Why zero divided by zero is undefined/indeterminate | Algebra II | Khan Academy

Multiple arguments for what we could get when we divide zero by zero. We will later see that this can be considered indeterminate

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[soul]

nothing devided by nothing. hmmm. what does it give. i wonder i wonder...

[gnouveli]

in physic, this is called *_something from nothing_*
0 / 0 = amazingly it would be something

[khanacademy]

You are correct that 0/0 is often considered an indeterminate form. Indeterminate forms are a subset of undefined forms (so it is not incorrect to say that 0/0 is undefined). The what makes something indeterminate is a beyond the scope of the level I wanted to do this video at.

[cornholio]

Zero divided by zero will give you a quantum number which means the answer could be 1 and 0 at the same time!

[...]

My best explanation is to make an equation and solve it:
0/0=x
*0   *0
0=0
which means that any value of x will work in the equation, so 0/0 can equal any number

[HellaPerformance]

It's undefined because it can be thought of as all real numbers. In calculus you learn that 0/0 can mean absolutely anything, but what it really means can only be determined by looking at the step done before arriving at the answer 0/0. If you take 0/0 and set it equal to x, what does x equal? Well, multiply both sides by zero and you have 0x = 0. What values of x make that equation true?

[NomadicIsaac]

It seems like noone here has taken a Calc course from a theory-loving math professor. You'd be surprised at the work that goes into proving the "simplest" of equations. The careful thought that goes into higher level math is truly an art.

[hareecionelson]

fun trivia: relativistic momentum is defined as mv/SqRt(1-(v/c)^2)

a photon has 0 mass and travels at the speed of light. Plug this into the momentum formula and you return (0*3x10^8)/(1-1) = 0/0

but by assuming the momentum does in fact exist and using algebra, we can derive pc = hf, or p = h/λ

this is proof that 0/0 could take any value, since every photon has a momentum of 0/0, but every photon also has its own discrete momentum depending on its wavelength.

0/0 can be anything it wants

[denbyk]

yes, it is.
if you consider the definition of the division, you have:
r * 0 = 0 indeterminate
where r = 0/0 could be any number.
for the same reason r multiplied by 0 can't be any number different from 0, so r = a / 0 is undefined (impossible?). It simply doesn't have sense to ask what's the result cause the function y(x) = a / x is not defined in the point x = 0

[batataooo]

Why isn’t it just 0 tho? If you have nothing and add it to nothing it stays being nothing with no change happening, so if you have nothing to give to no one, it just stays nothing also since 0-0=0 0+0=0 0*0=0 we know too that * is a product of +, for example: 2*7 = 2+2+2+2+2+2+2 or 7+7, division is the same but with -, for example 12:3 is 12-3-3-3-3 until it arrives at and answer (it can at some point become a part of the just a part of the 3), so, 0:0 is 0-0 zero times, therefore 0:0 is 0 since 0-0 happening 0 times is just the 0 itself, why isn’t it zero. Also about the getting close to zero divided by close to zero isn’t really right since it’s more then zero by more then zero so it shouldn’t be the same for the 0:0 since it’s not 0.1:0.1 it’s 0:0

[idlesuggest]

The one correct answer is the number "error" XD

[ezpz]

What if there is a negative 0 and a positive 0 rather than the "Neutral" 0 that we think of now?
1/(-0) = negative infinity, and 1/(+0) = positive infinity.
To me it is the explanation that makes the most sense.

[AvgasStew]

Feeling dumb is actually the first real sign of intelligence. As someone wise once told me. "The more I understand, the less I know. Where as the less I know the less I understand". So if you feel you don't know enough (aka you feel dumb) your smart enough to understand what you don't know.

[TheAllboutwin]

Can we think of this as the limit as (x, y) tends to (0, 0) of f(x, y)= x/y? Because that would approach two different values from two different paths so therefore 0/0 is undefined?

[ryanalving]

I think first we must step back and examine the two axioms presented:

Any number / itself = 1

0 / any number = 0

Now, I'll start at the second one. Why is it that 0 over any # is zero? It must be because you cannot fit any amount of something into nothing.
A point with diameter 0 cannot fit any fraction of a circle with diameter # > 0 into itself. However, a point with diameter 0 can manifestly fit a point of diameter 0 into itself.

Now, consider a circle of any diameter. How many points of diameter 0 can you fit inside said circle? You can obviously fit an infinite amount of them into any sized circle, in contrast to a point which can fit only a single point into itself.

In rational equations, the vertical asymptotes are points where the denominator is zero; and on either side f(x) ----》infinity or - infinity (the number of 0 diameter points that can be fit into any space).

