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Course: Algebra (all content)>Unit 1

Lesson 13: Division by zero

The problem with dividing zero by zero

One can argue that 0/0 is ​0, because 0 divided by anything is 0. Another one can argue that 0/0 is ​1, because anything divided by itself is 1. And that's exactly the problem! Whatever we say 0/0 equals to, we contradict one crucial property of numbers or another. To avoid "breaking math," we simply say that 0/0 is undetermined. Created by Sal Khan.

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• Is 0/0 = 1? Or is it undefined?

Couldnt you say that the numbers are like a dimension where you could join the infinitys'?

Like for example, if you wrote a number line on a piece of paper, with positive infinity at the top and negative infinity at the bottom, couldn't you fold the paper so that negative and positive infinity are the same?
• 0/0 is undefined. If substituting a value into an expression gives 0/0, there is a chance that the expression has an actual finite value, but it is undefined by this method. We use limits (calculus) to determine this finite value. But we can't just substitute and get an answer.

Example: the limit as x approaches 0 of x/x = 1, but the expression x/x is undefined at x = 0.

In geometry, we do not consider lines to meet at infinity. We consider them to go on forever in opposite directions, never meeting. In physics, of course, space is curved, and a real "line" might, in fact, be a closed figure (which is somewhat like "meeting at infinity")... but not in geometry.

Your thinking is great - keep it up.
• In my opinion, it seems that 0/0 could be equal not only to 0 and/or 1, but actually to any number.

Let's say that 0/0 = x
Following the principles of division and multiplication, we can re-arrange the equation like this:
0x = 0
From here it becomes obvious that this equation is true for any x, because 0 multiplied by anything is still equal to 0:
0 * 0 = 0; 0/0 = 0
0 * 1 = 0; 0/0 = 1
0 * 2 = 0; 0/0 = 2
... and so on

Is this also a valid way to prove that 0/0 is indeterminate?
• I'm just stating what Sal said in the video, but some people say that 0/0 is obviously 0, since 0/4, for example, is zero. They say zero divided by anything is zero. However, some say anything divided by zero is undefined, since 4/0 and 5/0 are and so on. Others say that 0/0 is obviously one because anything divided by itself, just like 20/20 is 1. All of these points of view are logical and reasonable, yet they contradict each other. These statements are impossible and don't work, since a mathematical expression consisting of only constants (numbers like 1, not variables like x and y) can only have one value. These thoughts can not merge, as 0 is not 1 or undefined, and vice versa. So 0/0 must be undefined.

Also, if you think about it more closely, (Sal also says this in the next video.) division must be able to be undone by multiplication. For example, 6 divided by 2 is 3, and it can be undone by multiplying 2 by 3 to get 6. If 0/0 is 1, then 1 times 0 is , so it is correct. If 0/0 is 0, then 0 times 0 is 0, so it is also correct. If 0/0 is undefined, then you can't multiply back. The first two can not be proved false using this method, nor can the latter, since it is not exactly defined as division anyway.
• If we have something and we divide it by 2, then don't we separate it into two pieces?
And if we have something and divide it by 1, then don't we "separate" it into one piece?
So if we have something and divide it by 0, wouldn't that be like making it disappear?
Wouldn't that be impossible, rather than undefined?
To take a thing and turn into nothing?
(So, my first hypothesis is that dividing by zero is not undefined, but is defined as impossible.)

(Now, to look at what we mean by "divide" or if we actually mean more than one thing.)
Divide (per somewhere online, I read), means to calculate how many times we could add one number to itself to get another number.
(For example, 10/5 = 2. We add 5 to itself 2 times to get 10.)

So with two divided by zero, we could continue adding zero to itself forever, but still not get to two.
Doesn't that make 2/0 impossible, not undefined?

1/3 says you have 1 number of thirds.
So with 3/0, would you have 3 number of "0ths"...
I've never heard of a "0th" and I don't think they exist.
0 (zero) of a pie is 0.
1/3 of a pie is 1/3.
1/2 of a pie is 1/2.
1/1 of a pie is a whole.
1/0 of a pie?

Since dividing by zero (i.e. having a "zeroth" as a denominator), I think, is impossible, then there is no such thing as a zeroth of a pie.

On a similar note, if you say you have zero halves (0/2),
Or you say you have zero thirds (0/3)
Then you have zero of those things.
But you could also read those as zero divided by two (0/2)
Or zero divided by three (0/3)

Divide (per some online definition) also means calculate how many times one number goes into another.
With 10/5, 5 goes into 10 two times. So 10/5 = 2.
But with 0/2, how does 2 go into 0?
(So, now it seems that 0/2 is undefined!) (Though, using a different meaning of division, we know we can have zero halves and that that simply equals 0!)
(So that is part of the reason why it seems to me that we are not being consistent in our meaning and usage of division, and that is affecting how we think about dividing by zero.)

(So, to restate my main point...)
I think that dividing by zero, regardless of what you mean by "divide," is impossible.
So next would be why this classic example meant to show that we can't divide by zero is actually flawed:
a/0 = b

Each side is multiplied by 0 in order to prepare to cancel out the zeros, like this:
(a/0) x 0 = b x 0

The problem with this is that a/0 is impossible, so when the zeros are "cancelled," what's really getting cancelled (on the left side) (along with the zero we added) is a part of an impossible number. If the number is impossible, then who are we to take a part of it out (the zero), and assume the remaining part is valid?
It is the entire number a/0 that is impossible, not just the 0 at the bottom.

