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Is it possible to divide by 0 as a rule? Why can't you divide by zero? A good example

“You can’t divide by zero!” - Most schoolchildren learn this rule by heart, without asking questions. All children know what “you can’t” 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 not possible.

The thing is that the four operations of arithmetic - addition, subtraction, multiplication and division - are actually unequal. Mathematicians recognize only two of them as valid - 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 does 5 – 3 mean? The student will answer this simply: you need to take five objects, take away (remove) three of them and see how many remain. But mathematicians look at this problem completely differently. There is no subtraction, there is only addition. Therefore, the notation 5 – 3 means a number that, when added to the number 3, will give the number 5. That is, 5 – 3 is simply an abbreviated notation of the equation: x + 3 = 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. Entry 8:4 can be understood as the result of dividing eight items into four equal piles. But in reality, it's just a shorthand 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, the result is always 0. This is an inherent property of zero, strictly speaking, part of its definition.

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

The most attentive readers in this place will certainly ask: is it possible to divide zero by zero? In fact, the equation 0 x = 0 can be solved safely. For example, we 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. Correct? So 0:0 = 1? But this way 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 choose 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 makes no sense. It turns out that even zero cannot be divided by zero. (In mathematical analysis there are cases when, thanks to additional conditions tasks can be given preference to one of possible options solutions to the equation 0 x = 0; In such cases, mathematicians talk about “unfolding uncertainty,” but such cases do not occur in arithmetic.)

This is the peculiarity of the division operation. More precisely, the operation of multiplication and the number associated with it have zero.

Well, the most meticulous ones, having read this far, may ask: why does it happen that you can’t divide by zero, but you can subtract zero? In a sense, this is where real mathematics begins. You can answer it only by becoming familiar with the formal mathematical definitions of numerical sets and operations on them. It's not that difficult, but for some reason it's not taught in school. But at lectures on mathematics at the university, first of all, they will teach you exactly this.

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Mathematicians have a specific sense of humor and some questions related to calculations are no longer taken seriously. It’s not always clear whether they are trying to explain to you in all seriousness why you can’t divide by zero or whether this is just another joke. But the question itself is not so obvious; if in elementary mathematics one can 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 They didn’t know this number; the reference system began with I.
For a long time, Arabs and Indians argued for the right to be called the progenitors of zero.
Studies of the Mayan culture have shown that this ancient civilization could well have been the first in terms of using zero.
Zero has nothing numerical value, even minimal.
It literally means nothing, the absence of things to count.

In the primitive system there was no particular need for such a figure; the absence of something could be explained using words. But with the emergence of civilizations, human needs also increased in terms of architecture and engineering.

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

Is it possible to divide by zero?

There are two diametrically opposed opinions:

At school, even in elementary grades, they teach that you should never divide by zero. This is explained extremely simply:

Let's imagine that you have 20 tangerine slices.
By dividing them by 5, you will give 4 slices to five friends.
Dividing by zero will not work, because the process of division between someone will not happen.

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

After all, in fact, in this way one can denote the fact of the absence of division. Why complicate mathematical calculations and also write down the absence of division?

Can zero be divided by a number?

From the point of view of applied mathematics, any division that involves a zero does not make much sense. But school textbooks are clear in their opinion:

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

The third point may cause slight bewilderment, since 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 are doing the 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, dividing zero by zero. Especially when we're talking about about computers and programming languages.

The need to divide zero by a number may arise when solving any equalities and searching 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 won’t end up with more than zero. Therefore, if you notice this treasured number in a huge formula, try to quickly “figure out” whether all the calculations will come down to a very simple solution.

If infinity is divided by zero

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

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

The numerators must contain 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 the use of zero. Nothing has changed with this sign; it still cannot be divided into, 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 find practical application, and they will provide some kind of grandiose breakthrough in science.

In the meantime, most mathematical geniuses only dream of worldwide recognition. The exception to these rules is our compatriot, Perelman. But he is known for solving a truly epoch-making problem with the proof of the Poinqueré conjecture and for his extravagant behavior.

Paradoxes and the meaninglessness of division by zero

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

Division is represented as inverse function of multiplication.
We can multiply any number by zero and get zero as an answer.
By the same logic, one could divide any number by zero.
Under such conditions, it would be easy to come to the conclusion that any number multiplied or divided by zero is equal to any other number on which this operation was performed.
We discard the mathematical operation and get most interesting conclusion- any number is equal to any number.

