“You can not divide by zero!” – most students learn this rule by heart, without asking questions. All children know what “it is impossible” and what will happen if, in response to it, ask: “Why?” But in fact, it is very interesting and important to know why it is impossible.
The thing is that the four actions of arithmetic – addition, subtraction, multiplication and division – are in fact unequal. Mathematicians recognize only two of them – addition and multiplication. These operations and their properties are included in the very definition of the concept of a number. All other actions are constructed in one way or another from these two.
Consider, for example, subtraction. What does 5 – 3 mean? The schoolboy will answer this simply: we must take five items, take away (remove) three of them and see how much remains. But mathematicians look at this task quite differently. There is no subtraction; there is 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 simply a shortened notation of the equation: x + 3 = 5. In this equation, there is no subtraction. There is only a problem – to find a suitable number.
The same is true for multiplication and division. Recording 8: 4 can be understood as the result of the division of eight subjects into four equal groups. But in reality, this is simply an abbreviated form of the equation 4 · x = 8.
This is where it becomes clear why it is impossible (or more precisely impossible) to divide by zero. The 5: 0 record is an abbreviation from 0 · x = 5. That is, this job is to find a number that, when multiplied by 0, gives 5. But we know that multiplying by 0 always produces 0. This is an inherent property of zero, strictly speaking, part of its definition.
A number that, when multiplied by 0, yields something other than zero, simply does not exist. That is, our task has no solution. (Yes, it happens, not every task has a solution.) So, the 5: 0 record does not correspond to any particular number, and it just does not mean anything and therefore does not make sense. The meaninglessness of this record is briefly expressed, saying that you can not divide by zero.
The most attentive readers in this place will certainly ask: is it possible to divide zero by zero? In fact, in fact the equation 0 · x = 0 is successfully solved. 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? Hence, 0: 0 = 1? But after all, you can take any number and get 0: 0 = 5, 0: 0 = 317, and so on.
But if any number is suitable, then we have no reason to choose one of them. That is, we can not say what number the record corresponds 0: 0. And if so, then we are forced to admit that this record also does not make sense. It turns out that zero can’t be divided evenly by zero. (In mathematical analysis, there are cases when, thanks to the additional conditions of the problem, one can prefer one of the possible solutions to the equation 0 · x = 0, in such cases, mathematicians talk about “uncovering uncertainty”, but in arithmetic, there are no such cases.)
This is a feature of the operation of division. More precisely, the operation of multiplication and the number associated with it are zero.
Well, the most meticulous, after reading up to this place, they can ask: why does it happen that you can not divide by zero, and you can subtract zero? In a sense, it is with this question that real mathematics begins. You can answer it only by getting acquainted with 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 the lectures on mathematics at the university, you will first of all be taught exactly this.