"Divide by zero is impossible!" all memorize this rule by heart, without hesitation. And, actually, why not?
The thing is that the four steps of arithmetic - addition, subtraction, multiplication, and division - in fact unequal. Mathematics recognize complete only two of them are addition and multiplication. These operations and their properties are included in the definition of the concept of number. All other actions are built one way or the other of these two.
Consider, for example, the subtraction. What does 5 - 3 ? The student will answer it simply: we must take five items to take (remove) the three of them and see how many will remain. But mathematicians look at this problem differently. There is no subtraction is just addition. Therefore, recording 5 - 3 means a number that when added to the number 3 will give the number 5 . That is, 5 - 3 is just shorthand for the equation: x + 3 = 5 . In this equation there is no subtraction. There's only task is to find an appropriate number.
Exactly the same is the case with multiplication and division. Entry 8 : 4 can be understood as the result of the division of the eight subjects in four equal piles. But in reality it's just a shortened form of the equation 4 x = 8 .
Here it becomes clear why it is impossible (if not impossible) to divide by zero. Entry 5 : 0 is reduced from 0 · x = 5 . That is, the task to find a number that when multiplied by 0 will give 5 . But we know that when multiplied by 0 is always 0 . This is an inherent property of zero, strictly speaking, part of its definition.
This number, which when multiplied by 0 will give something other than zero, simply does not exist. That is our problem has no solution. Hence, entry 5 : 0 does not correspond to any specific number, and it just means nothing and therefore does not make sense. The senselessness of this entry briefly expressed by the phrase "divide By zero is impossible".
The most attentive readers of this place will certainly be asked: is it possible zero divided by zero? In fact, the equation 0 · x = 0 successfully solved. For example, we can take x = 0 , and then we get 0 · 0 = 0 . Goes, 0 : 0=0 ? But let's not rush. Let us take x = 1 . Get 0 · 1 = 0 . Right? Then 0 : 0 = 1 ?
But so you can take any number and get a 0 : 0 = 5 , 0 : 0 = 317 , etc. And, if there is any number, then we have no reason to opt for any one of them. That is, we can't tell which number corresponds to an entry of 0 : 0 . If so, then we have to admit that this entry has no meaning. It turns out that zero cannot be divided even zero.
This feature has operations division. Or rather, the operations of multiplication and the associated number zero.