In addition to creating such incidents, division by zero has no practical significance , from the word in general. Even if it is possible to perform this action, it will not be possible to obtain 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 a single time. Simply put - no fission process occurs, therefore, there cannot be a result of this event.

Being in the same company as 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 irritation, because this is probably not the first time a person has been asked this. And not even in the tenth. So take care of your mathematician friends, don’t force them to repeat one explanation a hundred 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 a mathematical point of view:

At school they teach us all simple rule, which cannot be divided by zero. At the same time, when we ask the question: “Why?”, they answer us: “This is just a rule and you need to know it.” In this article I will try to explain to you why you cannot divide by zero. Why are those people wrong who say that you can divide by zero and then you get infinity?

Why can't you divide by zero?

Formally, in mathematics, there are only two actions. Addition and multiplication of numbers. So what about subtraction and division? Let's consider this example. 7-4=3, we all know that seven minus four will equal three. In fact, this example can, formally, be considered as a way to solve the equation x+4=7. That is, we select a number that, when added to four, will give 7. Then we won’t think long and realize that this number is equal to three. It's the same with division. Let's say 12/3. This will be the same as x*3=12.

We select a number that, when multiplied by 3, will give us 12. B in this case m that makes four. This is pretty obvious. What about examples like 7/0. What happens if we write seven divided by zero? This means that we seem to be solving an equation of the form 0*x=7. But this equation has no solution, because if zero is multiplied by any number, the result is always zero. That is, there is no solution. This is written either with the words there are no solutions, or with an icon that means an empty set.

In other words

This is the meaning of this rule. You can't divide by zero because the corresponding equation, zero times x equals seven or whatever number we're trying to divide by zero, has no solutions. The most attentive ones can say that if we divide zero by zero, it will turn out quite fair that if 0*X=0. Everything is great, we multiply zero by some number, we get zero. But then our solution can be any number. If we look at x=1, 0*1=0, x=100500, 0*100500=0. Any number will do here.

So why should we choose any one of them? We really don't have any considerations by which we can take one of these numbers and say that these are solutions to the equations. Therefore, there are infinitely many solutions and this is also an ambiguous problem in which it is believed that there are no solutions.

Infinity

Above I told you the reasons why you cannot divide, now I want to talk to you about. Let's try to approach the division by zero operation with caution. Let's first divide the number 5 by two. We know that the result will be a decimal fraction of 2.5. Now we will reduce the divisor and divide 5 by 1, it will be 5. Now we will divide 5 by 0.5. This is the same as five divided by one half, or the same as 5 * 2, then it will be 10. Please note that the result of division, that is, the quotient, increases: 2.5, 5, 10.

Now let's divide 5 by 0.1, this will be the same as 5*10=50, the quotient has increased again. At the same time, we decreased the divisor. If we divide 5 by 0.01, it will be the same as 5*100=500. Look. The smaller we make the divisor, the larger the quotient becomes. If we divide 5 by 0.00001, we get 500000.

Let's sum it up

What then is division by zero, if you look at it in this sense? Notice how we reduced our quotient? If you draw an axis, you can see on it that first we had a two, then a one, then 0.5, 0.1, and so on. We were getting closer and closer to zero on the right, but we never got to zero. We take a smaller and smaller number and divide our quotient by it. It's getting bigger and bigger. In this case they write that we divide 5 by X, where X is infinitely small. That is, it gets closer and closer to zero. Just in this case, when dividing five by X, we get infinity. Endlessly large number. This is where a nuance arises.

If we approach zero from the right, then this infinitesimal will be positive and we get plus infinity. If we approach X from the left, that is, if we first divide by -2, then by -1, by -0.5, by -0.1 and so on. We will get a negative quotient. And then five divided by x, where x will be infinitesimal, but on the left, will be equal to minus infinity. In this case they write: x tends to zero from the right, 0+0, showing that we tend to zero from the right. Let's say if we were aiming for a three on the right, in this case we write X is aiming on the left. Accordingly, we would aim for three on the left, writing this as x tends to 3-0.

