# Why can't you divide by zero? - TED-Ed

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• Published: 23 April 2018
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View full lesson: https://ed.ted.com/lessons/why-can-t-you-divide-by-zero

In the world of math, many strange results are possible when we change the rules. But there’s one rule that most of us have been warned not to break: don’t divide by zero. How can the simple combination of an everyday number and a basic operation cause such problems?

Lesson by TED-Ed, animation by Nick Hilditch.

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• TED-Ed   1 years back

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• Vincent RedDragon  1 months back

TED-Ed I just want to ask,when I tried 0 divided 0 in calculator,the result is ERROR. Why?

• Talote1983  2 months back

The voice is super annoying

• Andrew Long  2 months back

Why do they use anti matter for i its till has postive mass if it was negative we would have no matter on contacted of each other

• Alejandro Cardone  3 months back

At minute 1:56 the video shows a triple equality that isn't true. I know what you wanted to say, but you have to be careful with notation.

• Alan  1 days back

When God divided O, boobs were created.

• GlowstoneLove Pad  1 days back

1 does equal 2.

We have two variables, a and b. As long as they're the same and not zero, it's fine.
They could be both 3, both 7, or both 1371742, but they can't be both zero.

1. a = b
2. a^2 = a*b(multiply by a)
3. a^2 - b^2 = a*b - b^2(subtract b^2)
4. (a+b)*(a-b) = b*(a-b)(factor left side as a difference of squares: a^2 - b^2 = (a+b)*(a-b) and right side as a distributive property, a*b - b*c = b*(a-c))
5. a+b = b(divide by a-b)
6. b+b = b(since a = b, you can substitute a for b in an equation, and vice versa)
7. 2b = b(for every a, 2a = a + a)
8. 2 = 1(divide by b)

B is not zero, defined by the given a and b are not zero.

Now that we have that 2 = 1, we can create an infinite list of axioms, each inferred from the last one:

1. 2 = 1
2. 1 = 0(subtract 1)
3. 2 = 0(combine 1 and 2: if a = b and b = c, a = c)
5. 3 = 0(combine 4 and 2)
7. 4 = 0(combine 6 and 2)
... .... .....
z+1. m = 0(combine z and 2)
z+3. m+1 = 0(combine z+2 and 2)
... and so on.

Now we have an infinite list of things, if we continue this on infinitely.
Extract line 1 from the list, and all of the even numbered lines that are after 3(4, 6, 8... )
Now we have the list:

2. 1 = 0
3. 2 = 0
5. 3 = 0
7. 4 = 0
9. 5 = 0
... and so on.

Now renumber the lines.

1. 1 = 0
2. 2 = 0
3. 3 = 0
... and so on.

Now we have an infinite list saying that every positive integer = 0. Let's call this list l, and the items in it 1, 2, 3, 4...

By the rule if a = b, -a = -b(multiplying both sides by -1), since -0 = 0, we can create another list, called l'(l prime):

1'. -1 = 0
2'. -2 = 0
3'. -3 = 0
... and so on.

Now we have an infinite list saying that every negative integer = 0.

Also, if we have a fraction, we can substitute the top for zero, making that every fraction is zero too, call this l''(l double-prime):

1''. 1/2 = 0
2''. 1/3 = 0
3''. 2/3 = 0
... and so on, continuing with 1/4, 2/4, 3/4, 1/5, 2/5, 3/5... and so on...

This list, l''(l double prime) says that every fraction of two positive integers inbetween 0 and 1 is equal to zero too.

If you add 0 to 0 you get 0 also, and so if we make all of the mixed numbers = 0, in a list l'''(l triple prime).
Let's call infinity "t" for short:

1'''. 1 1/2 = 0
2'''. 2 1/2 = 0
... ... ...
t'''. t 1/2 = 0
t + 1'''. 1 1/3 = 0
... ... ... and so on... ... ...

And therefore, every fraction is equal to zero. Again, we have l quadruple prime(l'''') and l quintuple prime (l''''') saying that every negative fraction is also equal to zero:

1''''. -1/2 = 0
2''''. -1/3 = 0
3''''. -2/3 = 0
... ... ... and so on... ... ...

1'''''. -1 1/2 = 0
2'''''. -2 1/2 = 0
... ... ... and so on... ... ...

l'''' is just a negative of l'', and l''''' is a negative of l'''.

Since every fraction above 1 is just a mixed number, all of the fractions above 1 = 0 list is the same as l''', and the fractions below -1 are in l'''''.

Now we can merge all of these l lists together, in a superlist called q.

We have a list saying that every integer and fraction, positive and negative, is equal to zero now. Now that, since if a = b and b = c, a = c, every integer and fraction, positive and negative, is equal to eachother too!

