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## Integrated math 1

### Course: Integrated math 1>Unit 1

Lesson 6: Division by zero

# Why dividing by zero is undefined

As much as we would like to have an answer for "what's 1 divided by 0?" it's sadly impossible to have an answer. The reason, in short, is that whatever we may answer, we will then have to agree that that answer times 0 equals to 1, and that cannot be ​true, because anything times 0 is 0. Created by Sal Khan.

## Want to join the conversation?

• What is 0/0 then? Because aren't there several rules that yield different answers for example, any number divided by itself is one, zero divided by any number is zero, etc? At Sal says, "any non-zero number divided by zero is undefined". What if you divide zero by zero?

Thanks! •   The way I explained it to my 7th graders is this:
If 0/0 is equal to a number, then let's rewrite it as a multiplication.
0/0 = 0 ==> 0 * 0 = 0, so that works...
But wait!
0/0 = 1 ==> 0 * 1 = 0. Oops, that also works.
and so does 0/0 = 2 and 0/0 = 6 and 0/0 = any real number.
This is where it breaks down.

Because there are so many (in fact, an infinite number) of ways that this division could be converted into a valid multiplication, we can conclude that this division isn't valid or indeterminate, to use the correct terminology). For any operation, there should only be at most one operation that "undoes" it.

Hope this helps.
• Infinity is a strange thing. What is most strange about it is the fact that it isn't an actual number. Like what is the absolute value of negative infinity. One could argue that the answer is simply infinity, but since infinity doesn't really exist in the first place, can it have an opposite? What actually is an example of negative infinity? •   You ask for an example of negative infinity, but what is an example of infinity? Because infinity is not a real number (it basically means 'goes on and on forever') then infinity would be the largest number in existence (this would not be possible because numbers go on forever), so negative infinity would be the smallest number in existence (this is also not possible because number also go on forever in negative numbers) so there is no example of infinity or negative infinity.
• Playing a bit of devil's advocate, so please take this post as more or a philosophical discussion than a question...

Just wondering why this is not defined as an imaginary number or an "imaginary ratio". Seems to me if we can define the square root of -1, which for real numbers is nonsensical, we could define 1/0.

Also, I am not convinced by the argument that it breaks down because we can have infinity or negative infinity. Why not simply state that infinity is an absolute (perhaps unobtainable) limit and that it is approached from either the positive or negative side? We do not give a sign to zero--we say that 0/-1 is the same is 0/1 so why not define infinity the same way?

I am of course not really trying to argue that 1/0 can be defined. If it could, far, far more brilliant people than me would have figured that out years ago, but I wanted to throw this out for discussion. • Sal, is it possible that 0 is the intersection of positive and negative numbers, since all other division results in some other non zero number and multiplication by zero is zero? Therefore infinity exists as an intersection. • Why couldn't it just be defined like this:

0=nothing

If i divide something by nothing, I'm left with that something. I simply didn't divide it.

Therefore, x/0=x

If I divide by .0000000001 or -.00000000001, I am then dividing by something again and therefore have an answer that is quantifiable. This could still allow us to hold true that dividing something by nothing simply means you did not divide it at all. • Okay, let's accept the fact the x/0=x but accepting it leads to all sorts of contradictions,
Consider V=S/T (Velocity=displacement divided by time)
say you have to cover 1m in 0 sec,which means in no time,then you ask what is the velocity, according to you it must be 1m/s. Does that make sense?
The contradiction becomes more apparent after you rearrange the equation by solving for time.
T=S/V (Time=displacement/velocity)
Say you have to cover 1 metre,with a velocity of 0m/s,but a velocity of 0m/s means that you are standing still,and if you are standing still how can you cover a given distance?Even if you stand motionless at a fixed point for an eternity you will never be able to cover any given distance,in this case asking the question 'what is the time if the velocity is 0 and distance 1' does not make sense,the solution is not infinity,the solution is not 1,there is no solution.
I guess in this universe dividing by 0 does not make sense but probably in a universe where you could cover distance without moving,it would make sense.
• Guys What is square root of Zero?? • Hi, I'm failing to understand the issue here? It seems to me that 2 different questions are trying to be represented here with the same mathematical formula:

1. What happens when you have nothing to divide?

If you have 7 and you want to divide by nothing then no division is going to happen, so it seems to me that 7/0 = 7

If you owe 1 and you want to divide by nothing then nothing is going to happen either -1/0 = -1 Again, no division is happening because you have nothing to divide.

2. If we are asking the question: How many times can I fit a number into another like how many times can we fit 0.1 into 1 then is obvious that the smaller the number the more times we can fit it, that is what happens when we do:

1/0.1 = 10

Obviously here the smaller the numbers we are using to divide the more times we can fit them and since we know that there are an infinite amount of numbers in between 0.1 & 1, then as long as we have something to divide, no matter how small, we get a result. But the assumption that because we are dividing by smaller and smaller numbers then 1/0=∞ seems wrong to me because the moment we are dealing with 0 we really have nothing to divide, therefore the division doesn't happen.

CONCLUSION:
So I'm failing to see the issue here and why there is a problem determining the value? Where is the problem with this line of thinking and what are the other ways of reasoning that contradict this? • I like to think of it this way.
When you are dividing say, 7/0, you are dividing it by nothing, but into a number of groups, which in this case would be no groups.
So if you divide 7 items into 0 groups, how many are there in each group? 0.

As for your second question. Let's consider factors.
So continuing with 7/0. If 7/0 is a question of how many 0's can we fit into 7, then once again the answer is 0.
Why? Because 0 cannot fit into 7, the only numbers that can are 1, and 7 (the factors of 7).

This is a great question, and I hope my answer helps clear it up a little!
• If infinity is not a real number, what exactly is it? • Hi,
Infinity is a concept. We cannot put a value on infinity because there will always be something bigger that we can come up with.

We tend to think of infinity as "the biggest number" but it's not, infinity is just an idea. If I give you my biggest number, you will be able to come up with a bigger number. Then I can come up with a number bigger than yours, and you come up with an even bigger number, and so on and so on forever.

By trying to put a value on infinity, we are limiting infinity which, by definition, is limitless. Hope that helps :-)  