How a function graph can help

The graph of a function, which we studied in school, helps us understand this better. The function is called an inverse relationship, and its graph is a hyperbola. The hyperbole looks like this: This is a curve whose asymptotes are the x-axis and the y-axis. Asymptote is a line that a curve tends to but never reaches. Such is the mathematical drama. We see that the closer we get to zero, the greater our value becomes. The smaller X becomes, that is, as X tends towards zero on the right, the game becomes larger and larger, and rushes to plus infinity. Accordingly, when x tends to zero from the left, when x tends to zero from the left, i.e. x tends to 0-0, we tend to minus infinity. Correctly it is written like this. Y tends to minus infinity, with X tending to zero on the left. Accordingly, we will write that i tends to plus infinity, and when x tends to zero on the right. That is, in essence, we do not divide by zero, we divide by an infinitesimal value.

And those who say that you can divide by zero, we simply get infinity, they simply mean that you can not divide by zero, but you can divide by a number close to zero, that is, by an infinitesimal value. Then we get plus infinity if we divide by an infinitesimal positive and minus infinity we divide by an infinitesimal negative.

I hope that this article helped you understand the question that has plagued most people since childhood, why you can’t divide by zero. Why are we forced to learn some rule, but nothing is explained. I hope the article helped you understand that you really cannot divide by zero, and those who say that you can divide by zero actually mean that you can divide by an infinitesimal value.

“You cannot divide by zero!” - most schoolchildren learn this rule by heart, without asking questions. All children know what “You can’t” is and what will happen if you ask in response to it: “Why?” But in fact, it’s very interesting and important to know why you can’t.

We'll look at subtraction, for example. What does 5 - 3 mean? The student will answer this simply: you need to take five objects, take away (remove) three of them and see how many remain. But mathematicians look at this problem completely differently. There is no subtraction, there is only addition. Therefore, the notation 5 - 3 means a number that, when added to the number 3, will give the number 5. That is, 5 - 3 is simply a shorthand notation of the equation: x 3 = 5. There is no subtraction in this equation. There is only a task - to find a suitable number.

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, we always get 0. This is an inherent property of zero, strictly speaking , part of its definition.

The most attentive readers in this place will certainly ask: is it possible to divide zero by zero? In fact, the equation 0 * x = 0 can be solved safely. For example, we 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 this way 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 choose any one of them. That is, we cannot say what number the entry 0:0 corresponds to. And if so, then we are forced to admit that this entry also makes no sense. It turns out that even zero cannot be divided by zero. (In mathematical analysis, there are cases when, thanks to additional conditions of the problem, one can give preference to one of the possible solutions to the equation 0 * x = 0; in such cases, mathematicians talk about “Revelation of Uncertainty”, but in arithmetic such cases do not occur. This is the peculiarity of There are division operations. Or rather, the multiplication operation and the number associated with it have zero.

Why can’t you divide by zero? “You can’t divide by zero!” - Most schoolchildren learn this rule by heart, without asking questions. All children know what “you can’t” 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 not possible. The thing is that the four operations of arithmetic - addition, subtraction, multiplication and division - are actually unequal. Mathematicians recognize only two of them as valid - 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 does 5 – 3 mean? The student will answer this simply: you need to take five objects, take away (remove) three of them and see how many remain. But mathematicians look at this problem completely differently. There is no subtraction, there is only addition. Therefore, the notation 5 – 3 means a number that, when added to the number 3, will give the number 5. That is, 5 – 3 is simply a shorthand notation of the equation: x + 3 = 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. Entry 8:4 can be understood as the result of dividing eight items 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, the result is always 0. This is an inherent property of zero, strictly speaking , part of its definition.There is no such number that when multiplied by 0 will give something other than zero. That is, our problem has no solution. (Yes, this happens; not every problem has a solution.) This means that the entry 5:0 does not correspond to any specific number, and it simply does not mean anything and therefore has no meaning. The meaninglessness of this entry is briefly expressed by saying that you cannot divide by zero.The most attentive readers in this place will certainly ask: is it possible to divide zero by zero? Indeed, the equation 0 x = 0 can be solved safely. For example, we 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. Correct? So 0:0 = 1? But this way 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 choose 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 makes no 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 solutions to the equation 0 x = 0; in such cases, mathematicians talk about “revealing uncertainty,” but such cases do not occur in arithmetic.) This is the peculiarity of the division operation. More precisely, the operation of multiplication and the number associated with it have zero. Well, the most meticulous ones, having read this far, may ask: why does it happen that you can’t divide by zero, but you can subtract zero? In a sense, this is where real mathematics begins. You can answer it only by becoming familiar with the formal mathematical definitions of numerical sets and operations on them. It's not that difficult, but for some reason it's not taught in school. But in mathematics lectures at the university, this is what you will be taught first of all.