So, if a/0 = b, and a/0 is impossible, then it seems that b is impossible too.

I've yet to finish a review of imaginary numbers and how they are used, while keeping the above in mind.
But I wonder if this is normal practice -- to pick and choose which parts of impossible things we use and which we don't. It seems that would really mess up our math.

I'm curious how this applies to the study of more advanced math, science, and astronomy.
Any thoughts on dividing by zero, or any of this?

4/10/17 I corrected this post and would love to hear/read anyone's thoughts on it!
• 1. Separating into n pieces means that you need n new pieces to get the original piece. If you separate into 0 pieces means that you need 0 of the new piece/s to get to the original 1. In other words, you're trying to find- meaning define- a number that when multiplied by zero you get 1. Which is impossible.
Note that I didn't call p/0 impossible. I called it impossible to define, which is why it's undefined.

2. (No 2, because in the question it's just a definition)

3. When you say "forever", it means that you would have to add an infinite amount of 0's to get to 2. But doing that would only give you 0, meaning you'll need to add a few more zeros- another infinite amount- and you'll stay at zero. And you keep going, and staying at zero. So yeah, it is impossible- impossible to define it's answer.

4-5. Interesting way of looking at it. But that doesn't prove that it's not undefined.

6. 0/p is 0, just like you say.

7. 2 does go into 0, just like it goes into 1 and into 0.3. in fact, it goes into 0 better than into them! Into zero it goes exactly 0 times (a whole number), but into 1 it goes only a half time and into 0.3 it goes only 0.15 times (both are fractions).

8-11. The flaw in your logic is that you aren't allowed to multiply by 0 when solving equations.

13. I also don't know much about how this relates to more advanced math, science, and astronomy.

4/26/17 I re-answered to the corrected post, and I still say the following:
Thoughts. Mathematicians are really comfortable with it being undefined, and they really don't have any motivation to change that.
• I'm just thinking about why x/x=1. Division is repeated subtraction. 10/5=2 cause 10-2-2-2-2-2=0. You subtract 2 five times from ten to get zero. You subtract ten one time from ten to get zero. You subtract zero zero times from zero to get zero. Therefore, 0/0=0. And then it works in reverse. 0*0=0. And the rule x/0=0 isn't broken. x/x=1 is ignored, but I already explained why. Also, it makes sense because zero isn't a number. Zero is a symbol that represents nothing. And nothing is undefined.
• Zero isn't a symbol. It is a number just how 1 or 45 is. Just how every number tells us the existence of something, zero tells us the existence of nothing.

As for 0/0, you can't define it as 0 because a lot of problems arise, some of which are far beyond the current scope here, and will be dealt with in a really interesting branch of Mathematics called Calculus.

And to add on, there's a reason why your logic doesn't work. So, okay, suppose we say 0/0 is 0. What is 0/1? That's also 0. What about 0/2? That's also zero. 0/1 = 0 says "I subtract 0 from 0 one time to get 0". 0/2 = 0 says "I subtract 0 from 0 twice to get 0". These are fair statements to make. But, subtracting 0 from 0, 0 times is a bit absurd, as that's similar to not doing anything at all. So, the idea of division being repeated subtraction breaks down.

Finally, I'll leave you with this. 0/0 has a special name in the context of limits. It's called an indeterminate form, and it pops up a lot in advanced Math, along with other indeterminate forms like infinity/infinity. Just know that 0/0 can be any number you want it to be. Think of it like this. Let 0/0 = x. So, we have x * 0 = 0. Now, what value of x satisfies this equation? Well, any number does! That's why 0/0 can be any number, but in the future, you'll learn methods to narrow down what number it is, as you'll need it to solve problems!
• Imagine having 0 cookies to give among 0 friends. How many cookies would each person get? See, it doesn`t make sense. And the cookie monster is sad that you have 0 cookies, and you are sad that you have 0 friends.
• From what I understand, whenever mathematicians can't find a good answer for something, they leave it undefined. Do they leave things undefined quickly? How many undefined things are there in the world? Can undefined cases even exist?
• In real life, there is no reason to divide something by nothing, as it is undefined in math, it has no real applications in the real world. But in theory, it can be undefined. Similarly, infinity is also just a concept in math, it cannot have a real application either, even if you ask how many atoms are in the universe, it is a certain number even though it might be a very large number. There are not a lot of undefined things in math, but all of Geometry is based on three undefined terms, a point, line and plane none of which can exist in the real world, but without the theory of these, real world applications would be more difficult. You act like having something undefined is just an arbitrary thing mathematicians do, it is not at all.
• i think 0/0 is 0 bcs if u have zero friends and zero cookies how many cookies will each friend get?
ZERO!
• The Multiplication Property of 0 tells us that any number times 0 = 0. If you turn that into an equation, you have:
x*0 = 0, where "x" is any number.
Solve the equation and your got x = 0/0. So, 0/0 can be made to equal any number. Since there is no single agreed upon solution, we say that 0/0 is indeterminate.
• If 0 x 5 = 0 then I divide both sides by 0, 0/0 = 5. So,
0 x y = 0, y = 0/0, so nothing divided by nothing can be anything?
• Essentially, yes. This is why 0/0 is considered indeterminate - there is no single agreed upon solution.