A strict ban on dividing by zero is imposed in the lower grades of the school. Children usually don鈥檛 think about its reasons, but actually knowing why something is forbidden is 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 full ones - addition and multiplication. They enter the very concept of number, and all other actions with numbers are somehow built on these two. That is, not only division by zero is impossible, but division in general.
Subtraction and Division
What is the rest of the actions missing? Again, from school it is known that, for example, subtracting four from seven means taking seven candies, eating four of them and counting those that remain. But mathematicians do not solve the problem of eating sweets and generally perceive them in a completely different way. For them, there is only addition, that is, the record 7 - 4 means a number, which in total with the number 4 will be 7. That is, for mathematicians 7 - 4 is a short record of the equation: x + 4 = 7. This is not a subtraction, but a task - find the number to be replaced by x.
The same applies to division and multiplication. Dividing ten into two, a junior student lays ten candies in two identical heaps. The mathematician sees the equation here: 2 路 x = 10.
So it turns out why division by zero is forbidden: it is simply impossible. The record 6: 0 should turn into the equation 0 路 x = 6. That is, you need to find a number that can be multiplied by zero and get 6. But it is known that multiplying by zero always gives zero. This is the essential property of zero.
Thus, there is no such number that, multiplying by zero, would give some number other than zero. This means that this equation has no solution, there is no such number that would correspond to the record 6: 0, that is, it does not make sense. They also talk about its meaninglessness when the division by zero is forbidden.
Is zero divided by zero?
Is it possible to divide zero by zero? The equation 0 路 x = 0 is straightforward, and you can take this same zero as x and get 0 路 0 = 0. Then 0: 0 = 0? But, if, for example, one is taken as x, it will also be 0 路 1 = 0. Any number can be taken as x and divided by zero, and the result will remain the same: 0: 0 = 9, 0: 0 = 51 and so 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 record 0: 0. That is, this record also does not make sense, and division by zero is still impossible: it not even divided by himself.
This is an important feature of the division operation, that is, multiplication and the number zero associated with it.
The question remains: why is it impossible to divide by zero, but can it be subtracted? We can say that real mathematics begins with this interesting question. To find the answer to it, you need to learn the formal mathematical definitions of numerical sets and get acquainted with the operations on them. For example, there are not only simple, but also complex numbers, division which differs from the division of ordinary. This is not part of the school curriculum, but university lectures in mathematics begin with just that.