaller the divisor, the larger the quotient will become. This is how infinity appears.

But physicists and mathematicians do not like infinity, because It is conventionally accepted that you cannot divide by zero. It turns out that the assumption is the impossibility of dividing by zero.

Let's turn to the basics of mathematics. There are four operations in arithmetic - addition, subtraction, multiplication and division. But they are not equal. Mathematicians consider only two of them to be basic actions: addition and multiplication, the rest are reverse actions, consequences of the main ones.

Consider the concept of "subtraction". To solve the example "5 - 3 \u003d ...", three of the five items must be removed, the remaining number will be the answer to our example. But, given that addition is considered the main action, let's slightly change our example by writing it in the form of addition: "x + 3 = 5". That is, to what number must three be added to make five?

The same is true with division. The expression "8: 4 = ..." follows from the expression "4 x = 8". How many times four must be taken to make eight?

And here it is, the answer! If 5: 0 is a variant of writing 0 x = 5, then it turns out that you need to find a number that, when multiplied by 0, will give 5. How many times do you need to take zero to get something more than nothing ?! But multiplying by 0 always results in 0, this fact lies in the very definition of zero! There is no number that, when multiplied by 0, gives something other than zero. It turns out that the problem has no solution, and the expression 5: 0 does not make sense. To reduce the number of meaningless tasks, it was accepted that you cannot divide by zero.

The most meticulous readers will certainly ask: But what about dividing zero by zero?

Let's figure it out. It turns out that the equation 0 x = 0 has a solution? Or an infinite number of solutions? "X" can be equal to one, two, and a million. So, with x=0, it turns out 0 0 = 0, then 0: 0=0? And if x=1, 0 1 =0, then 0: 0 = 1?! Or 0: 0 = 1000000?!

It turns out that we cannot find a solution to the expression "0: 0", which means that this expression does not have a solution either. So you can't divide zero by zero either.

You can come to such interesting conclusions by thinking about a fact known from primary school: you cannot divide by zero.

Interested? Did you read to the end? So, it is because of people like you that the next life anecdote appeared.

Why can't you divide by zero? You can multiply, and it also turns out zero.

- Why not? It is possible, only the result of such a division is infinity

Why not zero?

- Well, look: 2 * 0 - this is two take zero times, it will be zero. And 2/0 is “how many times zero fits in a deuce”, infinity.

- If 2/0=x, then 2=x*0 means 2=0. And if 2=0, then 2/0=0!

- Well, in order not to engage in such nonsense, mathematicians have adopted an unspoken agreement: you can’t divide by zero!

Each of us learned at least two unshakable rules from school: “zhi and shi - write with the letter I” and “ can't divide by zero". And if the first rule can be explained by the peculiarity of the Russian language, then the second one raises a completely logical question: “Why?”

Why can't you divide by zero?

It’s not entirely clear why they don’t talk about this in school, but in terms of arithmetic, the answer is very simple.

Let's take a number 10 and divide it by 2 . This implies that we have taken 10 any objects and arranged them according to 2 equal groups, that is 10: 2 = 5 (on 5 items in the group). The same example can also be written using the equation x * 2 = 10(and X here will be equal to 5 ).

Now, for a second, imagine that you can divide by zero, and try 10 divide by 0 .

You will get the following: 10:0=x, hence x * 0 = 10. But our calculations cannot be correct, since when multiplying any number by 0 always turns out 0 . In mathematics, there is no such number, which, when multiplied by 0 would give something other than 0 . Therefore, the equations 10:0=x and x * 0 = 10 don't have a solution. In view of this, they say that you cannot divide by zero.

When can you divide by zero?

There is a variant in which division by zero still makes some sense. If we divide zero itself, then we get the following 0: 0 = x, which means x * 0 = 0.

Let's pretend that x=0, then the equation does not raise any questions, everything converges perfectly 0: 0 = 0 , which means 0 * 0 = 0 .

But what if X≠ 0 ? Let's pretend that x = 9? Then 9 * 0 = 0 and 0: 0 = 9 ? And if x=45, then 0: 0 = 45 .

We can really share 0 on the 0 . But this equation will have an infinite number of solutions, since 0:0 = anything.

Why 0:0 = NaN

Have you ever tried to share 0 on the 0 on a smartphone? Since zero divided by zero gives absolutely any number, the programmers had to look for a way out of this situation, because the calculator cannot ignore your requests. And they found a kind of way out: when you divide zero by zero, you get NaN (not a number).

Why x:0=∞ a x: -0 = — ∞

If you try to divide any number by zero on your smartphone, the answer will be equal to infinity. The point is that in mathematics 0 sometimes viewed not as "nothing", but as "an infinitesimal quantity". Therefore, if any number is divided by an infinitesimal value, an infinitely large value will be obtained (∞) .

So is it possible to divide by zero?

The answer, as is often the case, is ambiguous. At school, it's best to cut yourself on the nose that can't divide by zero This will save you unnecessary complications. But if you enter the Faculty of Mathematics at the university, you still have to divide by zero.

The mathematical rule regarding division by zero was taught to all people in the first grade of a comprehensive school. “You can’t divide by zero,” they taught all of us and forbade, under pain of a slap in the back, to divide by zero and generally discuss this topic. Although some elementary school teachers still tried to explain why it is impossible to divide by zero using simple examples, these examples were so illogical that it was easier to just remember this rule and not ask too many questions. But all these examples were illogical for the reason that the teachers could not logically explain this to us in the first grade, since in the first grade we didn’t even know what an equation was, and logically this mathematical rule can be explained only with the help of equations.

Everyone knows that when dividing any number by zero, a void will come out. Why exactly emptiness, we will consider later.

In general, in mathematics, only two procedures with numbers are recognized as independent. This is addition and multiplication. The remaining procedures are considered derivatives of these two procedures. Let's look at this with an example.

Tell me, how much will it be, for example, 11-10? We will all instantly answer that it will be 1. And how did we find such an answer? Someone will say that it’s already clear that it will be 1, someone will say that he took 10 from 11 apples and calculated that it turned out to be one apple. From the point of view of logic, everything is correct, but according to the laws of mathematics, this problem is solved differently. It must be remembered that addition and multiplication are considered the main procedures, so you need to make the following equation: x + 10 \u003d 11, and only then x \u003d 11-10, x \u003d 1. Note that addition comes first, and only then, based on the equation, can we subtract. It would seem, why so many procedures? After all, the answer is so obvious. But only such procedures can explain the impossibility of dividing by zero.

For example, we are doing the following mathematical task: we want to divide 20 by zero. So 20:0=x. To find out how much it will be, you need to remember that the division procedure follows from multiplication. In other words, division is the derivative procedure of multiplication. Therefore, you need to make an equation from multiplication. So, 0*x=20. Here is the dead end. Whatever number we multiply by zero, it will still be 0, but not 20. This is where the rule follows: you cannot divide by zero. Zero can be divided by any number, but a number cannot be divided by zero.

This raises another question: is it possible to divide zero by zero? So 0:0=x means 0*x=0. This equation can be solved. Take, for example, x=4, which means 0*4=0. It turns out that if you divide zero by zero, you get 4. But even here everything is not so simple. If we take, for example, x=12 or x=13, then the same answer will come out (0*12=0). In general, no matter what number we substitute, 0 will still come out. Therefore, if 0: 0, then infinity will turn out. Here's some simple math. Unfortunately, the procedure for dividing zero by zero is also meaningless.

In general, the number zero in mathematics is the most interesting. For example, everyone knows that any number to the zero power gives one. Of course, we don’t meet such an example in real life, but with division by zero, life situations come across very often. So remember that you can't divide by zero.