Division by zero: why not?

Strict prohibition on division by zero is imposed even in the junior classes of the school. Children usually do not think about its causes, but in fact to know why something is forbidden is both interesting and useful.

Arithmetic operations

Arithmetical actions that are studied in school are unequal from the point of view of mathematicians. They recognize as full only two of these operations - addition and multiplication. They enter into the very concept of a number, and all other actions with numbers somehow are built on these two. That is, it is impossible not only to divide by zero, but also to divide in general.

Subtraction and division

What is missing for the rest of the action? Again, from school it is known that, for example, subtract from seven four means to take seven sweets, four of them to eat and count those that will remain. But mathematicians do not solve the problem of eating sweets and generally perceive them completely differently. For them, there is only addition, that is, record 7 - 4 means a number that, in the sum with the number 4, will be 7. That is, for mathematicians 7-4 it is a short record of the equation: x + 4 = 7. This is not a subtraction, but a task - find the number that you want to put in place of x.

The same applies to division and multiplication. Dividing ten by two, the junior high school student places ten sweets into two identical groups. The mathematician here also sees the equation: 2 · x = 10.

This is why it is forbidden to divide by zero: it is simply impossible. The record 6: 0 should be transformed into the equation 0 · x = 6. That is, it is required to find a number that can be multiplied by zero and get 6. But it is known that zero multiplication always yields zero. This is an essential property of zero.

Thus, there is no such number, which, multiplying by zero, would give some number different from zero. Hence, 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. Its meaninglessness is also said when it is forbidden to divide by zero.

Is zero divided by zero?

Is it possible to divide zero by zero? The equation 0 · x = 0 does not cause difficulties, and we can take this zero for x and get 0 · 0 = 0. Then 0: 0 = 0? But if, for example, we take 0 as 1, we also get 0 · 1 = 0. We can take any number and divide by x at zero, and the result will remain the same: 0: 0 = 9, 0: 0 = 51 and so Further.

Thus, it is possible to insert absolutely any number into this equation, and it is impossible to choose any specific one, it is impossible to determine what number is indicated by the record 0: 0. That is, this record also does not make sense, and division by zero is still impossible: it Is not even divisible by itself.

This is an important feature of the operation of division, that is, multiplication and the associated number zero.

The question remains: why can not you divide by zero, but can you subtract it? It can be said 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 on mathematics begin with this.