# “You cannot divide by zero!” - but why...?

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“You cannot divide by zero!” - Most students memorize this rule, without questioning why. All kids know what is "impossible" and what will happen if, they ask: "Why?" But in fact it is very interesting and important to know why you can't divide by zero.

The thing is that the four actions of arithmetic - addition, subtraction, multiplication and division - are actually unequal. Mathematicians recognize only two of them as full-fledged: addition and multiplication. These operations and their properties are included in the very definition of the concept of number. All other actions are built in one way or another from these two.

Consider, for example, subtraction. What does "$${\color{darkblue}5 - 3}$$" mean? A student will simply say: this means you start with five items, remove three of them and see how many are left. But mathematicians look at this problem in a completely different way. There is no subtraction, there is only addition. Therefore, writing "$${\color{darkblue}5 - 3}$$" means a number that, when added to $$3,$$ gives the number $$5.$$ That is, "$${\color{darkblue}5 - 3}$$" is just an abbreviated notation for the following equation: $$({\color{darkblue}5 - 3})+3=5\Leftrightarrow {\color{darkblue}\bf x} + 3 = 5.$$ There is no subtraction in this equation. There is only one task - to find the right number.

This is exactly the same case with multiplication and division. A ratio like "$${\color{darkgreen}8: 4}$$" can be understood as distributing eight objects into four equal heaps. But in reality, this is simply an abbreviated form of the equation $$4 \times {\color{green}\bf x} = 8.$$

Here it becomes clear why it is impossible (or rather impossible) to divide by zero. The notation "$${\color{purple}5: 0}$$" is an abbreviation of $$0 \times {\color{purple}\bf x} = 5.$$ That is, this task is to find a number that, when multiplied by $$0,$$ will give $$5.$$ But we know that when multiplied by $$0$$ it always turns out to be $$0.$$ This is an integral property of zero, strictly speaking, and is part of its definition.

Such a number, which when multiplied by $$0$$ will give something other than zero, simply does not exist. That is, our equation has no solution. (Yes, this happens; not every problem has a solution. ) So, a "$${\color{purple}5: 0}$$"
ratio does not correspond to any specific number, and it simply does not mean anything and therefore does not make sense. The meaninglessness of this entry is briefly expressed by saying that it is impossible to divide by zero.

The most attentive readers in this place will certainly ask: is it possible to divide zero by zero? Indeed, the equation $$0 \times {\color{darkred}\bf x} = 0$$ is solvable. For example, you can take $${\color{darkred}\bf x} = 0,$$ and then we get $$0 \times {\color{darkred} 0} = 0.$$ It turns out that $$0: 0 = {\color{darkred} 0} ?$$ But let's not rush. Let's try to take $${\color{darkred}\bf x} = 1.$$ Get $$0 \times {\color{darkred} 1} = 0.$$ Is that right? So $$0: 0 = {\color{darkred} 1}?$$ But you can take any number in this way and get $${\color{darkred} 0: 0} = 15, {\color{darkred} 0: 0} = 314,$$ etc.

But if any number works, then we have no reason to choose any particular one of them as the answer. That is, we cannot say what number corresponds to the ratio "$${\color{darkred} 0: 0}$$." And if so, then we are forced to admit that this ratio also does not make sense. It turns out that even zero cannot be divided by zero. (In mathematical analysis, there are cases when, thanks to additional conditions of the problem, one of the possible solutions to the equation $$0 \times {\color{darkred}\bf x} = 0$$ can be preferred more than other ones; in such cases, mathematicians speak of “Indeterminate form”, but there are no such cases in arithmetic.)

This is a feature of the division operation. Or, if we are to say it more precisely, the operation of multiplication and the number zero associated with it.

Well, the most meticulous ones, having read up to this point, may ask: why is it that you cannot divide by zero, but you can subtract zero? In some sense, it is from this question that real mathematics begins. You can answer it only by familiarizing yourself with the formal mathematical definitions of numerical sets and operations on them. It is not so difficult, but you might like to learn a lot more before that.

• @nastya thanks

• Here's another explanation from Beast Academy: Math Guide 4B. It's mainly for younger kids, but it's funny, so

• I personally think it's more a perspective problem. There's tons of people out there who have their own views, and none are really wrong.

• Guys, ask siri. It's gonna be hilarious!

• @debbie said in Overheard (July 2020):

## Overheard from the Friday 7/31/20 live stream

"We should get a negative second timer." -- T., in response to Prof. Loh being late

"Bananas are extinct." --- E.

"Strange corner of the internet that lacks green shirts." --- T.

"Highest like-to-dislike ratio I've ever seen!" --- T.

"Yayyyy there are still live streams! Even if it's only once a week, it's better than nothing!" --- M.

"This guy also runs the free site expii if you're stuck with no classes; not as advanced as his "daily challenge," but it's something." --- S.

"I am an oldie." --- C.

"Join the Nassau County Gavel Club." --- J.

"Who here remembers the "Earth disco ball" Po-Shen Loh discussed a long time back?" --- T.

"Who remembers the mobius hamster wheel he drew?" --- A.

"DA GREEN EQUASION." --- I.

"Mayonnaise is the best instrument." --- T.

"Infinity signs but screwed." --- A. (describing Prof. Loh's diagram of the absolute value function problem asked in the livestream)

"Why did you do a video about football?" --- A. (regarding Prof. Loh's video here)

"Everyone ask Siri: What is $$\frac{0}{0}?$$ Siri will be savage." --- S.

"Siri says I have no friends." --- T.

"Siri is a bully. He should get detention." --- A.

"Siri can be both a female and a male." --- A.

"I loved daily AMA...." --- A.

• This post is deleted!

• @Da-Parasite it only works if you do 0/0

• you are correct =D

• @Potato2017 @Da-Parasite
It kind of works... not really?
Siri said “Somewhere between infinity, negative infinity and undefined.”
I can’t show you a photo because it said the server went wrong.

• Try using a tablet, i guess?