Posted on Mar 9, 2022 * 475*

**“You can’t divide by zero!” - everyone memorizes this rule by heart without thinking about it. Why can’t they?**

It’s because the four actions of arithmetic-addition, subtraction, multiplication, and division-are actually unequal. Mathematicians recognize only two of them - addition and multiplication - as valid. These operations and their properties are included in the very definition of the concept of number. All other operations are constructed in either from these two.

Consider, for example, subtraction. What does 5 - 3 mean? A schoolboy will answer it simply: you have to take five objects, subtract (remove) three of them and see how many remain. But mathematicians look at this problem very differently. There is no subtraction, only addition. That’s why the notation 5 - 3 means the number that when added to the number 3 gives the number 5. So 5 - 3 is just a shortened notation of the equation: x + 3 = 5. There is no subtraction in this equation. There is only finding the right number.

It is the same with multiplication and division. The entry 8 : 4 can be understood as the result of dividing eight objects into four equal piles. But in reality it is just a shortened form of the equation 4 - x = 8.

This is where it becomes clear why you can’t (or rather can’t) divide by zero. The entry 5 : 0 is an abbreviation of 0 - x = 5. It is a task to find a number which, when multiplied by 0, yields 5. But we know that multiplying by 0 always yields 0. This is an inherent property of zero, strictly , part of its definition.

There is no number that, when multiplied by 0, would yield anything other than zero. Our problem has no solution. This means that the notation 5 : 0 does not correspond to any specific number, and it simply means nothing and therefore has no meaning. The meaninglessness of this entry is briefly expressed by the phrase “You cannot divide by zero”.

The most attentive readers will inevitably ask at this point: can zero be divided by zero? The equation 0 - x = 0 is safely solved. For example, we can take x = 0, and then we get 0 - 0 = 0. So 0 : 0 = 0? But let’s not be in a hurry. Let us try to take x = 1. We get 0 - 1 = 0. Right? So 0 : 0 = 1?

But we can take any number and get 0 : 0 = 5 , 0 : 0 = 317 and so on. And, if any number fits, then we have no reason to choose any of them. We cannot say to which number the entry 0 : 0 corresponds. And since this is the case, we are forced to admit that this entry makes little sense either. It turns out that even zero cannot be divided by zero.

This is the peculiarity of the division operation. More precisely, the multiplication operation and the associated number zero have such a peculiarity.