Since therefore the numbers picked as a and b are zero too, therefore this could never exist since then a and b would violate the zero-rule. But then this could never have been proved, and therefore the rules of math are back to regular. Therefore the numbers picked as a and b would not be zero anymore, this could be proved.

So this could be proved, which means that a and b are zero, which means this could not be proved, which means that a and b are no longer zero, which means this could be proved again. It's a paradox.

• Because your teacher keeps telling you so, it's simple

• Mannem Nageswarao  3 days back

What is 0÷0

• Yinit Belloraaa  3 days back

Any number that is devided by zero is zero

Example 10/0= zero not infinity

• GlowstoneLove Pad  1 days back

if a / 0 = 0, 0 * 0 = a
So what you're saying is that the product of 0 and 0 is any number, therefore the square root of any number is 0?

• 321 Cubing  3 days back

Is this not just advanced algebra and what they try to find out

• ra s.  4 days back

I broke a math rule, but failed the exam.

• ifigenio buenrostro  4 days back

Maybe current mathematics can't divide by zero but surely future mathematics will

• PraWin Shidore  4 days back

0 is amazing by the way 🤔😁😜

• Morning Star  4 days back

After watching this ,I clicked Like but and app said "Removed from liked videos".

I was like wt!!! I already watched that many times ago and literally forgot.😅

• rianma 123  5 days back

You have a 20 cm ruler
You divided it by 3 doesn't mean you cut the ruler into 3 pieces of ruler that each piece is 20:3 cm long.
It means you cut the ruler into 3 pieces of ruler and you can just keep 1 piece.
So a 20cm ruler divide by 3 =a 20:3cm ruler
Not = 3 20:3 rulers
So when your mum tells you to divide a bread by 3, you cut the bread into 3 pieces and throw 2 pieces away.

• Mr. Formal  5 days back

infinity ÷ 0 is less than or equal to infinity.

• Aditya Kant Kushwaha  6 days back

• Dude Trust Me  6 days back

As an English education major this finally put me to sleep. Thanks math!

• KT7rocks Jr  6 days back

Infinity divided by 0

Thanos:DID YOU SAY INFINITY?

• Eunhyouk Shin  7 days back

1:56 This notation can be quite misleading

• Remember infinity is a SYMBOL for a number we can't count to/it's an infinite amount of zeros. So it doesn't =infinity (saying infinity is a number)it is infinity

• Francesco Sorce  7 days back

3:30
there's the fallacy

you have said "let the symbol infty be a number so that infty *0 =1" and you also said "assuming we know nothing about infinity"

then why should 2 infty =infty?

that is a preconceived notion of how infinity works

• Francesco Sorce  7 days back

there are other ways to show this method isn't valid, just that this one in particular was not it

for example, let y so that 0y=1

1+1=2
0y+0y=2

2 0 y = 2

(2 0) y = 2

0 y = 2

but 0y =1

however note, this disproved EITHER associativity OR the existence of y, not necessarily the latter.

• Thijmen Willems  1 weeks back

John has four apples, he distributes them fairly with Mike, how many apples do they each have? Two.

John has zero apples, he distributes them fairly with Mike, how many apples do they each have? Zero because there are no apples to distribute.

there are four apples on the table, they are not divided because there are zero people to distribute them, how many apples does everyone have? zero because there are zero people to divide them. 4 ÷ 0 = 0.

SOLVED
(Google calculator says that 4 ÷ 0 equals infinity)

• Naff YT  1 weeks back

• Hdtomo  1 weeks back

They is an answer to any ÷0. That is an infinity. And if it's 1÷0 it's infinity because it's not how much 0 is in 1. It's just the opposite of times. Look it's hard to explain but it's still infinity. Because reasons

• Maxwell Baron  1 weeks back

Easy answer to the title question without watching the video is because you can’t turn nothing into something.

• Imagination TM  1 weeks back

You can't divide 0 by 0 because that's just like saying I have 0 dollars and I divide it by 0 people. Check on our calculator

• actually
once i asked my math teacher in fifth grade: What is 1 divided by 0? (said it before that became a math dilemma)
the next day he told me this:
"Any number times 0 would be 1"
Edit: acctually happened

• Yash Rai  1 weeks back

Its equal to 0 dude

• Fake Person  1 weeks back

1:30 . 10÷2 means if you divide ten twice, how much is a single piece's value. Not how many times 2 can be subtracted from 10. I know, I'm being a snob. But there's a difference.

