The strict ban on division by zero is imposed even in the lower grades of the school. Children usually do not think about its reasons, but actually knowing why something is forbidden is both interesting and useful.
Arithmetic operations
The arithmetic operations that are studied at school are unequal from the point of view of mathematicians. They recognize only two of these operations as full-fledged - addition and multiplication. They are included in the very concept of a number, and all other operations 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 else is missing? Again, it is known from school that, for example, subtracting four from seven means taking seven sweets, eating four of them and counting those that remain. But mathematicians do not solve problems by eating sweets and generally perceive them in a completely different way. For them, there is only addition, that is, the entry 7 - 4 means a number that, in total with the number 4, will be equal to 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 replace x.
SameThe same goes for division and multiplication. Dividing ten by two, the elementary school student arranges ten candies into two identical piles. The mathematician also sees the equation here: 2 x=10.
So it turns out why division by zero is forbidden: it is simply impossible. Recording 6: 0 should turn into the equation 0 x=6. That is, you need to find a number that can be multiplied by zero and get 6. But it is known that multiplication by zero always gives zero. This is the essential property of zero.
Thus, there is no such number, which, multiplied by zero, would give some number other than zero. This means that this equation does not have a solution, there is no such number that would correlate with the notation 6: 0, that is, it does not make sense. It is said to be meaningless when division by zero is prohibited.
Does zero divide by zero?
Can zero be divided by zero? The equation 0 x=0 does not cause difficulties, and you can take this same zero for x and get 0 x 0=0. Then 0: 0=0? But, if, for example, we take one for x, it will also turn out 0 1=0. You can take any number you want for x and divide by zero, and the result will remain the same: 0: 0=9, 0: 0=51 and so next.
Thus, absolutely any number can be inserted into this equation, and it is impossible to choose any specific number, it is impossible to determine which number is indicated by the notation 0: 0. That is, this notation also does not make sense, and division by zero still impossible: it is not even divisible by itself.
Such an importanta 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 subtract it? We can say that real mathematics begins with this interesting question. To find the answer to it, you need to know the formal mathematical definitions of numerical sets and get acquainted with operations on them. For example, there are not only prime, but also complex numbers, the division of which differs from the division of ordinary ones. This is not part of the school curriculum, but university lectures in mathematics begin with this.