If the numerator and denominator of the equation both contain a common zero (e.g. [(x-3)(x+4)] / [(x-3)(x+7)] ) the procedure is to cancel the common zero and solve (e.g. f(3)=x+4/x+7 )
And this represents a "hole" in the graph (a point where the graph is non continuous). However, it is only discoverable by cancelling 0/0 into 1. (e.g. 1(x+4) / 1(x+7) )
If it were to be accepted that
0/0 =1
Such discontinuous equations would become continuous. Further, we can deduce that
any #/0 = + or - infinity from the same logic and equations. Thus making said equations likewise continuous at the asymptotes.

It must further be noted, that our equations regarding to the "singularity" of the universes origin are said to break down at a point of infinite density.
E.g.
D = M/V where M = all mass and V = 0; meaning that we have a real world corrolary for the idea of
#/0 = infinity, the singularity. So we can see that to make all things continuous, we need simply adjust our axioms according to our observations.

[dgzlqyg]

19강 : Algebra 1 - Unit1 - lesson 6 - 2/3
Unit 1 Algebra foundation
L6 Division by zero
2/3 The problem with dividing zero by zero (=Why zero divided by zero is undefined/indeterminate)

11. And it didn't even matter 앤딧 니븐 매럴 0:57
11. and I'd 애낟 1:00
13. this seems like a 디 심자이커 1:10
11. these were 디줠 0:58
19. these are 디절 2:07

1. 0:01 when 0:02 why 0:05
In the last video we saw why when we take any non-zero number divided by zero why Mathematicians have left that as being undefined.
지난 동영상에서 0이 아닌 수를 0으로 나누었을 때 왜 결괏값을 정의하지 못하는지에 대해 알아봤습니다.
😢2. 0:09
But it might have raised a question in your brain.
버릿 마이르 리즈더 퀘스춴 인유얼 브레인
여기서 하나의 의문점이 생겼을 것입니다.
ㆍㆍㆍ
raise a question 의문을 제기하다, 질문을 꺼내다[제기하다].
raise a question (whether…) (…의 여부에) 의문을 제기하다, (…의 여부를) 문제삼다
raise 레이즈(미, 영), 리즈(인도) 1. (무엇을 위로) 들어올리다[올리다/들다] 5. (안건·문제 등을) 제기[언급]하다

3. 0:11 0:12
What about 0 divided by 0?
0 ÷ 0은 얼마일까?
(ㅈㄷㅂㅇ) 0을 0으로 나누면 어떨까요?
4. 0:15 0:17
Isn't there an argument why that could be defined?
이즌데언 알규먼트 와이댓 쿠드비 드빠인드?
0 ÷ 0을 정의할 수 있을까?
(ㅈㄷㅂㅇ) 그것이 정의될 수 있는 이유에 대한 논쟁이 없나요?
why 관계부사. 1. [제한적 용법] …한 (이유) 2.
[선행사 없이 명사절을 이끌어] …하는 이유

5. 0:18
So we're going to think about 0 divided by 0.
그럼 우리는 0을 0으로 나누는 것에 대해 생각해 보겠습니다.
6. 0:22
Well there's a couple of lines of reasoning here.
월데저 커플오브 라인저 리즈닝 히얼
여기에는 몇 가지 추론이 있습니다.
lines of (줄 지어) 늘어서 있는
reasoning 추리, 추론
7. 0:24 0:25 and 0:28
One, you could start taking numbers closer to zero and dividing them by themselves.
첫 번째는 0에 가까운 두 숫자를 택하여 서로 나누어볼 수 있습니다.

8. 0:30 0:34
So for example, you take 0.1 divided by 0.1. Well, that's gonna be 1.
9. 0:36 ; 0:38 / 0:43
Let's get even closer to zero: 0.001 divided by 0.001. Well, that also equals 1.
10. 0:44 0:45; 0:46 / well 0:54
Let's get super close to zero: divided by Well once again, that also equals 1.
super 부. 매우; 극도로.

😐11. 0:57 0:58 / 0:59 and 1:00 1:01
And it didn't even matter whether these were positive or negative. I could make these negative and I'd still get the same result.
앤딧 니븐 매럴 웨더 디줠 파저리 올 네거립.아이큳 메익 띠즈 네거립 애낟 스틸 겟떠 쎄임 리절트
두 수의 부호가 동시에 음수로 바뀌어도 같은 결과를 얻을 수 있습니다.
(ㅈㄷㅂㅇ) 이 수들이 양수인지 음수인지는 중요하지 않았습니다.
ㆍㆍㆍ
even 부사.
옥스퍼드 1. (예상 밖이나 놀라운 일을 나타내어) …도[조차] 3. 더 정확히 말하면, (심지어) …하기까지 하게
동아출판
[사실·극단적인 사례 등을 강조하여] …까지도, …조차(도), …마저
YBM
1. [예외적인 것의 강조]
…조차(도), …이라도, …까지도.

12. 1:03 1:06
Negative this thing divided by negative this thing still gets me to one.