• solus 22  2 weeks back

*number comparaison*
1 one
10 ten
100 hundred
10^3 thousand
10^6 million
10^100 googol
10^10^100 googolplex
g:64 graham number
aleph 0 infinity
aleph 1 infinity of infinity
*1 divided by 0*

• Roni Shaheen  2 weeks back

3:30 ∞ + ∞ =2∞ so 0x2∞ =2

• Bry Bry  2 weeks back

How about 5 ÷ 0 =
\$5 ÷ 0 friends = \$0
\$0 given to your friends XD

• Kowi Kowi  2 weeks back

but somehow you must explain that the 5 dollar suddenly stop existing

• Bry Bry  2 weeks back

If you cant divide by 0
5 × 0 = 0

0 ÷ 0 = 5

• Gnome Ann  2 weeks back

Bry Bry, you are right, 0/0 can be 5. But also...

1 * 0 = 0
0/0 = 1

It can be 1, too. In fact, it can give any number in existence. Since it can be any number, and we don’t know which one, we call it indeterminate. We need more information to find its limit.

• Richard Hough  2 weeks back

10 divided by 2 means that 10 items are divided into 2 groups of 5 of these things. This would indicate that 10 divided by 0 means there are 0 groups, thus dividing by 0 equals 0?

i thought the answer was always ERROR

• Nobody  2 weeks back

my math teacher make me divide by zero im not confused

• _ShadowShot _  2 weeks back

Anything to the power of zero is one.
But something cannot come of nothing.
So my question is...

What is zero to the power of zero?

• Gnome Ann  2 weeks back

_ShadowShot _, 0^0 is indeterminate/undefined. By itself it has no definable value.

• Disappointed to see you break the rule of equals. 2x3=6x0.5=3 isn't correct; not all of them are the same.

Poor form Ted talks, poor form

• Austin Liu  2 weeks back

Infinity times infinity is infinity squared 3:20

• Gnome Ann  2 weeks back

Austin Liu, but infinity to the power of any positive number is infinity.

• Agata Malecka  2 weeks back

Divide \$10by 0 u still got \$10 You don't have more money also divide 10 \$ by 0.00001 it can not be logically more than \$10

• Anonymoose  2 weeks back

That's... that's not how division works.

• Mr. Anonymous  2 weeks back

3:28 Wouldn't ∞ plus ∞ equal 2∞?

• Mr. Anonymous  2 weeks back

@Gnome Ann I understand that you mean that ∞ + ∞ would still algebraically equal ∞. But 2∞ doesn't only mean that its value is ∞. It more importantly denotes that ∞ can be written twice, like ∞ + ∞ which actually does matter in this case because it would then set the equation true, and 0 + 2∞ would = 2.

• Gnome Ann  2 weeks back

Mr. Anonymous, it does, but writing out 2∞ is unnecessary, since infinity times any number except zero is equal to infinity, so:

∞ + ∞ = ∞(1 + 1) = 2∞ = ∞

(Note that it is not really formal math to do operations on infinity like this, if we wanted to be more technical, we would use aleph-0 or limits).

• TheProgrammingJedi  2 weeks back

Excellent explanation. Thank you.

• Deva Preethi  2 weeks back

Where do we use division in basic day to day life?? Like dividing things between people. If you have 10 candies and want to divide it equally to 2 people, you will divide 10 by 2, so you will give 5 to each. If you have 3 apple and want to give equal pieces to 2 people, you will give 1 each (3/2, Quotient= 1 reminder=1), 1 will be reminder but if you choose to absolute you further divide that remaining apple into 2 and give half each (3/2=1.5, 2*1.5 = 3), so the answer will be 1 and half each . So if you are having 20 KG mango, you give equal kg of fruits to 0 people, how will you divide 20kg with 0 people? You don't divide because there is no one to give it to, reminder is 20kg.(20/0=?) So what must be multiplied to 0 to get reminder 20 itself. Answer is 0. But 0*0 is not equal to 20. ughhhhh.

• Gopal Roy  2 weeks back

0 discovered in India and explored with Einstein in 20th century, breaking rules of Newton.

So breaking rules will continue in future also..

• Gopal Roy  2 weeks back

Alot better explained.. The rules applies in limit and calculas

• Yes, fun worlds to explore, while completly pointless by any means, apart from the fun about nonsense.
Really?!
Mathematicians hate to be unemployed ... that might be TRUE!

• BangGamer 1338 YT  3 weeks back

0:0=1?

• seHun eXo  3 weeks back

well all i knew is that MITOCHONDRION is the powerhouse of the cell.

• Moepowerplant  3 weeks back

3:40 looks like heavy quantum stuff... Is it possible to divide by zero in a quantum world?

• Ethan Ton  3 weeks back

10 ÷ 0 = 0

There is no multiplication
0 × 0 = 0
So 0 equals nothing and if 10 × 0 would be 0 because there would be nothing there
Just a theory