13. 1:07 this 1:10 1:12 to be 1:15
So based on this logic you might say, "Hey, well this seems like a pretty reasonable argument for zero divided by zero to be defined as being equal to one."
디 심자이커 프리리 리즈너블 알규먼
따라서 0÷0=1이라는 주장도 충분히 합리적이라고 볼 수 있습니다.
(ㅈㄷㅂㅇ) 따라서 이 논리를 바탕으로 여러분은 "이봐, 이것은 0을 0으로 나눈 것이 1과 같다고 정의되는 꽤 합리적인 주장인 것 같습니다."라고 말할 수도 있습니다.
argument 2. 논거, 주장 1. 논쟁, 언쟁, 말다툼 3. 논의

14. 1:18 1:19 well 1:20 1:22 not 1:24 1:25 1:27
But someone could come along and say, "Well, no, what happens if we divided zero by numbers closer and closer to zero; / not a number by itself, / but zero by smaller and smaller numbers, or numbers closer and closer to zero."
낫 어 넘버바잇 쎌펏 지로 바이 ~ 넘벌즈올 넘버즈 클로설 ~
그러나 만약 0을 0이 아닌 0에 가까운 숫자로 나눈다면 결과가 달라집니다.
(ㅈㄷㅂㅇ, me) 그러나 누군가 와서 이렇게 말할 수도 있습니다. "그럼 우리가 나누면 어떻게 될까요? 0은 0에 점점 더 가까운 숫자로, / 숫자 자체는 아니지만 / 0은 점점 더 작은 숫자로, 또는 0에 점점 더 가까운 숫자로 나눠집니다."
(ㅍㅍㄱ) 우리가 0을 점점 더 0에 가까운 숫자로 나누면 어떤 일이 일어날까요? ~"
ㆍㆍㆍ
come along 도착하다; 생기다[나타나다]
by itself 그것만으로

15. 1:29 1:38 1:44
And so they say, "For example, 0 divided by 0.1, well that's just going to be zero.
Zero divided by 0.001, well that's also going to be to zero.
0 divided by is also going to be equal to zero."
예를 들어 0 나누기 0.1은 0입니다. 0 나누기 0.001도 0이고요. 0 나누기 0이 됩니다.

16. 1:51 whether 1:52
And it didn't matter whether we (were) divided/dividing by a positive or negative number.
애닛 띠든매러
여기서도 나누는 수가 양수인지 음수인지는 중요하지 않습니다.

17. 1:54
Make all of these negatives, you still get the same answer.
나누는 수를 모두 음수로 바꾸어도 같은 결과를 얻기 때문입니다.
18. 1:57 1:59 2:01
So this line of reasoning tells you that it's completely legitimate, to think at least that maybe 0 divided by 0 could be equal to 0.
텔쥬 대릿츠 컴플리을리 리짓밋
따라서 이 논리로 주장하면 0÷0=0이 됩니다.
(ㅈㄷㅂㅇ) 따라서 이러한 추론은 적어도 0을 0으로 나눈 값이 0과 같을 수 있다고 생각하는 것이 완전히 타당하다는 것을 알려줍니다.
ㆍㆍㆍ
legitimate 레지티멋. 정당한, 타당한, 적법한, 이치에 맞는, 합리적인, 논리적인

19. 2:07
And these are equally valid arguments.
디절 이퀄리 밸리 알규먼츠
이 주장 역시 충분히 합리적인 논리라고 볼 수 있네요.
(ㅈㄷㅂㅇ) 그리고 이것들은 똑같이 유효한 주장입니다.

.

[BoutinMathieu]

I really wish I had math classes given from you, thanks to be that clear!

[hisxmark]

Think about it. Any division is a ratio. Geometrically it can be represented as the slope of a vector. Any vector (0, n) has a slope of infinity, but that need not bother us because the arctangent of infinity is "pi"/2 radians. And the vector (0, 0) is a point. It has no slope. Keep that in mind and division by zero is not really a problem.

It all makes sense.

[poprockssuck]

The only way to represent zero in the denominator is to have it approach zero, while approaching zero can be thought of as approaching the reciprocal of infinity.. However, there are different rates at which one can approach infinity, i.e. linearly, exponentially, etc. A term can be said to be indeterminate because one must know the rate at which the limits of the numerator and denominator approach infinity. So zero divided by zero is an indeterminate form because it's too vague, so to speak.

[zachmartinez]

L'Hopital's Rule is a specific part of calculus but when you derive the basics differentiating most functions you are getting rid of the delta x in the denominator. As long as you vertically shift the function (which will not change the numerical solution to f'(x)) so that f(0)=0then without canceling terms or simplifying you have (f(0)-f(0+delta x))/delta x. If you initially set the delta x to zero as you would as the first step to find the limit as delta x approaches 0 you are left with 0/0